UNIVERSITY
OF

ASfRONOMY UBRARV

JEr %i bris

L5 BR AR Y

OF THE
ASTRONOMICAL SOCIETY
FACiFJC

241

H YDEOD YN AMIC S .

Sonton: C. J. CLAY AND SONS,
CAMBRIDGE UNIVEKSITY PEESS WAREHOUSE,

AVE MARIA LANE.
©laggoto: 263, ARGYLE STREET.

ILetpjig : F. A. BROCKHAUS.
fo lorfc : MACMILLAN AND CO.

HYDRODYNAMICS

BY

HORACE LAMB, M.A., F.R.S.

PBOFESSOB OF MATHEMATICS IN THE OWENS COLLEGE,

VICTORIA UNIVERSITY, MANCHESTER ;
FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

CAMBRIDGE :

AT THE UNIVERSITY PRESS.
1895

[All Rights reserved.]

ASffiONOMY LIBRARY

Cambridge:

PRINTED BY J. & C. F. CLAY,
AT THE UNIVERSITY PRESS.

ASTRONOMY

PREFACE.

rilHIS book may be regarded as a second edition of a " Treatise
-•- on the Mathematical Theory of the Motion of Fluids,"
published in 1879, but the additions and alterations are so ex
tensive that it has been thought proper to make a change in the
title.

I have attempted to frame a connected account of the principal
theorems and methods of the science, and of such of the more
important applications as admit of being presented within a
moderate compass. It is hoped that all investigations of funda
mental importance will be found to have been given with sufficient
detail, but in matters of secondary or illustrative interest I have
often condensed the argument, or merely stated results, leaving
the full working out to the reader.

In making a selection of the subjects to be treated I have
been guided by considerations of physical interest. Long analytical
investigations, leading to results which cannot be interpreted,
have as far as possible been avoided. Considerable but, it
is hoped, not excessive space has been devoted to the theory of
waves of various kinds, and to the subject of viscosity. On the
other hand, some readers may be disappointed to find that the
theory of isolated vortices is still given much in the form in which
it was. left by the earlier researches of von Helrnholtz and Lord
Kelvin, and that little reference is made to the subsequent
investigations of J. J. Thomson, W. M. Hicks, and others, in this
field. The omission has been made with reluctance, and can be
justified only on the ground that the investigations in question
L. b

M6772O1

VI PREFACE.

derive most of their interest from their bearing on kinetic theories
of matter, which seem to lie outside the province of a treatise like
the present.

I have ventured, in one important particular, to make a serious
innovation in the established notation of the subject, by reversing
the sign of the velocity -potential. This step has been taken not
without hesitation, and was only finally decided upon when I found
that it had the countenance of friends whose judgment I could
trust ; but the physical interpretation of the function, and the
far-reaching analogy with the magnetic potential, are both so much
improved by the change that its adoption appeared to be, sooner
or later, inevitable.

I have endeavoured, throughout the book, to attribute to their
proper authors the more important steps in the development of
the subject. That this is not always an easy matter is shewn by
the fact that it has occasionally been found necessary to modify
references given in the former treatise, and generally accepted as
correct. I trust, therefore, that any errors of ascription which
remain will be viewed with indulgence. It may be well,
moreover, to warn the reader, once for all, that I have allowed
myself a free hand in dealing with the materials at my disposal,
and that the reference in the footnote must not always be taken
to imply that the method of the original author has been
closely followed in the text. I will confess, indeed, that my
ambition has been not merely to produce a text-book giving a
faithful record of the present state of the science, with its
achievements and its imperfections, but, if possible, to carry it a
step further here and there, and at all events by the due coordina
tion of results already obtained to lighten in some degree the
labours of future investigators. I shall be glad if I have at least
succeeded in conveying to my readers some of the fascination
which the subject has exerted on so long a line of distinguished
writers.

In the present subject, perhaps more than in any other depart
ment of mathematical physics, there is room for Poinsot's warning

PREFACE. Vll

" Gardens nous de croire qu'une science soit faite quand on 1'a
reduite a des formules analytiques." I have endeavoured to make
the analytical results as intelligible as possible, by numerical
illustrations, which it is hoped will be found correct, and by the
insertion of a number of diagrams of stream-lines and other
curves, drawn to scale, and reduced by photography. Some of
these cases have, of course, been figured by previous writers, but
many are new, and in every instance the curves have been
calculated and drawn independently for the purposes of this work.

I am much indebted to various friends who have kindly taken
an interest in the book, and have helped in various ways, but who
would not care to be specially named. I cannot refrain, however,
from expressing my obligations to those who have shared in the
tedious labour of reading the proof sheets. Mr H. M. Taylor has
increased the debt I was under in respect of the former treatise by
giving me the benefit, so long as he was able, of his vigilant
criticism. On his enforced retirement his place was kindly taken
by Mr R. F. Gwyther, whose care has enabled me to correct many
errors. Mr J. Larmor has read the book throughout, and has
freely placed his great knowledge of the subject at my
disposal ; I owe to him many valuable suggestions. Finally, I
have had the advantage, in the revision of the last chapter, of
Mr A. E. H. Love's special acquaintance with the problems there
treated.

Notwithstanding so much friendly help I cannot hope to have
escaped numerous errors, in addition to the few which have been
detected. I shall esteem myself fortunate if those which remain
should prove to relate merely to points of detail and not of
principle. In any case I shall be glad to have my attention called
to them.

HORACE LAMB.

May, 1895.

CONTENTS.

CHAPTER I.
THE EQUATIONS OF MOTION.

ART. PAGE

I, 2. Fundamental property of a fluid . . . . . . 1

3-8. 'Eulerian' form of the equations of motion. Dynamical

equations, equation of continuity . . . . . 3

9. Physical equations . . 7

10. Surface-conditions . . . . . . . . . 8

II. Equation of energy . . . .10

12. Impulsive generation of motion . . . ' . . . 12

13, 14. 'Lagrangian' forms of the dynamical equations, and of the

equation of continuity 14

15. Weber's Transformation . . 15

16, 17. Extension of the Lagrangian notation. Comparison of the

two forms . 16

CHAPTER II.
INTEGRATION OF THE EQUATIONS IN SPECIAL CASES.

18. Velocity-potential. Lagrange's theorem 18

19, 20. Physical meaning of <£. Geometrical properties . . .19
21. Integration of the equations when a velocity-potential exists ;

pressure-equation . . . . . . . .21

22-24. Steady motion. Deduction of the pressure-equation from

the principle of energy. Limit to the velocity . . 22

25. Efflux of liquids ; vena contracta 26

26. Efflux of gases 28

27-30. Examples of rotating fluid : uniform rotation ; Kankine's ' com
bined vortex ' ; electro-magnetic rotation .... 29

CONTENTS.

CHAPTER III.
IRROTATIONAL MOTION.

ART. PAGE

31. Analysis of the motion of a fluid element .... 33

32, 33. ' Flow ' and « Circulation.' Stokes' Theorem .... 35

34. Constancy of circulation in a moving circuit . . . .38

35, 36. Irrotational motion in simply-connected spaces ; <£ single-

valued 39

37-39. Case of an incompressible fluid ; tubes of flow. <£ cannot
be a maximum or minimum. Mean value of 0 over a
spherical surface . .40

40, 41. Conditions of determinateness of <f> . . . . .44

42-46. Green's theorem. Dynamical interpretation. Formula for
kinetic energy. Lord Kelvin's theorem of minimum
energy. . . . . . . . ^ . . . 47

47-51. Multiply-connected regions. Irrotational motion ; cyclic

constants .......... 53

52. Conditions of determinateness for the motion of an incom
pressible fluid in a cyclic region 58

53-55. Lord Kelvin's extension of Green's theorem ; dynamical in
terpretation. Energy of an irrotationally moving liquid
in a cyclic space 60

56-58. ' Sources ' and ' sinks.' Double-sources. Surface-distributions

of simple and double sources ...... 63

CHAPTER IV.

MOTION OF A LIQUID IN TWO DIMENSIONS.

59. Stream-function. . . . . ... . .69

60, 61. Irrotational motion. Kinetic energy . . . . .71

62. Connection with the theory of the complex variable. Con

jugate functions . . .73

63, 64. Simple types of motion, acyclic and cyclic .... 78

65. Inverse formulae. Examples . . . . . . .81

67-70. General formulae ; Fourier's method. Motion of a cylinder,

without and with circulation of the fluid round it . 84

71. Inverse methods. Motion due to translation of a solid.

Elliptic cylinder. Flow past an oblique lamina ; couple-
resultant of fluid pressures 91

72. Motion due to a rotating solid. Rotating prismatic vessel ;

cases where the section is an ellipse, a triangle, or a
circular sector. Rotating elliptic cylinder in infinite
liquid 95

CONTENTS.

XI

ART.

73-80.

81.

Discontinuous motions. Investigations of von Helmholtz

and Kiruhlioft*. Applications to Borda's mouth-piece,

the vena contracta, the impact of a stream on a lamina.

, Bobyleff's problem .......

Flow in a curved stratum .

100
114

CHAPTER V.

IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN
THREE DIMENSIONS.

82,83. Spherical harmonics ; Maxwell's theory ; poles. . . 117

84. Transformation of the equation v20 = 0to polar coordinates. 120

85, 86. Zonal harmonics. Hypergeometric series. Legendre's

functions. Velocity-potential of a double-source.
Functions of the second kind . . . . .121

87. Tesseral and sectorial harmonics . . . . . 125

88. Conjugate property of surface harmonics ; expansions . 127

89. 90. Hydrodynamical applications. Impulsive pressure over a

sphere. Prescribed normal velocity over a sphere.

Energy 127

91, 92. Motion of a sphere in an infinite liquid. The effect of
the fluid is on the inertia of the sphere. Sphere
in a liquid with a concentric spherical boundary . 130
93-96. Stokes' stream-function. General formula) for stream-
function. Stream-lines of a sphere. Image of double-
source in a sphere. Rankine's method . . .133
97, 98. Motion of two spheres in a liquid ; kinematic formulae . 139

99. Sphere in cyclic region 143

100-103. Ellipsoidal harmonics for ovary ellipsoid. Motion of a
liquid due to an ovary ellipsoid (translation and

rotation) 145

104-106. Ellipsoidal harmonics for planetary ellipsoid. Flow through
a circular aperture. Motion of fluid due to a planet
ary ellipsoid (translation and rotation). Stream-lines
of a circular disk . . . . . . . .150

107. Motion of fluid in ellipsoidal case (unequal axes) . . 155

108. General expression of v20 = 0 in orthogonal coordinates . 156

109. Confocal quadrics ; ellipsoidal coordinates .... 158
110-112. Motion of fluid due to a varying ellipsoidal boundary . 159

Flow through an elliptic aperture. Translation and rota
tion of an ellipsoid.
113. References to other researches . 166

Xll CONTENTS.

CHAPTER VI.

ON THE MOTION OF SOLIDS THROUGH A LIQUID:
DYNAMICAL THEORY.

ART. PAGE

114, 115. Determination of <£ for the acyclic motion due to a single

solid in an infinite liquid ; kinematical formulae . 167

116. Theory of the ' impulse ' 169

117-120. Dynamical equations relative to moving axes. Expression
for the energy; coefficients of inertia. Formulae for
'impulse.' Reciprocal formulae. Impulsive pressures
of fluid on solid . 170

121. Equations of motion. Components of fluid pressure on

moving solid. Three directions of permanent transla
tion. Stability . . .. »/ . > . -^ . ., 176

122. Steady motions. Case where the 'impulse' reduces to a

couple ....•; . . . . . . 178

123. Simplification of the expression for the energy in certain

cases . . . 181

124-126. Motion of a solid of revolution with its axis in one plane ;
stability. Stability increased by rotation about axis.
Steady motion of a solid of revolution . . .184

127. Motion of an 'isotropic helicoid' 191

128. Motion of a hollow body filled with liquid . . .192
129-131. Motion of a perforated solid, when there is cyclic motion

through the apertures. Meaning of 'impulse' in
this case. Steady motion of a ring ; stability . . 192
132, 133. Equations of motion in generalized coordinates; Hamil-

tonian principle. Derivation of Lagrange's equations 197

134. Application to Hydrodynamics 201

135, 136. Motion of a sphere near a plane boundary. Motion of

two spheres in the line of centres .... 205

137. Modification of Lagrange's equations in the case of cyclic

motion 207

138, 139. Alternative investigation ; flux-coordinates. Equations of

motion of a 'gyrostatic system' 211

140. Motion of a sphere in a cyclic region . . . .217

141. Pressures on solids held at rest. Cases of thin cores,

and tubes. Comparison with electro-magnetic pheno
mena 218

CONTENTS.

Xlll

CHAPTER VII.
VORTEX MOTION.

ART. PAGE

142. ' Vortex-lines ' and ' vortex- filaments '; kinematical proper

ties 222

143. Persistence of vortices 224

144-146. Conditions of determinateness of vortex-motion. Deter
mination of motion in terms of expansion and rotation.
Electro-magnetic analogy 227

147, 148. Case of a single isolated vortex. Velocity-potential due

to a vortex . 231

149. 'Vortex-sheets' 234

150-153. 'Impulse' and energy of a vortex system. . . . 236

154, 155. Rectilinear vortices. Special Problems .... 243
156. Vortex- pair; 'impulse' and energy. Kirchhoff's form of

the theory .248

157,158. Stability of a cylindrical vortex. Kirchhoff's elliptic vortex 250

159. Vortices in a curved stratum of fluid . . . . 253

160,161. Circular vortices; energy and 'impulse.' Stream-function 254

162. Isolated vortex-ring. Stream-lines. Velocity of transla

tion 257

163. Mutual influence of vortex-rings. Image of a vortex in

a sphere 260

164. General conditions for steady motion of a fluid. Examples.

Hill's spherical vortex . . . . . . .262

CHAPTER VIII.

TIDAL WAVES.

165. Introduction. Recapitulation of the general theory of

small oscillations 266

166-170. Waves in canal of uniform section. Equations of motion.
Integration and interpretation. Wave- velocity. Mo
tion in terms of initial circumstances. Physical
meaning of the various approximations . . .271

171. Energy of a wave-system. In progressive waves it is

half potential and half kinetic 278

172. Artifice of steady motion 279

173. Superposition of waves. Reflection 280

174-176. Effect of disturbing forces. Free and forced oscillations

in a canal of finite length . . . . . .281

177. Canal theory of the tides. Disturbing potential . . 286

XIV CONTENTS.

ART. PAGE

178, 179. Tides in equatorial canal, and in a canal parallel to equator ;

semi-diurnal tides . . . . . . 287

180. Canal coincident with a meridian ; change of mean level ;

fortnightly tide 290

181, 182. Waves in a canal of variable section. General laws of

Green and Lord Rayleigh. Problems of simple-
harmonic oscillations in a variable canal. Exaggera
tion of tides in shallow seas and estuaries . . 291
183, 184. Waves of finite amplitude. Change of type. Tides of

the second order 297

185. Wave-propagation in two horizontal dimensions. Equa

tions of motion and of continuity .... 301

186. Oscillations of a rectangular sheet of water . . . 303

187. 188. Circular sheet of uniform depth ; free and forced oscilla

tions ; Bessel's Functions 304

189. Circular sheet of variable depth . . . . . 312

190-193. Sheet of water covering a symmetrical globe ; free and
forced oscillations. Effect of the mutual attraction
of the particles of water. Case of an ocean limited
by meridians or parallels ; transition to plane pro
blem 314

194-198. Equations of motion relative to rotating axes. Adaptation
to case of infinitely small relative motions. Free
oscillations ; ' ordinary ' and * secular ' stability. Forced
oscillations . . . . . . . . .322

199, 200. Application to Hydrodynamics. Tidal oscillations of a
rotating sheet of water. Plane sheet of water ;
general equations . . . . . . . .331

201-205. Examples : long straight canal ; circular sheet ; circular

basin of variable depth 334

206-208. Tidal oscillations on a rotating globe. General equations.
Case of small ellipticity. Adaptation to simple-
harmonic motion 343

209-211. Tides of long period. Integration of the equations.

Numerical results 348

212. Diurnal tides. Evanescence of diurnal tide in a^spherical

ocean of uniform depth . . . . . 355

213, 214. Semi-diurnal tides. Special law of depth. Laplace's

solution for uniform depth ; numerical results . . 356
215. Stability of the ocean 362

APPENDIX : On Tide-generating Forces 364

CONTENTS.

XV

CHAPTER IX.

SURFACE WAVES.

ART. 1'AOE

216. Statement of problem. Surface- conditions . . . 370

217. Application to canal of unlimited length. Standing waves.

Motion of particles . . . . • -.-.-• .« 372

218. Progressive waves. Wave- velocity. Elliptic orbits of

particles. Numerical tables. . . . . . 374

219. Energy of wave-system . . . . . . . 378

220. General solution in terms of initial circumstances . . 379

221. Group-velocity. Wave-resistance ^ , . . . 381

222. Artifice of steady motion. Stationary undulations . . 384
223-225. Oscillations of the common surface of two liquids, or of

two currents. Instability .. 385

226-229. Surface disturbance of a stream. Effect of a simple-
harmonic application of pressure. Effect of a line of
pressure. Effect of a pressure-point. Ship-waves.

Case of finite depth 393

230. Effect of inequalities in the bed of a stream . . . 407

231-235. Waves of finite amplitude. Stokes' wave of permanent
type. Momentum. Gerstner's rotational waves.
Solitary wave. Connection with general dynamical
theory . . . . . . . . . 409

236. Standing waves in limited masses of water. Case of

uniform depth . . . « . . . . 424

237, 238. Standing waves with variable depth. Oscillations across

a canal of triangular section. Approximate determina
tion of longest period for the case of semicircular
section 426

239, 240. General theory of waves in uniform canal. Special pro
blems ; triangular section 429

241-243. Gravitational oscillations of a liquid globe. Method of

energy. Ocean of uniform depth .... 436

244. Capillarity; surface-condition . . . . » • 442

245. Capillary waves on the common surface of two liquids . 443

246. 247. Waves due to joint action of gravity and cohesion. Mini

mum wave-velocity. Group-velocity. Waves on the
common boundary of two currents ; stability . . 445

248-250. Surface disturbance of a stream. Simple-harmonic dis
tribution of pressure. Effect of a local pressure.
Waves and ripples. Fish-line problem . . . 450

251, 252. Vibrations of a cylindrical jet. Instability of a jet for

varicose disturbance 457

253. Vibrations of a spherical globule, and of a bubble . . 461

XVI CONTENTS.

CHAPTER X.

WAVES OF EXPANSION.

ART. PAGE

254. Coefficient of cubic elasticity 464

255-258. Plane waves. Velocity of sound in liquids and gases ;

Laplace's theory. Energy of plane waves . . . 464

259-262. Waves of finite amplitude. Condition for permanency of
type. Change of type in progressive waves. In
vestigations of Riemann, Earnshaw, and Rankine.
Question as to the possibility of waves of discon
tinuity . . . 470

263, 264. Spherical waves. Determination of motion in terms of

initial conditions 477

265. General equation of sound-waves. Poisson's integral . 480

266. Simple-harmonic vibrations. Case of symmetry ; radial

vibrations in a spherical envelope ; simple source of
sound 483

267-270. General solution in spherical harmonics. Vibrations of air
in spherical envelope. Waves in a spherical sheet of
air. Waves propagated outwards from a spherical
surface. Ball-pendulum ; correction for inertia, and
dissipation-coefficient 484

271, 272. Isothermal oscillations of an atmosphere of variable

density. Plane waves . . . . . . .491

273, 274. Atmospheric tides. References to further researches . 494

CHAPTER XI.

VISCOSITY.

275, 276. General theory of dissipative forces. One degree of free
dom. Periodic imposed force. Retardation or accele
ration of phase 496

277. Application to tides in equatorial canal. Tidal friction . 499

278, 279. General theory resumed. Several degrees of freedom.

Frictional and gyrostatic terms. Dissipation-function.
Oscillations of a dissipative system about a con
figuration of absolute equilibrium .... 503

280. Effect of gyrostatic terms. Two degrees of freedom ;

disturbance of long period ...... 506

281-283. Viscosity of fluids. Specification of stress. Formulae of

transformation 508

CONTENTS.

XV11

ART. PAGE

284. Hypothesis that the stresses are linear functions of the

rates of strain. Coefficient of viscosity . . . 511

285. Boundary conditions. Question as to slipping of a fluid

over a solid 514

286. General equations of motion of a viscous fluid. Inter

pretation 'j ;. 514

287. Dissipation of energy by viscosity. Expressions for the

dissipation-function 517

288-290. Problems of steady motion. Flow of a viscous liquid

through a crevice, and through a tube of circular

section ; Poiseuille's laws. Question as to slipping.

Eesults for other forms of section . . . . . 519
291, 292. Steady rotation of a cylinder. Rotation of a sphere . 523
293-295. Motion of a viscous fluid when inertia is neglected.

General solution in spherical harmonics. Steady

motion of a sphere ; resistance ; terminal velocity.

Limitations to the solution . . . . . . 526

296. Steady motion of an ellipsoid . . . \. . . 534

297. General theorems of von Helmholtz and Korteweg . . 536
298-300. Periodic laminar motion. Oscillating plane. Periodic

tidal force 538

301, 302. Effect of viscosity on water-waves. Method of dissipation-
function. Direct method 544

303. Effect of surface-forces. Generation and maintenance of

waves by wind . . . . . . . . 549

304. Calming effect of oil on water-waves ..... 552
305-309. Periodic motion with a spherical boundary. General

solution in spherical harmonics. Applications. Decay
of motion in a spherical vessel. Torsional oscillations
of a shell containing liquid. Effect of viscosity on
the vibrations of a liquid globe. Torsional oscillations
of a sphere surrounded by liquid. Oscillations of a
ball-pendulum ........ 555

310. Effect of viscosity on sound-waves ..... 570

311-312. Instability of linear flow when the velocities exceed certain
limits. Law of resistance in pipes. Reynolds' ex
periments. Skin-resistance of ships. References to
theoretical investigations 572

XV111 CONTENTS.

CHAPTER XII.
EQUILIBRIUM OF ROTATING MASSES OF LIQUID.

ART. PAGE

313-316. Formulae relating to attraction of ellipsoids. Maclaurin's
and Jacobi's ellipsoids. References to other forms of
equilibrium . 580

31 7. Motion of a liquid mass with a varying ellipsoidal surface.

Finite oscillations about the spherical form. Case

of rotation ......... 589

318. Various problems relating to the motion of a liquid

ellipsoid with invariable axes 592

319. 320. Ordinary and Secular stability. Stability of Maclaurin's

ellipsoid. References 594

LIST OF AUTHORS CITED 599

INDEX 602

ADDITIONS AND CORRECTIONS.

Page 109, equation (10). Lord Kelvin maintains that the type of motion here
contemplated, with a surface of discontinuity, and a mass of ' dead water ' in the rear
of the lamina, has no resemblance to anything which occurs in actual fluids ; and
that the only legitimate application of the methods of von Helmholtz and Kirchhoff
is to the case of free surfaces, as of a jet. Nature, 1. 1. , pp. 524, 549, 573, 597 (1894).

Page 111, line 20, for a = 0 read a = 90°.
,, 132, equation (4), dele u2.

,, 156, equation (3), footnote. The author is informed that this solution
was current in Cambridge at a somewhat earlier date, and is due to Dr Ferrers.

Page 305, footnote. To the list of works here cited must now be added : Gray
and Mathews, A Treatise on Bessel Functions and their Applications to Physics,
London, 1895.

Page 376, line 22, for (g\!2irf read (S*?/X)*.

,, 381, line 10. Reference should be made to Scott Russell, Brit. Ass.
Rep., 1844, p. 369.

Page 386, equation (2). Attention has recently been called to some obser
vations of Benjamin Franklin (in a letter dated 1762) on the behaviour of surfaces
of separation of oil and water (Complete Works, 2nd ed., London, n. d., t. ii.,
p. 142). The phenomena depend for their explanation on the fact that the natural
periods of oscillation of the surface of separation of two liquids of nearly equal
density are very long compared with those of a free surface of similar extent.

Page 423, line 16, for the minimum condition above given read the con
dition that 8 ( V - T0) = 0, or 5 (V + K) = 0.

Page 449, footnote, for Art. 302 read Art. 303.

„ 482, line 16, for 0 read 0.

,, 487, footnote. The solution of the equation (1) of Art. 266 in spherical
harmonics dates from Laplace, " Sur la diminution de la dur6e du jour, par le
refroidissement de la Terre," Conn, des Terns pour VAn 1823, p. 245 (1820).

Page 491. Dele lines 9—18 and footnote.

HYDBODYNAMICS.

CHAPTER I.

THE EQUATIONS OF MOTION.

1. THE following investigations proceed on the assumption
that the matter with which we deal may be treated as practically
continuous and homogeneous in structure ; i. e. we assume that
the properties of the smallest portions into which we can conceive
it to be divided are the same as those of the substance in bulk.

The fundamental property of a fluid is that it cannot be in
equilibrium in a state of stress such that the mutual action
between two adjacent parts is oblique to the common surface.
This property is the basis of Hydrostatics, and is verified by the
complete agreement of the deductions of that science with ex
periment. Very slight observation is enough, however, to convince
us that oblique stresses may exist in fluids in motion. Let us
suppose for instance that a vessel in the form of a circular
cylinder, containing water (or other liquid), is made to rotate
about its axis, which is vertical. If the motion of the vessel be
uniform, the fluid is soon found to be rotating with the vessel as
one solid body. If the vessel be now brought to rest, the motion
of the fluid continues for some time, but gradually subsides, and
at length ceases altogether ; and it is found that during this
process the portions of fluid which are further from the axis lag
behind those which are nearer, and have their motion more
rapidly checked. These phenomena point to the existence of
mutual actions between contiguous elements which are partly
tangential to the common surface. For if the mutual action were
everywhere wholly normal, it is obvious that the moment of
momentum, about the axis of the vessel, of any portion of fluid

L. 1

2 THE EQUATIONS OF MOTION. [CHAP. I

bounded by a surface of revolution about this axis, would be
constant. We infer, moreover, that these tangential stresses are
not called into play so long as the fluid moves as a solid body, but
only whilst a change of shape of some portion of the mass is going
on, and that their tendency is to oppose this change of shape.

2. It is usual, however, in the first instance to neglect the
tangential stresses altogether. Their effect is in many practical
cases small, and independently of this, it is convenient to divide
the not inconsiderable difficulties of our subject by investigating
first the effects of purely normal stress. The further consideration
of the laws of tangential stress is accordingly deferred till
Chapter XL

If the stress exerted across any small plane area situate at a
point P of the fluid be wholly
normal, its intensity (per unit
area) is the same for all aspects
of the plane. The following proof
of this theorem is given here for
purposes of reference. Through
P draw three straight lines PA,
PB, PC mutually at right angles,
and let a plane whose direction-
cosines relatively to these lines
are I, m, n, passing infinitely close to P, meet them in A, B, C.
Let p, p1} p2, p3 denote the intensities of the stresses* across the
faces ABC, PBG, PGA, PAB, respectively, of the tetrahedron
PABC. If A be the area of the first-mentioned face, the areas
of the others are in order /A, wA, r&A. Hence if we form the
equation of motion of the tetrahedron parallel to PA we have
PI . ZA = pi . A, where we have omitted the terms which express
the rate of change of momentum, and the component of the
extraneous forces, because they are ultimately proportional to the
mass of the tetrahedron, and therefore of the third order of
small linear quantities, whilst the terms retained are of the second.
We have then, ultimately, p=pi, and similarly p=p2 = ps, which
proves the theorem.

* Reckoned positive when pressures, negative when tensions. Most fluids are,
however, incapable under ordinary conditions of supporting more than an ex
ceedingly slight degree of tension, so that p is nearly always positive.

1-4] EULERIAN EQUATIONS. 3

3. The equations of motion of a fluid have been obtained in
two different forms, corresponding to the two ways in which the
problem of determining the motion of a fluid mass, acted on by
given forces and subject to given conditions, may be viewed.
We may either regard as the object of our investigations a
knowledge of the velocity, the pressure, and the density, at all
points of space occupied by the fluid, for all instants; or we
may seek to determine the history of any particle. The equations
obtained on these two plans are conveniently designated, as by
German mathematicians, the ' Eulerian ' and the ' Lagrangian '
forms of the hydrokinetic equations, although both forms are in
reality due to Elder*.

The Eulerian Equations.

4. Let u, v, w be the components, parallel to the co-ordinate
axes, of the velocity at the point (x, y, z) at the time t. These
quantities are then functions of the independent variables x, yy z, t.
For any particular value of t they express the motion at that
instant at all points of space occupied by the fluid; whilst for
particular values of x, y, z they give the history of what goes on
at a particular place.

We shall suppose, for the most part, not only that u, v, w are
finite and continuous functions of x} y, z, but that their space-
derivatives of the first order (du/dx, dv/dx, dw/dx, &c.) are
everywhere finite f; we shall understand by the term 'continuous
motion,' a motion subject to these restrictions. Cases of excep
tion, if they present themselves, will require separate examination.
In continuous motion, as thus defined, the relative velocity of

* "Principes ge"neraux du mouvement des fluides." Hist, de I'Acad. de Berlin,
1755.

" De principiis motus fluidorum." Novi Comm. Acad. Petrop. t. xiv. p. 1 (1759).

Lagrange gave three investigations of the equations of motion ; first, incidentally,
in connection with the principle of Least Action, in the Miscellanea Taurinensia,
t. ii., (1760), Oeuvres, Paris, 1867-92, t. i.; secondly in his " Memoire sur la Theorie
du Mouvement des Fluides ", Nouv. mem. de VAcad. de Berlin, 1781, Oeuvres, t. iv.;
and thirdly in the Mecanique Analytique. In this last exposition he starts with the
second form of the equations (Art. 13, below), but translates them at once into the
' Eulerian ' notation.

t It is important to bear in mind, with a view to some later developments
under the head of Vortex Motion, that these derivatives need not be assumed to be
continuous.

1—2

4 THE EQUATIONS OF MOTION. [CHAP. I

any two neighbouring particles P, P will always be infinitely
small, so that the line PP' will always remain of the same order
of magnitude. It follows that if we imagine a small closed surface
to be drawn, surrounding P, and suppose it to move with the
fluid, it will always enclose the same matter. And any surface
whatever, which moves with the fluid, completely and permanently
separates the matter on the two sides of it.

5. The values of ut v, w for successive values of t give as it
were a series of pictures of consecutive stages of the motion, in
which however there is no immediate means of tracing the
identity of any one particle.

To calculate the rate at which any function F (x, y, z, t) varies
for a moving particle, we remark that at the time t + St the
particle which was originally in the position (a?, y, z) is in the
position (x -f u$t, y + v$t, z 4- iu$t), so that the corresponding value

F (x + uSt, y + vSt, z + wSt, t + St)

.

doc dy dz at

If, after Stokes, we introduce the symbol D/Dt to denote a
differentiation following the motion of the fluid, the new value
of F is also expressed by F + DF/Dt . St, whence
DF dF dF dF dF

-jI -j -J- -J- ............... (1).

dt dx dy dz

6. To form the dynamical equations, let p be the pressure, p
the density, X, Y, Z the components of the extraneous forces
per unit mass, at the point (x, y, z) at the time t. Let us
take an element having its centre at (x, y, z), and its edges &p,
by, Sz parallel to the rectangular co-ordinate axes. The rate at
which the ^-component of the momentum of this element is
increasing is pSac&ySzDu/Dt', and this must be equal to the
^-component of the forces acting on the element. Of these the
extraneous forces give p&x&ySzX. The pressure on the yz-f&ce
which is nearest the origin will be ultimately

(p — \dp\dx . Sac) §y§z*,

* It is easily seen, by Taylor's theorem, that the mean pressure over any face
of the element dxdySz may be taken to be equal to the pressure at the centre of
that face.

4-7] EQUATION OF CONTINUITY. 5

that on the opposite face

(p + ^dp/dx . Sx) SySz.

The difference of these gives a resultant — dp/dx. SxtySz in the
direction of ^-positive. The pressures on the remaining faces are
perpendicular to x. We have then

p§x§y§z^- = pSxtySz X — — BxBySz.

Substituting the value of Du/Dt from (1), and writing down
the symmetrical equations, we have

(2).

7. To these dynamical equations we must join, in the first
place, a certain kinematical relation between u, v, w, p, obtained
as follows.

If v be the volume of a moving element, we have, on account
of the constancy of mass,

du
di*
dv

du

U dx
dv

du
v -j--f
dy

dv

du
W ~dz~
dv

I dp
^ pdx'

F_i*

di
dw
~dt~*

dx

dw

U —j r

dx

V dy
dw

fl-jT +

dy

° dz~~
dw
W~dz~

pdy'

Z--^
p dz'

To calculate the value of 1/v.Dv/Dt, let the element in question
be that which at time t fills the rectangular space &pfy£* having
one corner P at (x, y, z), and the edges PL, PM, PN (say) parallel
to the co-ordinate axes. At time t + &t the same element will
form an oblique parallelepiped, and since the velocities of the
particle L relative to the particle P are du/dx . &», dv/dx . Sx,
dw/dx.Bx, the projections of the edge PL become, after the
time &,

/-i du *.\ * dv ~ 5. dw ~ s
1 -h -5- dn &r, -j-St. So?, 3- Bt . &r,

V dx J dx dx

respectively. To the first order in &t, the length of this edge is
now

, , du 2.\ *
1 + -j- o< I ox,
dx J

0 THE EQUATIONS OF MOTION. [CHAP. I

and similarly for the remaining edges. Since the angles of the
parallelepiped differ infinitely little from right angles, the volume
is still given, to the first order in St, by the product of the three
edges, i.e. we have

Dv .,, f, /du dv dw

= \1 + (f+f+^nt

( \d% dy dz)
1 Z)v du dv dw

Hence (1) becomes

Dp (du dv dw\

-7^7 -r p -; -- h j -- r~j- I — U .................. (o).

Dt r \dx dy dz)
This is called the ' equation of continuity.'

rp, . du dv dw

The expression -7- + -y- + -^-

c?^1 c?y c?5

which, as we have seen, measures the rate of increase of volume
of the fluid at the point (x, y, z\ is conveniently called the
'expansion' at that point.

8. Another, and now more usual, method of obtaining the
above equation is, instead of following the motion of a fluid
element, to fix the attention on an element Sxby&z of space, and
to calculate the change produced in the included mass by the
flow across the boundary. If the centre of the element be at
(#, y, z), the amount of matter which per unit time enters
it across the i/^-face nearest the origin is

and the amount which leaves it by the opposite face is

The two faces together give a gain

per unit time. Calculating in the same way the effect of the flow

7-9] PHYSICAL EQUATIONS. 7

across the remaining faces, we have for the total gain of mass,
per unit time, in the space SxSySz, the formula

id. pu d.pv d. pw\ ~ 5, 5.
- —f- + —£- + — f— &%&*.
\ dx dy dz J

Since the quantity of matter in any region can vary only in
consequence of the flow across the boundary, this must be
equal to

whence we get the equation of continuity in the form

dj> d^u djn d^ = () .....
dt doc dy dz

9. It remains to put in evidence the physical properties of
the fluid, so far as these affect the quantities which occur in our
equations.

In an ' incompressible ' fluid, or liquid, we have Dp/Dt - 0, in
which case the equation of continuity takes the simple form

^ + J + ^=0 ........................ (1).

dx dy dz

It is not assumed here that the fluid is of uniform density,
though this is of course by far the most important case.

If we wished to take account of the slight compressibility of
actual liquids, we should have a relation of the form

p = K(p-p0)/p0 ........................ (2),

or p/p0 = l+p/K ........................... (3),

where tc denotes what is called the ' elasticity of volume.'

In the case of a gas whose temperature is uniform and constant
we have the ' isothermal ' relation

where p0, p0 are any pair of corresponding values for the tempera
ture in question.

In most cases of motion of gases, however, the temperature is
not constant, but rises and falls, for each element, as the gas is
compressed or rarefied. When the changes are so rapid that we can
ignore the gain or loss of heat by an element due to conduction
and radiation, we have the 'adiabatic' relation

8 THE EQUATIONS OF MOTION. [CHAP. I

where p0 and p0 are any pair of corresponding values for the
element considered. The constant 7 is the ratio of the two
specific heats of the gas; for atmospheric air, and some other
gases, its value is 1*408.

10. At the boundaries (if any) of the fluid, the equation of
continuity is replaced by a special surface-condition. Thus at a
fixed boundary, the velocity of the fluid perpendicular to the
surface must be zero, i.e. if I, m, n be the direction-cosines of the
normal,

lu + mv + nw = 0 (1).

Again at a surface of discontinuity, i.e. a surface at which the
values of u, v, w change abruptly as we pass from one side to the
other, we must have

I (X - u.2) + m(vl — v2) + n(w1-w.2) = 0 (2),

where the suffixes are used to distinguish the values on the two
sides. The same relation must hold at the common surface of a
fluid and a moving solid.

The general surface-condition, of which these are particular
cases, is that if F(xy y, z, t) = 0 be the equation of a bounding
surface, we must have at every point of it

DF/Dt = 0 (3).

The velocity relative to the surface of a particle lying in it must
be wholly tangential (or zero), for otherwise we should have a
finite flow of fluid across it. It follows that the instantaneous
rate of variation of F for a surface-particle must be zero.

A fuller proof, given by Lord Kelvin*, is as follows. To find the
rate of motion (v) of the surface F(x, y, z, t) = 0, normal to itself,
we write

where I, in, n are the direction-cosines of the normal at (x, y, z),
whence

dF dF ndF\ dF^Q

dx H dy n dz ) dt

dF dF dF _

Since 6, m, n = -j- , -j- , -7- , -r *f»

c?« dy a*

:; (W. Thomson) "Notes on Hydrodynamics," Camb. and Dub. Math. Journ.
Feb. 1818. Mathematical and Physical Papers, Cambridge, 1882..., t. i., p. 83.

9-10] SURFACE-CONDITION.

P

where R =

dxj \dy

1 d^
this gives v = ~~Ti ~di ........................... *•

At every point of the surface we must have

i> = lu+ mv + mu,

which leads, on substitution of the above values of I, m, n, to the
equation (3).

The partial differential equation (3) is also satisfied by any surface
moving with the fluid. This follows at once from the meaning of the operator
DjDt. A question arises as to whether the converse necessarily holds ; i. e.
whether a moving surface whose equation ^=0 satisfies (3) will always
consist of the same particles. Considering any such surface, let us fix our
attention on a particle P situate on it at time t. The equation (3) expresses
that the rate at which P is separating from the surface is at this instant zero ;
and it is easily seen that if the motion be continuous (according to the definition
of Art. 4), the normal velocity, relative to the moving surface F, of a particle
at an infinitesimal distance £ from it is of the order £, viz. it is equal to G£
where G is finite. Hence the equation of motion of the particle P relative to
the surface may be written

This shews that log £ increases at a finite rate, and since it is negative infinite
to begin with (when £=0), it remains so throughout, i.e. £ remains zero for
the particle P.

The same result follows from the nature of the solution of

dF dF dF dF , ,..

lfc+^ + ^ + ^=0 ........................... (1)'

considered as a partial differential equation in F*. The subsidiary system
of ordinary differential equations is

, dx dii dz ,..»

dt = — = ^- = — .............................. (11),

u v w

in which a, y, z are regarded as functions of the independent variable t.
These are evidently the equations to find the paths of the particles, and their
integrals may be supposed put in the forms

where the arbitrary constants a, 6, c are any three quantities serving to
identify a particle; for instance they may be the initial co-ordinates. The
general solution of (i) is then found by elimination of a, 6, c between (iii) and

F=+(a,b,c) ................................. (iv),

where \^ is an arbitrary function. This shews that a particle once in the
surface F=0 remains in it throughout the motion.

* Lagrange, Oeuvres, t. iv., p. 706.

10 THE EQUATIONS OF MOTION. [CHAP. I

Equation of Energy.

11. In most cases which we shall have occasion to consider
the extraneous forces have a potential ; viz. we have

cm cm <m

A = --- =— , Y — --- 7 , /j — -- 7— ............ (I).

doc dy dz

The physical meaning of H is that it denotes the potential energy,
per unit mass, at the point (x, y, z\ in respect of forces acting at
a distance. It will be sufficient for the present to consider the
case where the field of extraneous force is constant with respect to
the time, i.e. d£l/dt = 0. If we now multiply the equations (2) of
Art. 6 by u, v, w, in order, and add, we obtain a result which
may be written

If we multiply this by 8x81/80, and integrate over any region, we
find, since DjDt . (p 8x81/82) = 0,

where

V = ffftlpdxdydx ...(3),

i.e. T and V denote the kinetic energy, and the potential energy
in relation to the field of extraneous force, of the fluid which at the
moment occupies the region in question. The triple integral on
the right-hand side of (2) may be transformed by a process which
will often recur in our subject. Thus, by a partial integration,

ud dxdydz = 1 1 [pu] dydz — 1 1 Ip -T- dxdydz,

where [pit] is used to indicate that the values of pu at the points
where the boundary of the region is met by a line parallel to x are
to be taken, with proper signs. If /, m, n be the direction-cosines
of the inwardly directed normal to any element 8S of this
boundary, we have 8y8z=±l8S, the signs alternating at the
successive intersections referred to. We thus find that

Sf[pu] dydz = —fJpuldS,

11] ENERGY. 11

where the integration extends over the whole bounding surface.
Transforming the remaining terms in a similar manner, we obtain

^ (T + V) = I fp (lu + mo + nw) dS

dv

In the case of an incompressible fluid this reduces to the
form

.(5).

Since lu + mv + nw denotes the velocity of a fluid particle in the
direction of the normal, the latter integral expresses the rate at
which the pressures p$S exerted from without on the various
elements SS of the boundary are doing work. Hence the total
increase of energy, kinetic and potential, of any portion of
the liquid, is equal to the work done by the pressures on its
surface.

In particular, if the fluid be bounded on all sides by fixed
walls, we have

lu + mv + nw = 0

over the boundary, and therefore

T+F= const (6).

A similar interpretation can be given to the more general
equation (4), provided p be a function of p only. If we write

E = -

then E measures the work done by unit mass of the fluid against
external pressure, as it passes, under the supposed relation between
p and p, from its actual volume to some standard volume. For
example*, if the unit mass were enclosed in a cylinder with a
sliding piston of area A, then when the piston is pushed outwards
through a space &x, the work done is p A . S#, of which the factor
A§x denotes the increment of volume, i.e. of p~l. We may there-

* See any treatise on Thermodynamics. In the case of the adiabatic relation
we find

1 (2.6)

7-1 \P Pot

12 THE EQUATIONS OF MOTION. [CHAP. I

fore call E the intrinsic energy of the fluid, per unit mass. Now
recalling the interpretation of the expression

du dv dw
dx dy dz

given in Art. 7 we see that the volume-integral in (4) measures the
rate at which the various elements of the fluid are losing intrinsic
energy by expansion* ; it is therefore equal to — DWjDt,

where W = fffEpda;dydz, (8).

Hence -^ (T + V+ W) = ! fp (lu + mv + nw) dS (9).

The total energy, which is now partly kinetic, partly potential in
relation to a constant field of force, and partly intrinsic, is therefore
increasing at a rate equal to that at which work is being done on
the boundary by pressure from without.

Impulsive Generation of Motion.

12. If at any instant impulsive forces act bodily on the
fluid, or if the boundary conditions suddenly change, a sudden
alteration in the motion may take place. The latter case may
arise, for instance, when a solid immersed in the fluid is suddenly
set in motion.

Let p be the density, u, v, w the component velocities immedi
ately before, u', v', w' those immediately after the impulse, X' , Yf, Z'
the components of the extraneous impulsive forces per unit mass,
OT the impulsive pressure, at the point (x, y, z). The change of
momentum parallel to x of the element defined in Art. 6 is then
p§x§y§z(u'—ii)\ the ^--component of the extraneous impulsive
forces is p&x§y&zX', and the resultant impulsive pressure in the
same direction is — dv/dx . Sx&y&t. Since an impulse is to be
regarded as an infinitely great force acting for an infinitely short
time (r, say), the effects of all finite forces during this interval are
neglected.

* Otherwise,

(du dv dw
Tx + Ty + -S

11-12]

Hence,

IMPULSIVE MOTION.

„ dl

or

, 1

Similarly,

w —w= Z — -7-.
p dz

(i).

These equations might also have been deduced from (2) of Art. 6, by
multiplying the latter by 8t, integrating between the limits 0 and T, putting

r=[TYdt, Z' = [TZdt, w= [ pdt,
Jo jo Jo

and then making r vanish.

In a liquid an instantaneous change of motion can be produced
by the action of impulsive pressures only, even when no impulsive
forces act bodily on the mass. In this case we have X', Y', Z' = 0,
so that

'_ = _i^
p dx '

V — V = --- -y- ,

p dy

—w = --- ^- .
p dz

(2).

If we differentiate these equations with respect to #, y, z,
respectively, and add, and if we further suppose the density to
be uniform, we find by Art. 9 (1) that

d V

_
~

The problem then, in any given case, is to determine a value of CT
satisfying this equation and the proper boundary conditions* ; the
instantaneous change of motion is then given by (2).

* It will appear in Chapter in. that the value of or is thus determinate, save as
to an additive constant.

14 THE EQUATIONS OF MOTION. [CHAP. I

The Lagrangian Equations.

13. Let a, b, c be the initial co-ordinates of any particle of
fluid, x, y, z its co-ordinates at time t. We here consider a?, y, z as
functions of the independent variables a, 6, c, t; their values in
terms of these quantities give the whole history of every particle
of the fluid. The velocities parallel to the axes of co-ordinates of
the particle (a, 6, c) at time t are dx/dt, dyjdt, dz/dt, and the
component accelerations in the same directions are d-x/dfi, d2y/dt2,
dzzjdt2. Let p be the pressure and p the density in the neigh
bourhood of this particle at time t; X, Y, Z the components
of the extraneous forces per unit mass acting there. Consider
ing the motion of the mass of fluid which at time t occupies
the differential element of volume Sx&ySz, we find by the same
reasoning as in Art. 6,

d?x _ I dp
~

__
dt* pdy

d*z _ „ 1 dp
H&~ ~^dz'

These equations contain differential coefficients with respect to
oc, y, z, whereas our independent variables are a, b, c, t. To
eliminate these differential coefficients, we multiply the above
equations by dxjda, dy/da, dz/da, respectively, and add ; a second
time by dxjdb, dy/db, dz/db, and add ; and again a third time by
dxjdc, dy/dc, dz/dc, and add. We thus get the three equations

_ _

d? l da \& ~ da p / da p da ~

d^x \dx (d*y v\ dy (d*z \dz Idp

-X + -Y + -z + --

+ --

c+ pdc~

These are the ' Lagrangian ' forms of the dynamical equations.

14. To find the form which the equation of continuity
assumes in terms of our present variables, we consider the
element of fluid which originally occupied a rectangular parallel-

13-15]

LAGRANGIAN EQUATIONS.

15

epiped having its centre at the point (a, b, c), and its edges
£&, 56, Sc parallel to the axes. At the time t the same element
forms an oblique parallelepiped. The centre now has for its
co-ordinates x, y, z\ and the projections of the edges on the
co-ordinate axes are respectively

dx

dy

dz ~

j- oa ;
da

db

dx
dc

db ' db

dc

dz

-T

dc

The volume of the parallelepiped is therefore

dx

dy

dz

da'

da'

da

dx

dy

dz

db'

db'

db

dx

dy

dz

dc'

dc '

dc

or, as it is often written,

d (a, 6, c)
Hence, since the mass of the element is unchanged, we have

d (a?, y, z) = ^

where pQ is the initial density at (a, 6, c).

In the case of an incompressible fluid p=p0, so that (1)

becomes

d (x, y, z)
d (a, b, c)

.(2).

Weber s Transformation.

15. If as in Art. 11 the forces X, Y, Z have a potential
the dynamical equations of Art. 13 may be written

___,
dt'2 da dt~ da dP da da p da '

16 THE EQUATIONS OF MOTION. [CHAP. I

Let us integrate these equations with respect to t between the
limits 0 and t. We remark that

p d?x dx , _ Vdx dxV _ [fdx
L~dVda "dtda. J.dt

fdx d*x

dadt

dxdx *

,
*

where u0 is the initial value of the ^-component of velocity of the
particle (a, b, c). Hence if we write

we find

dx dx dy dy dz dz dy \

_1_ «/ «£; I -it /V .

dt da dt da dt da da '

dx dx dy dy dz dz d%

dx dx dy dy dz dz dy

I «<_ *J ^i. MSf A*i — - _ ^V

dt dc dt dc dt dc dc '
These three equations, together with

.(2)

and the equation of continuity, are the partial differential equa
tions to be satisfied by the five unknown quantities xt y, z, p, % ;
p being supposed already eliminated by means of one of the rela
tions of Art. 9.

The initial conditions to be satisfied are

x =. a, y — l>, z = c, % = 0.

16. It is to be remarked that the quantities a, b, c need not
be restricted to mean the initial co-ordinates of a particle ; they
may be any three quantities which serve to identify a particle,
and which vary continuously from one particle to another. If
we thus generalize the meanings of a, b, c, the form of the
dynamical equations of Art. 13 is not altered ; to find the form
which the equation of continuity assumes, let x0, yQi ZQ now denote
the initial co-ordinates of the particle to which a, b, c refer.
The initial volume of the parallelepiped, whose centre is at

* H. Weber, " Ueber eine Transformation der hydrodynamischen Gleichungen ",
Crelle, t. Ixviii. (1868).

15-17] COMPARISON OF METHODS. 17

(#o, 2/o, #0) and whose edges correspond to variations Set, &b, &c of
the parameters, a, b, c, is

i

a (a, 6, c)

so that we have

d (x, y, z) _ rf(fl?0, y0> z0) m

pd(a,b,c)~po d(a,b,c)

or, for an incompressible fluid,

d (x, y, z) d(a;Qty0) z0) ,

d(a,b,c) d(a,b,c)

17. If we compare the two forms of the fundamental equations to which
we have been led, we notice that the Eulerian equations of motion are linear
and of the first order, whilst the Lagrangian equations are of the second order,
and also contain products of differential coefficients. In Weber's transfor
mation the latter are replaced by a system of equations of the first order, and
of the second degree. The Eulerian equation of continuity is also much
simpler than the Lagrangian, especially in the case of liquids. In these
respects, therefore, the Eulerian forms of the equations possess great ad
vantages. Again, the form in which the solution of the Eulerian equations
appears corresponds, in many cases, more nearly to what we wish to know
as to the motion of a fluid, our object being, in general, to gain a knowledge
of the state of motion of the fluid mass at any instant, rather than to trace
the career of individual particles.

On the other hand, whenever the fluid is bounded by a moving surface,
the Lagrangian method possesses certain theoretical advantages. In the
Eulerian method the functions u, v, w have no existence beyond this surface,
and hence the range of values of #, y, z for which these functions exist varies
in consequence of the motion which is itself the subject of investigation. In
the other method, on the contrary, the range of the independent variables
«, 6, c is given once for all by the initial conditions'*.

The difficulty, however, of integrating the Lagrangian equations has
hitherto prevented their application except in certain very special cases.
Accordingly in this treatise we deal almost exclusively with the Eulerian
forms. The simplification and integration of these in certain cases form the
subject of the following chapter.

* H. Weber, I.e.

CHAPTER II.

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES.

18. IN a large and important class of cases the component
velocities u, v, w can be expressed in terms of a single function
<f>, as follows :

d(f> dd> dd> ,,x

u = — f't v = — -71-, w = — f (1).

dv' dy' dz

Such a function is called a 'velocity-potential/ from its analogy
with the potential function which occurs in the theories of
Attractions, Electrostatics, &c. The general theory of the
velocity-potential is reserved for the next chapter; but we give
at once a proof of the following important theorem :

If a velocity-potential exist, at any one instant, for any
finite portion of a perfect fluid in motion under the action of
forces which have a potential, then, provided the density of the
fluid be either constant or a function of the pressure only, a
velocity-potential exists for the same portion of the fluid at all
instants before or after*.

In the equations of Art. 15, let the instant at which the

* Lagrange, " Me'moire sur la Th6orie du Mouvement des Fluides," Nouv.
mim. de VAcad. de Berlin, 1781 ; Oeuvres, t. iv. p. 714. The argument is repro
duced in the Mecanique Anatytique.

Lagrange's statement and proof were alike imperfect ; the first rigorous demon
stration is due to Cauchy, " Memoire sur la Th6orie des Ondes," Mem. de VAcad.
roy. des Sciences, t. i. (1827) ; Oeuvres Completes, Paris, 1882..., lre Se"rie, t. i. p. 38 ;
the date of the memoir is 1815. Another proof is given by Stokes, Camb. Trans, t.
viii. (1845) (see also Math, and Pliys. Papers, Cambridge, 1880..., t. i. pp. 106, 158,
and t. ii. p. 36), together with an excellent historical and critical account of the
whole matter,

18-19] VELOCITY-POTENTIAL. 19

velocity-potential $0 exists be taken as the origin of time ; we

have then

u0da + v0db + w0dc = — c£</>0,

throughout the portion of the mass in question. Multiplying the
equations (2) of Art. 15 in order by da, db, dc, and adding,
we get

ffi^X~*~ dt ^ ^~dt^Z~ (u^a + Vodb + wodc) = - dx,
or, in the ' Eulerian ' notation,

udoc -f vdy + wdz = — d ((/>0 + %) = — c&£, say.

Since the upper limit of t in Art. 15 (1) may be positive or negative,
this proves the theorem.

It is to be particularly noticed that this continued existence
of a velocity-potential is predicated, not of regions of space,
but of portions of matter. A portion of matter for which a
velocity-potential exists moves about and carries this property
with it, but the part of space which it originally occupied may,
in the course of time, come to be occupied by matter which
did riot originally possess the property, and which therefore
cannot have acquired it.

The class of cases in which a velocity-potential exists in
cludes all those where the motion has originated from rest under
the action of forces of the kind here supposed ; for then we have,
initially,

u0da + v0db + w0dc = 0,

or <£>0 = const.

The restrictions under which the above theorem has been proved must
be carefully remembered. It is assumed not only that the external forces
X, F, Z, estimated at per unit mass, have a potential, but that the density
p is either uniform or a function of p only. The latter condition is violated
for example, in the case of the convection currents generated by the unequal
application of heat to a fluid ; and again, in the wave-motion of a hetero
geneous but incompressible fluid arranged originally in horizontal layers of
equal density. Another important case of exception is that of * electro-magnetic
rotations.'

19. A comparison of the formulae (1) with the equations
(2) of Art. 12 leads to a simple physical interpretation of 0.

Any actual state of motion of a liquid, for which a (single-valued)
velocity-potential exists, could be produced instantaneously from rest

2—2

20 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

by the application of a properly chosen system of impulsive pressures.
This is evident from the equations cited, which shew, moreover,
that <f> = tr/p + const. ; so that w = p<f> 4- C gives the requisite sys
tem. In the same way ^ = — p(f> + C gives the system of impulsive
pressures which would completely stop the motion. The occur
rence of an arbitrary constant in these expressions shews, what is
otherwise evident, that a pressure uniform throughout a liquid
mass produces no effect on its motion.

In the case of a gas, <f> may be interpreted as the potential
of the external impulsive forces by which the actual motion at
any instant could be produced instantaneously from rest.

A state of motion for which a velocity-potential does not exist
cannot be generated or destroyed by the action of impulsive
pressures, or of extraneous impulsive forces having a potential.

20. The existence of a velocity-potential indicates, besides,
certain kinematical properties of the motion.

A 'line of motion' or 'stream-line'* is defined to be a line
drawn from point to point, so that its direction is everywhere that
of the motion of the fluid. The differential equations of the
system of such lines are

dx = dy==dz

U V W " " \ /'

The relations (1) shew that when a velocity-potential exists the
lines of motion are everywhere perpendicular to a system of sur
faces, viz. the ' equipotential ' surfaces <£ = const.

Again, if from the point (x, y, z) we draw a linear element Ss
in the direction (I, m, ri), the velocity resolved in this direction is
lu + mv + nw, or

_ d<t> dx d$dy d(f> dz , . , _ d(/>
dx ds dy ds dz ds ' ds '

The velocity in any direction is therefore equal to the rate of
decrease of </> in that direction.

Taking 8s in the direction of the normal to the surface </> = const,
we see that if a series of such surfaces be drawn corresponding to

* Some writers prefer to restrict the use of the term ' stream-line ' to the case of
steady motion, as defined in Art. 22.

19-21] VELOCITY-POTENTIAL. 21

equidistant values of </>, the common difference being infinitely
small, the velocity at any point will be inversely proportional to the
distance between two consecutive surfaces in the neighbourhood
of the point.

Hence, if any equipotential surface intersect itself, the velocity is zero at
the intersection. The intersection of two distinct equipotential surfaces would
imply an infinite velocity.

21. Under the circumstances stated in Art. 18, the equations
of motion are at once integrable throughout that portion of the
fluid mass for which a velocity-potential exists. For in virtue

of the relations

dv _dw dw __ du du _ dv
dz ~ dy ' dx dz ' dy dx'

which are implied in (1), the equations of Art. 6 may be written

d26 du dv dw d£l 1 dp s s
-T-^ t + u^- + Vj- + w-j-=- j --- f» fee., &c.
dxdt dx dx dx dx p dx

These have the integral

where q denotes the resultant velocity (u2 + v2 + w2)*, and F (t) is
an arbitrary function of t. It is often convenient to suppose this
arbitrary function to be incorporated in the value of d<f)/dt ; this
is permissible since, by (1), the values of u, v, w are not thereby
affected.

Our equations take a specially simple form in the case of an
incompressible fluid ; viz. we then have

with the equation of continuity

which is the equivalent of Art. 9 (1). When, as in many cases
which we shall have to consider, the boundary conditions are
purely kinematical, the process of solution consists in finding a
function which shall satisfy (5) and the prescribed surface-con
ditions. The pressure p is then given by (4), arid is thus far

22 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

indeterminate to the extent of an additive function of t. It
becomes determinate when the value of p at some point of the
fluid is given for all values of t.

Suppose, for example, that we have a solid or solids moving through a
liquid completely enclosed by fixed boundaries, and that it is possible (e.g. by
means of a piston) to apply an arbitrary pressure at some point of the
boundary. Whatever variations are made in the magnitude of the force ap
plied to the piston, the motion of the fluid and of the solids will be absolutely
unaffected, the pressure at all points instantaneously rising or falling by
equal amounts. Physically, the origin of the paradox (such as it is) is that
the fluid is treated as absolutely incompressible. In actual liquids changes
of pressure are propagated with very great, but not infinite, velocity.

Steady Motion.

22. When at every point the velocity is constant in magnitude
and direction, i.e. when

dt=°' ^=°' ^~ = ° (1)

everywhere, the motion is said to be ' steady.'

In steady motion the lines of motion coincide with the paths
of the particles. For if P, Q be two consecutive points on a line
of motion, a particle which is at any instant at P is moving in the
direction of the tangent at P, and will, therefore, after an infinitely
short time arrive at Q. The motion being steady, the lines of
motion remain the same. Hence the direction of motion at Q
is along the tangent to the same line of motion, i.e. the particle
continues to describe the line.

In steady motion the equation (3) of the last Art. becomes

— = — II — J<?24- constant (2).

P

The equations may however in this case be integrated to a
certain extent without assuming the existence of a velocity-
potential. For if 8s denote an element of a stream-line, we have
u = q dx/ds, &c. Substituting in the equations of motion and
remembering (1), we have

du _ cZO 1 dp
^ ds dx p dx'

21-23] STEADY MOTION. 23

with two similar equations. Multiplying these in order by
dx/ds, dy/ds, dzjds, and adding, we have

du dv dw _ c£ft __ 1 dp
ds ds ds ds p ds9

or, integrating along the stream-line,

P

This is similar in form to (2), but is more general in that it
does not assume the existence of a velocity-potential. It must
however be carefully noticed that the 'constant ' of equation (2) and
the ' C ' of equation (3) have very different meanings, the former
being an absolute constant, while the latter is constant along any
particular stream-line, but may vary as we pass from one stream
line to another.

23. The theorem (3) stands in close relation to the principle
of energy. If this be assumed independently, the formula may be
deduced as follows*. Taking first the particular case of a liquid,
let us consider the portion of an infinitely narrow tube, whose
boundary follows the stream-lines, included between two cross
sections A and B, the direction of motion being from A to B. Let
p be the pressure, q the velocity, ft the potential of the external
forces, a the area of the cross section, at A, and let the values
of the same quantities at B be distinguished by accents. In each
unit of time a mass pqa at A enters the portion of the tube
considered, whilst an equal mass pq'a' leaves it at B. Hence
qa = q'a. Again, the work done on the mass entering at A is
pqa per unit time, whilst the loss of work at B is p'qa'. The
former mass brings with it the energy pqa (^ q2 + ft), whilst the
latter carries off energy to the amount pqo-'(^q2 -f ft'). The
motion being steady, the portion of the tube considered neither
gains nor loses energy on the whole, so that

pqo- + pqo- (%q2 + ft) =p'q'a' + pq'a' (%q'z + ft').
Dividing by pqa (= pq'a'}, we have

* This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica,
Argeiitorati, 1738.

24 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

or, using C in the same sense as before,

(4),

which is what the equation (3) becomes when p is constant.

To prove the corresponding formula for compressible fluids, we
remark that the fluid entering at A now brings with it, in addition
to its energies of motion and position, the intrinsic energy

-K)— M?-

per unit mass. The addition of these terms to (4) gives the
equation (3).

The motion of a gas is as a rule subject to the adiabatic law

PlP» = (p/Po)y ........................... (5),

and the equation (3) then takes the form

• ~i-p = -n-^+G .................. («)•

24. The preceding equations shew that, in steady motion,
and for points along any one stream-line*, the pressure is,
ccvteris paribus, greatest where the velocity is least, and vice versa.
This statement, though opposed to popular notions, becomes
evident when we reflect that a particle passing from a place of
higher to one of lower pressure must have its motion accelerated,
and vice versd"^.

It follows that in any case to which the equations of the last
Art. apply there is a limit which the velocity cannot exceed J. For
instance, let us suppose that we have a liquid flowing from a
reservoir where the motion may be neglected, and the pressure is
po, and that we may neglect extraneous forces. We have then, in
(4), C = PQ/P, and therefore

Now although it is found that a liquid from which all traces

* This restriction is unnecessary when a velocity-potential exists.

t Some interesting practical illustrations of this principle are given by Froude,
Nature, t. xiii., 1875.

J Cf. von Helmholtz, " Ueber discontinuirliche Fliissigkeitsbewegungen," Berl.
Monatsber., April, 1868 ; Phil. Mag., Nov. 1868 ; Gesammelte Abhandlungen, Leipzig,
1882-3, t. i., p. 146.

23-24] LIMITING VELOCITY. 25

of air or other dissolved gas have been eliminated can sustain a
negative pressure, or tension, of considerable magnitude, this is not
the case with fluids such as we find them under ordinary conditions.
Practically, then, the equation (7) shews that q cannot exceed

'

If in any case of fluid motion of which we have succeeded
in obtaining the analytical expression, we suppose the motion
to be gradually accelerated until the velocity at some point reaches
the limit here indicated, a cavity will be formed there, and the
conditions of the problem are more or less changed.

It will be shewn, in the next chapter, that in irrotational /?/././/
motion of a liquid, whether 'steady' or not, the place of least
pressure is always at some point of the boundary, provided the
extraneous forces have a potential II satisfying the equation

This includes, of course, the case of gravity.

The limiting velocity, when no extraneous forces act, is of course that with
which the fluid would escape from the reservoir into a vacuum. In the case
of water at atmospheric pressure it is the velocity ' due to ' the height of the
water-barometer, or about 45 feet per second.

In the general case of a fluid in which p is a given function
of p we have, putting O = 0 in (3),

P
For a gas subject to the adiabatic law, this gives

7 -

= ;p1(co2-c2) ........................... (10),

if c, = (yp/p)*, = (dp/dp)*, denote the velocity of sound in the gas
when at pressure p and density p, and c0 the corresponding velocity
for gas under the conditions which obtain in the reservoir. (See
Chap, x.) Hence the limiting velocity is

or, 2-214 c0, if 7 =1-408.

26 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

25. We conclude this chapter with a few simple applications
of the equations.

Efflux of Liquids.

Let us take in the first instance the problem of the efflux
of a liquid from a small orifice in the walls of a vessel which
is kept filled up to a constant level, so that the motion may be
regarded as steady.

The origin being taken in the upper surface, let the axis of z
be vertical, and its positive direction downwards, so that £l = —gz.
If we suppose the area of the upper surface large compared with
that of the orifice, the velocity at the former may be neglected.
Hence, determining the value of C in Art. 23 (4) so that p — P (the
atmospheric pressure), when z — 0, we have

At the surface of the issuing jet we have p — P, and therefore

9" = 20* .............................. (2),

i.e. the velocity is that due to the depth below the upper surface.
This is known as Torricellis Theorem.

We cannot however at once apply this result to calculate the
rate of efflux of the fluid, for two reasons. In the first place, the
issuing fluid must be regarded as made up of a great number of
elementary streams converging from all sides towards the orifice.
Its motion is not, therefore, throughout the area of the orifice,
everywhere perpendicular to this area, but becomes more and
more oblique as we pass from the centre to the sides. Again,
the converging motion of the elementary streams must make the
pressure at the orifice somewhat greater in the interior of the
jet than at the surface, where it is equal to the atmospheric
pressure. The velocity, therefore, in the interior of the jet will
be somewhat less than that given by (2).

Experiment shews however that the converging motion above
spoken of ceases at a short distance beyond the orifice, and that (in
the case of a circular orifice) the jet then becomes approximately
cylindrical. The ratio of the area of the section S' of the jet at this
point (called the 'vena contracta') to the area 8 of the orifice is called
* This result is due to D. Bernoulli, /. c. ante p. 23.

25] EFFLUX OF LIQUIDS. 27

the ' coefficient of contraction.' If the orifice be simply a hole in
a thin wall, this coefficient is found experimentally to be about '62.

The paths of the particles at the vena contracta being nearly
straight, there is little or no variation of pressure as we pass from
the axis to the outer surface of the jet. We may therefore assume
the velocity there to be uniform throughout the section, and to have
the value given by (2), where z now denotes the depth of the vena
contracta below the surface of the liquid in the vessel. The rate

of efflux is therefore

&' ........................... (3).

The calculation of the form of the issuing jet presents difficulties which
have only been overcome in a few ideal cases of motion in two dimensions. (See
Chapter iv.) It may however be shewn that the coefficient of contraction must,
in general, lie between \ and 1. To put the argument in its simplest form,
let us first take the case of liquid issuing from a vessel the pressure in which,
at a distance from the orifice, exceeds that of the external space by the
amount P, gravity being neglected. When the orifice is closed by a plate, the
resultant pressure of the fluid on the containing vessel is of course nil. If
when the plate is removed, we assume (for the moment) that the pressure on
the walls remains sensibly equal to P, there will be an unbalanced pressure
PS acting on the vessel in the direction opposite to that of the jet, and
tending to make it recoil. The equal and contrary reaction on the fluid
produces in unit time the velocity q in the mass pqS' flowing through the
4 vena contracta,' whence

PS=pqW ....................................... (i).

The principle of energy gives, as in Art. 23,

so that, comparing, we have S' = $S. The formula (1) shews that the
pressure on the walls, especially in the neighbourhood of the orifice, will in
reality fall somewhat below the static pressure P, so that the left-hand side
of (i) is too small. The ratio S'/S will therefore in general be >i.

In one particular case, viz. where a short cylindrical tube, projecting
inwards, is attached to the orifice, the assumption above made is sufficiently
exact, and the consequent value £ for the coefficient then agrees with
experiment.

The reasoning is easily modified so as to take account of gravity (or other
conservative forces). We have only to substitute for P the excess of the static
pressure at the level of the orifice over the pressure outside. The difference
of level between the orifice and the ' vena contracta ' is here neglected *.

* The above theory is due to Borda (Mem. de VAcad. des Sciences, 1766), who
also made experiments with the special form of mouth-piece referred to, and found
SJS' = 1'942. It was re-discovered by Hanlon, Proc. Land. Math. Soc. t. iii. p. 4,
(1869) ; the question is further elucidated in a note appended to this paper by
Maxwell. See also Froude and J. Thomson, Proc. Glasgow Phil. Soc. t. x., (1876).

28 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

Efflux of Oases.

26. We consider next the efflux of a gas, supposed to flow
through a small orifice from a vessel in which the pressure is
PQ and density p0 into a space where the pressure is plt We assume
that the motion has become steady, and that the expansion takes
place according to the adiabatic law.

If the ratio PQ/PI of the pressures inside and outside the vessel do not
exceed a certain limit, to be indicated presently, the flow will take place in
much the same manner as in the case of a liquid, and the rate of discharge
may be found by putting p =PI in Art. 24 (9), and multiplying the resulting
value of q by the area >S" of the vena contracta. This gives for the rate of
discharge of mass

<«••

It is plain, however, that there must be a limit to the applicability of this
result; for otherwise we should be led to the paradoxical conclusion that
when £>! = (), i.e. the discharge is into a vacuum, the flow of matter is nil.
The elucidation of this point is due to Prof. Osborne Reynolds f. It is easily
found by means of Art. 24 (8), that qp is a maximum, i.e. the section of an
elementary stream is a minimum, when q/2 = dp/dp, that is, the velocity of the
stream is equal to the velocity of sound in gas of the pressure and density
which prevail there. On the adiabatic hypothesis this gives, by Art. 24 (10),

2

and therefore, since c2 oc py \

p ( 2 \y^l p ( 2 \v-i

or, if y= 1-408,

p = -634Po, j? = '527^0 ........................... (iv).

If p± be less than this value, the stream after passing the point in question,
widens out again, until it is lost at a distance in the eddies due to viscosity.
The minimum sections of the elementary streams will be situate in the
neighbourhood of the orifice, and their sum S may be called the virtual
area of the latter. The velocity of efflux, as found from (ii), is

The rate of discharge is then =qpS, where q and p have the values just

* A result equivalent to this was given by de Saint Venant and Wantzel,
Journ. de VEcole Polyt., t. xvi., p. 92 (1839).

t " On the Flow of Gases," Proc. Manch. Lit. and Phil. Soc., Nov. 17, 1885 ;
Phil. Mag., March, 1876. A similar explanation was given by Hugoniot, Comptes
Rendus, June 28, July 26, and Dec. 13, 1886. I have attempted, above, to condense
the reasoning of these writers.

26-28] EFFLUX OF GASES. 29

found, and is therefore approximately independent* of the external pressure
pl so long as this falls below -527jD0. The physical reason of this is (as pointed
out by Reynolds) that, so long as the velocity at any point exceeds the velocity
of sound under the conditions which obtain there, no change of pressure can
be propagated backwards beyond this point so as to affect the motion further
up the stream.

These conclusions appear to be in good agreement with experimental results.

Under similar circumstances as to pressure, the velocities of efflux of
different gases are (so far as y can be assumed to have the same value for
each) proportional to the corresponding velocities of sound. Hence (as we
shall see in Chap, x.) the velocity of efflux will vary inversely, and the rate
of discharge of mass will vary directly, as the square root of the density f.

Rotating Liquid.

27. Let us next take the case of a mass of liquid rotating,
under the action of gravity only, with constant and uniform angular
velocity co about the axis of z, supposed drawn vertically upwards.

By hypothesis,

u = — coy, v = cox, w = 0,
X = 0, F=0, Z=-g.

The equation of continuity is satisfied identically, and the dynamical
equations of Art. 6 become

1 dp 1 dp A 1 dp ,,<

_ 0,2^, = _ _ ^ -tfy = ---f, 0 = - - -£ - a . . ..(1).
pdx pdy* pdz

These have the common integral

^ = %co2(a)2 + y*)-gz + const ............. (2).

The free surface, p = const., is therefore a paraboloid of revolution
about the axis of z, having its concavity upwards, and its latus
rectum =

0. dv du 0

Since ^ --- ;- = 2o>,

dx dy

a velocity-potential does not exist. A motion of this kind could
not therefore be generated in a ' perfect ' fluid, i.e. in one unable
to sustain tangential stress.

28. Instead of supposing the angular velocity co to be uni
form, let us suppose it to be a function of the distance r from the

* The magnitude of the ratio pjp1 will of course have some influence on the
arrangement of the streams, and consequently on the value of S.
t Cf. Graham, Phil. Trans., 1846.

30 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

axis, and let us inquire what form must be assigned to this
function in order that a velocity-potential may exist for the
motion. We find

dv du dco

-; --- ^-=20) + r-r- ,
dx dy dr

and in order that this may vanish we must have o>r2 = //,, a
constant. The velocity at any point is then = /tt/r, so that the
equation (2) of Art. 22 becomes

2 = const. -i£ ...................... (1),

if no extraneous forces act. To find the value of < we have

- _

dr~ rd6~ r'

whence (/> = - p6 + const. = — fi tan"1 - 4- const .......... (2).

x

We have here an instance of a ' cyclic ' function. A function
is said to be ' single-valued ' throughout any region of space when
we can assign to every point of that region a definite value of the
function in such a way that these values shall form a continuous
system. This is not possible with the function (2) ; for the value
of (/>, if it vary continuously, changes by -'2-TTyu, as the point to
which it refers describes a complete circuit round the origin. The
general theory of cyclic velocity-potentials will be given in the
next chapter.

If gravity act, and if the axis of z be vertical, we must add to
(1) the term — gz. The form of the free surface is therefore that
generated by the revolution of the hyperbolic curve a?z = const.
about the axis of z.

By properly fitting together the two preceding solutions we
obtain the case of Rankine's 'combined vortex.' Thus the
motion being everywhere in coaxial circles, let us suppose the
velocity to be equal to wr from r = 0 to r = a, and to wa?/r for
r > a. The corresponding forms of the free surface are then
given by

28-29]

ROTATING FLUID.

31

these being continuous when r = a. The depth of the central
depression below the general level of the surface is therefore

29. To illustrate, by way of contrast, the case of external
forces not having a potential, let us suppose that a mass of liquid
filling a right circular cylinder moves from rest under the action
of the forces

the axis of z being that of the cylinder.

If we assume u— - a>y, v = (a.v, w = 0, where « is a function of t only, these
values satisfy the equation of continuity and the boundary conditions. The
dynamical equations become

da) n I dp

--- -

dt
da>

n

Ax + By
J

pdx*

I dv

Differentiating the first of these with respect to y, and the second with
respect to x and subtracting, we eliminate p, and find

.(ii).

The fluid therefore rotates as a whole about the axis of z with constantly
accelerated angular velocity, except in the particular case when B = B. To
find p, we substitute the value of dw/dt in (i) and integrate ; we thus get

where 2/3

32 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II

30. As a final example, we will take one suggested by the
theory of 'electro-magnetic rotations.'

If an electric current be made to pass radially from an axial wire, through
a conducting liquid (e.g. a solution of CuS04), to the walls of a metallic
containing cylinder, in a uniform magnetic field, the external forces will be
of the type

Assuming u= -to?/, v = o>.r, w = 0, where o> is a function of r and t only, we
find

d<0 2 _ \lX

Eliminating p, we obtain

2dt+rdrdt = °'
The solution of this is

where F and / denote arbitrary functions. Since o) = 0 when t = Q, we have
and therefore

^oo-^(o) x

where X is a function of t which vanishes for £ = 0. Substituting in (i), and
integrating, we find

Since p is essentially a single-valued function, we must have d\/dt=n, or
\ = fj.t. Hence the fluid rotates with an angular velocity which varies
inversely as the square of the distance from the axis, and increases con
stantly with the time.

* If C denote the total flux of electricity outwards, per unit length of the axis,
and 7 the component of the magnetic force parallel to the axis, we have /*= 7(7/2717).
For the history of such experiments see Wiedemann, Lehre v. d. Elektricitat , t. iii.
p. 163. The above case is specially simple, in that the forces X, Y, Z, have a
potential (ft = - /j. tan"1 y/x), though a ' cyclic ' one. As a rule, in electro -magnetic
rotations, the mechanical forces X, Y, Z have not a potential at all.

CHAPTER III.

IEROTATIONAL MOTION.

31. THE present chapter is devoted mainly to an exposition
of some general theorems relating to the kinds of motion already
considered in Arts. 18 — 21; viz. those in which udx + vdy + wdz
is an exact differential throughout a finite mass of fluid. It is
convenient to begin with the following analysis, due to Stokes*,
of the motion of a fluid element in the most general case.

The component velocities at the point (x, y, z) being u, v, w, the
relative velocities at an infinitely near point (x + x, y + y, z + z) are

If we write

du du
= -7- x+ -y- y -
dx dy

dv dv

du

f<fez>

dv

dx dy

dw dw
= —. x +-7— y -
dx dy

h dz*'
dw

l~^z-

du
-,-,

dx

7 dv

0=-j-
dy

du dw

dw

dv

H/U/M/ \AJVU\ i

du dw

1 (dv du\

= i [ — - + -— 1

2 \dx d)

dy

—

dy

equations (1) may be written

u = ax + hy +gz + rfL — fy,
v = hx + by +fz + Jx - f z,
w = tfx + fy H- cz

(2).

* "On the Theories of the Internal Friction of Fluids in Motion,
Phil. Trans., t. viii. (1845) ; Math, and Pliys. Papers, t. i., p. 80.

34 IRROTATIONAL MOTION. [CHAP. Ill

Hence the motion of a small element having the point (x, y, z)
for its centre may be conceived as made up of three parts.

The first part, whose components are u, v, w, is a motion of
translation of the element as a whole.

The second part, expressed by the first three terms on the
right-hand sides of the equations (2), is a motion such that every
point is moving in the direction of the normal to that quadric
of the system

ax2 + 6y2 + cz2 + 2/yz + 2#zx + 2/ixy = const (3),

on which it lies. If we refer these quadrics to their principal axes,
the corresponding parts of the velocities parallel to these axes will be

u' = aV, v' = 6'y', w' = c'z' (4),

if a'x'2 + 6'y'2 + c'z/2 = const.

is what (3) becomes by the transformation. The formulae (4) express
that the length of every line in the element parallel to x' is being
elongated at the (positive or negative) rate a, whilst lines parallel
to y' and z' are being similarly elongated at the rates b' and c'
respectively. Such a motion is called one of pure strain and the
principal axes of the quadrics (3) are called the axes of the strain.

The last two terms on the right-hand sides of the equations (2)
express a rotation of the element as a whole about an instan
taneous axis; the component angular velocities of the rotation
being £ rj, £

This analysis may be illustrated by the so-called 'laminar' motion of a
liquid in which

u = 2py, v=0, w = 0,

so that a, b, c, /, g, |, ,7 = 0, h = p, £=-/*.

If A represent a rectangular fluid element bounded by planes parallel to
the co-ordinate planes, then B represents the change produced in this in a
short time by the strain, and C that due to the strain plus the rotation.

31-32] RELATIVE MOTION IN A FLUID ELEMENT. 35

It is easily seen that the above resolution of the motion is
unique. If we assume that the motion relative to the point
(x, y, z) can be made up of a strain and a rotation in which the
axes and coefficients of the strain and the axis and angular
velocity of the rotation are arbitrary, then calculating the relative
velocities u, v, w, we get expressions similar to those on the right-
hand sides of (2), but with arbitrary values of a, b, c,f, g, h, f, 77, f.
Equating coefficients of x, y, z, however, we find that a, b, c, &c.
must have respectively the same values as before. Hence the direc
tions of the axes of the strain, the rates of extension or contraction
along them, and the axis and the angular velocity of rotation, at
any point of the fluid, depend only on the state of relative motion
at that point, and not on the position of the axes of reference.

When throughout a finite portion of a fluid mass we have
f , 77, f all zero, the relative motion of any element of that portion
consists of a pure strain only, and is called ' irrotational/

32. The value of the integral

f(udx + vdy + wdz),

[f dx dy dz\ 7

or I (u -j- + v -/- + w -y- }ds,

J\ds ds ds J

taken along any line ABCD, is called* the 'flow' of the fluid
from A to D along that line. We shall denote it for shortness by
I (ABCD).

If A and D coincide, so that the line forms a closed curve, or
circuit, the value of the integral is called the 'circulation' in that
circuit. We denote it by I (ABC A). If in either case the inte
gration be taken in the opposite direction, the signs of dx/ds,
dy/ds, dzjds will be reversed, so that we have

I(AD) = -I(DA), and I (ABC A) = - I (ACBA).
It is also plain that

/ (ABCD) = I (AB) + / (BC) + / (CD).

Let us calculate the circulation in an infinitely small circuit
surrounding the point (x, y, z). If (x + x, y + y, z + z) be a
point on the circuit, we have, by Art. 31 (2),
uox + vdy + wdz = Jd (ax2 + 6y2 + cz2 + 2/yz + 2#zx + 2/zxy)
+ ? (ydz - zdy) + 77 (zdx - xdz) + f (xdy - ydx). . .(I).

* Sir W. Thomson, "On Vortex Motion." Edin. Trans., t. xxv. (1869).

3—2

36 IRROTATIONAL MOTION. [CHAP. Ill

Hence, integrating round a small closed circuit,

f(udx + vdy + vrdz)

= ZI(jd* - zdy) + 7?/(zdx - xdz) + £/(xdy - jdx). . .(2).
The coefficients of f , ?;, f in this expression are double the pro
jections of the area of the circuit on the co-ordinate planes, these
projections being reckoned positive or negative according to the
direction of the integrations. In order to have a clear under
standing on this point, we shall in this book suppose that the
axes of co-ordinates form a right-handed system ; thus if the axes
of x and y point E. and N. respectively, that of z will point ver
tically upwards*. Now let &S be the area of the circuit, and let
I, m, n be . the direction-cosines of the normal to &S drawn in the
direction which is related to that in which the circulation round
the circuit is estimated, in the manner typified by a right-handed
screwf. The formula (2) then shews that the circulation in the
circuit is given by

2(J£*tiwf+n{>«8 ....(3),

or, twice the product of the area of the circuit into the component
angular velocity of the fluid about the normal.

33. Any finite surface may be divided, by a double series
of lines crossing ft, into infinitely small elements. The sum
of the circulations round the boundaries of these elements, taken
all in the same sense, is equal to the circulation round the origi
nal boundary of the surface (supposed for the moment to consist
of a single closed curve). For, in the sum in question, the flow

along each side common to two elements comes in twice, once
for each element, but with opposite signs, and therefore disap-

* Maxwell, Proc. Lond. Math. Soc., t. iii., pp. 279, 280.

t See Maxwell, Electricity and Magnetism, Oxford, 1873, Art. 23.

32-33] CIRCULATION. 37

pears from the result. There remain then only the flows along
those sides which are parts of the original boundary ; whence the
truth of the above statement.

Expressing this analytically we have, by (3),

j(udx + vdy + wdz) = 2 // (1% + mrj + n£) dS (4),

or, substituting the values of f, 77, f from Art. 31,

j(udx + vdy + wdz)

fdw dv\ fdu dw\ fdv du\] ia /rx*

-J--T- )+m \j---j~ )+n(j--j- nds (5) ;

\dy dzj \dz dscj \dx dyj)

where the single-integral is taken along the bounding curve,
and the double-integral over the surface. In these formula
the quantities I, m, n are the direction-cosines of the normal
drawn always on one side of the surface, which we may term the
positive side ; the direction of integration in the second member
is then that in which a man walking on the surface, on the
positive side of it, and close to the edge, must proceed so as to
have the surface always on his left hand.

The theorem (4) or (5) may evidently be extended to a surface
whose boundary consists of two or more closed curves, provided
the integration in the first member be taken round each of
these in the proper direction, according to the rule iust given.

-//I-

Thus, if the surface-integral in (5) extend over the shaded portion
of the annexed figure, the directions in which the circulations
in the several parts of the boundary are to be taken are shewn by

* This theorem is attributed by Maxwell to Stokes, Smith's Prize Examination
Papers for 1854. The first published proof appears to have been given by Hankel,
Zur allgem. Theorie der Beweguny der Fliissigkeiten, Gottingen, 1861, p. 35. That
given above is due to Lord Kelvin, I.e. ante p. 35. See also Thomson and Tait, Natu
ral Philosophy, Art. 190 (j), and Maxwell, Electricity and Magnetism, Art. 24.

38 IRROTATIONAL MOTION. [CHAP. Ill

the arrows, the positive side of the surface being that which faces
the reader.

The value of the surface-integral taken over a closed surface is
zero.

It should be noticed that (5) is a theorem of pure mathe
matics, and is true whatever functions u, v, w may be of x, y, z,
provided only they be continuous over the surface*.

34. The rest of this chapter is devoted to a study of the
kinematical properties of irrotational motion in general, as defined
by the equations

f=o, 77=0, r=o.

The existence and properties of the velocity-potential in the
various cases that may arise will appear as consequences of this
definition.

The physical importance of the subject rests on the fact that
if the motion of any portion of a fluid mass be irrotational at any
one instant it will under certain very general conditions continue
to be irrotational. Practically, as will be seen, this has already
been established by Lagrange's theorem, proved in Art. 18, but
the importance of the matter warrants a repetition of the investi
gation, in the Eulerian notation, in the form originally given by
Lord Kelvin -f\

Consider first any terminated line AB drawn in the fluid, and
suppose every point of this line to move always with the velocity
of the fluid at that point. Let us calculate the rate at which the
flow along this line, from A to B, is increasing. If Sx, Sy, §z be
the projections on the co-ordinate axes of an element of the line,

D Diis DSx

we have Dt(u^ = ^Dt^ ^U^t '

Now DSxjDt, the rate at which 8x is increasing in consequence of
the motion of the fluid, is equal to the difference of the velocities
parallel to x at its two ends, i.e. to &u ; and the value of DujDt is
given in Art. 6. Hence, and by similar considerations, we find, if
p be a function of p only, and if the extraneous forces X, Y, Z
have a potential H,

-r\ £

y- (uSas + v8y + wBz) = — — — SH + uSu + v$v + wSw.

-L/ij p

* It is not necessary that their differential coefficients should be continuous,
t I.e. ante p. 35.

33-35] CIRCULATION. 39

Integrating along the line, from A to B, we get

~r\ rB r dr) ~~\ -B

-jr- I (udx + vdij + wdz) = — I — — O + ^q2\ (1),

UtJA L -> P _U

or, the rate at which the flow from A to B is increasing is equal
to the excess of the value which —fdp/p — £l + ^q2 has at B over
that which it has at A. This theorem comprehends the whole
of the dynamics of a perfect fluid. For instance, equations (2) of
Art. 15 may be derived from it by taking as the line AB the in
finitely short line whose projections were originally 8a, 86, Sc,
and equating separately to zero the coefficients of these in
finitesimals.

If H be single-valued, the expression within brackets on the
right-hand side of (1) is a single-valued function of a, y, z.
Hence if the integration on the left-hand be taken round a closed
curve, so that B coincides with A, we have

•gr I (udx + vdy + wdz) = 0 (2),

or, the circulation in any circuit moving with the fluid does not
alter with the time.

It follows that if the motion of any portion of a fluid mass be
initially irrotational it will always retain this property ; for other
wise the circulation in every infinitely small circuit would not
continue to be zero, as it is initially, by virtue of Art. 33 (4).

35. Considering now any region occupied by irrotationally-
moving fluid, we see from Art. 33 (4) that the circulation is zero
in every circuit which can be filled up by a continuous surface
lying wholly in the region, or which is in other words capable of
being contracted to a point without passing out of the region.
Such a circuit is said to be ' reducible.'

Again, let us consider two paths ACB, ADB, connecting two
points A, B of the region, and such that either may by con
tinuous variation be made to coincide with the other, without ever
passing out of the region. Such paths are called ' mutually
reconcileable.' Since the circuit AGED A is reducible, we have
I (ACB DA) = 0, or since I(BDA) = - 1 (ADB),

i.e. the flow is the same along any two reconcileable paths.

40 IRROTATIONAL MOTION. [CHAP. Ill

A region such that all paths joining any two points of it are
mutually reconcileable is said to be ' simply-connected.' Such a
region is that enclosed within a sphere, or that included between
two concentric spheres. In what follows, as far as Art. 46, we con
template only simply-connected regions.

36. The irrotational motion of a fluid within a simply-con
nected region is characterized by the existence of a single-valued
velocity-potential. Let us denote by — </> the flow to a variable
point P from some fixed point A} viz.

rp
</> = — I (udx 4- vdy + wdz) (1).

J A

The value of $ has been shewn to be independent of the path
along which the integration is effected, provided it lie wholly
within the region. Hence </> is a single- valued function of the
position of P ; let us suppose it expressed in terms of the co
ordinates (#, y, z) of that point. By displacing P through an
infinitely short space parallel to each of the axes of co-ordinates
in succession, we find

dcf) d(f) d<j> ,_.

i.e. (f> is a velocity-potential, according to the definition of Art. 18.

The substitution of any other point B for A, as the lower limit
in (1), simply adds an arbitrary constant to the value of <£, viz. the
flow from A to B. The original definition of <£ in Art. 18, and the
physical interpretation in Art. 1 9, alike leave the function indeter
minate to the extent of an additive constant.

As we follow the course of any line of motion the value of </>
continually decreases ; hence in a simply-connected region the
lines of motion cannot form closed curves.

37. The function <£ with which we have here to do is, together
with its first differential coefficients, by the nature of the case,
finite, continuous, and single-valued at all points of the region
considered. In the case of incompressible fluids, which we now
proceed to consider more particularly, </> must also satisfy the
equation of continuity, (5) of Art. 21, or as we shall in future
write it, for shortness,

V24> = o (i),

35-37] VELOCITY-POTENTIAL. 41

at every point of the region. Hence <£ is now subject to mathe
matical conditions identical with those satisfied by the potential of
masses attracting or repelling according to the law of the inverse
square of the distance, at all points external to such masses ; so
that many of the results proved in the theories of Attractions,
Electrostatics, Magnetism, and the Steady Flow of Heat, have also
a hydrodynamical application. We proceed to develope those
which are most important from this point of view.

In any case of motion of an incompressible fluid the surface-
integral of the normal velocity taken over any surface, open or
closed, is conveniently called the 'flux' across that surface. It is
of course equal to the volume of fluid crossing the surface per unit
time.

When the motion is irrotational, the flux is given by

d(j> 7C/
7 ao,
dn

where SS is an element of the surface, and Bn an element of the
normal to it, drawn in the proper direction. In any region ,

occupied wholly by liquid, the total flux across the boundary /'is / ^"

• J\ 3 &t /•£, £v

zero, i.e. ' ^

the element Sn of the normal being drawn always on one side (say
inwards), and the integration extending over the whole boundary.
This may be regarded as a generalized form of the equation of
continuity (1).

The lines of motion drawn through the various points of an
infinitesimal circuit define a tube, which may be called a tube of
flow. The product of the velocity (q) into the cross-section (<r, say)
is the same at all points of such a tube.

We may, if we choose, regard the whole space occupied by the
fluid as made up of tubes of flow, and suppose the size of the tubes
so adjusted that the product qa- is the same for each. The flux
across any surface is then proportional to the number of tubes
which cross it. If the surface be closed, the equation (2) ex
presses the fact that as many tubes cross the surface inwards as
outwards. Hence a line of motion cannot begin or end at a point
of the fluid.

42 11UIOTATIONAL MOTION. [CHAP. Ill

38. The function <f> cannot be a maximum or minimum at a
point in the interior of the fluid ; for, if it were, we should have
d^/dn everywhere positive, or everywhere negative, over a small
closed surface surrounding the point in question. Either of these
suppositions is inconsistent with (2).

Further, the absolute value of the velocity cannot be a maximum
at a point in the interior of the fluid. For let the axis of x be taken
parallel to the direction of the velocity at any point P. The equa
tion (1), and therefore also the equation (2), is satisfied if we write
d(f>/dx for (f>. The above argument then shews that d<p/dx cannot
be a maximum or a minimum at P. Hence there must be some
point in the immediate neighbourhood of P for which d(f>/dx has
a numerically greater value, and therefore a fortiori, for which

is numerically greater than dcfr/dx, i.e. the velocity of the fluid at
some neighbouring point is greater than at P*.

On the other hand, the velocity may be a minimum at some point of the
fluid. The simplest case is that of a zero velocity ; see, for example, the figure
of Art. 69, below.

39. Let us apply (2) to the boundary of a finite spherical
portion of the liquid. If r denote the distance of any point from
the centre of the sphere, Stzr the elementary solid angle subtended
at the centre by an element 88 of the surface, we have

dcj)/dn = — dfyjdr,
and BS = r*Svr. Omitting the factor r2, (2) becomes

!!'

.. dr
or

dr

Since l/4tir.ff$dar or l/4?rr2 .//</> dS measures the mean value of
$ over the surface of the sphere, (3) shews that this mean value is
independent of the radius. It is therefore the same for any sphere,
concentric with the former one, which can be made to coincide

* This theorem was enunciated, in another connection, by Lord Kelvin, Phil.
May., Oct. 1850; Reprint of Papers on Electrostatics, tfc., London, 1872, Art. 665.
The above demonstration is due to Kirchhoff, Vorlesnngen iiber mathematische
Pliysik, Mechanik, Leipzig, 1876, p. 186. For another proof see Art. 44 below.

38-39] FLUX ACROSS A SPHERICAL SURFACE. 43

with it by gradual variation of the radius, without ever passing
out of the region occupied by the irrotationally moving liquid.
We may therefore suppose the sphere contracted to a point, and so
obtain a simple proof of the theorem, first given by Gauss in his
memoir* on the theory of Attractions, that the mean value of (f>
over any spherical surface throughout the interior of which (1)
is satisfied, is equal to its value at the centre.

The theorem, proved in Art. 38, that (f> cannot be a maximum
or a minimum at a point in the interior of the fluid, is an obvious
consequence of the above.

The above proof appears to be due, in principle, to Frost f. Another
demonstration, somewhat different in form, has been given by Lord RayleighJ.
The equation (1), being linear, will be satisfied by the arithmetic mean of any
number of separate solutions ^, <£2, <£3,.... Let us suppose an infinite number
of systems of rectangular axes to be arranged uniformly about any point P as
origin, and let <p11 <£2, <£3,... be the velocity-potentials of motions which are
the same with respect to these systems as the original motion $ is with
respect to the system x, y, z. In this case the arithmetic mean (0, say) of the
functions 01? <£2, </>3,,.. will be a function of r, the distance from P, only.
Expressing that in the motion (if any) represented by $, the flux across any
spherical surface which can be contracted to a point, without passing out of
the region occupied by the fluid, would be zero, we have

' dr~
or 0 = const.

Again, let us suppose that the region occupied by the irrota-
tionally moving fluid is 'periphractic/§ i.e. that it is limited
internally by one or more closed surfaces, and let us apply (2) to
the space included between one (or more) of these internal
boundaries, and a spherical surface completely enclosing it and
lying wholly in the fluid. If 4>7rM denote the total flux into
this region, across the internal boundary, we find, with the
same notation as before,

jjdr^311

* " Allgemeine Lehrsiitze, u. s. w.," Eesultate aus den Beobachtungen des mag~
netischen Vereins, 1839 ; Werke, Gottingen, 1870—80, t. v., p. 199.

t Quarterly Journal of Mathematics, t. xii. (1873).

£ Messenger of Mathematics, t. vii., p. 69 (1878).

§ See Maxwell, Electricity and Magnetism, Arts. 18, 22. A region is said to be
' aperiphractic ' when every closed surface drawn in it can be contracted to a point
without passing out of the region.

44 IRROTATIONAL MOTION. [CHAP. Ill

the surface-integral extending over the sphere only. This may be
written

1 d , M

j

That is, the mean value of (/> over any spherical surface drawn
under the above-mentioned conditions is equal to M/r + C, where
r is the radius, M an absolute constant, and C a quantity which is
independent of the radius but may vary with the position of the
centre *.

If however the original region throughout which the irrotational
motion holds be unlimited externally, and if the first derivative (and
therefore all the higher derivatives) of (f> vanish at infinity, then G
is the same for all spherical surfaces enclosing the whole of the
internal boundaries. For if such a sphere be displaced parallel
to #-f, without alteration of size, the rate at which C varies in
consequence of this displacement is, by (4), equal to the mean
value of d(f)/dx over the surface. Since d$/dx vanishes at infinity,
we can by taking the sphere large enough make the latter mean
value as small as we please. Hence C is not altered by a displace
ment of the centre of the sphere parallel to x. In the same way
we see that G is not altered by a displacement parallel to y or z ;
i.e. it is absolutely constant.

If the internal boundaries of the region considered be such
that the total flux across them is zero, e.g. if they be the surfaces
of solids, or of portions of incompressible fluid whose motion is
rotational, we have M = 0, so that the mean value of <£ over any
spherical surface enclosing them all is the same.

40. (a) If </> be constant over the boundary of any simply-
connected region occupied by liquid moving irrotationally, it has
the same constant value throughout the interior of that region.
For if not constant it would necessarily have a maximum or a
minimum value at some point of the region.

* It is understood, of course, that the spherical surfaces to which this statement
applies are reconcileable with one another, in a sense analogous to that of Art. 35.
f Kirchhoff, Mechanik, p. 191.

39-40] CONDITIONS OF DETERMINATENESS. 45

Otherwise : we have seen in Arts. 36, 37 that the lines of
motion cannot begin or end at any point of the region, and that they
cannot form closed curves lying wholly within it. They must
therefore traverse the region, beginning and ending on its bound
ary. In our case however this is impossible, for such a line always
proceeds from places where <£ is greater to places where it is less.
Hence there can be no motion, i.e.

_ - -
^~' 3y ' Tz~

and therefore <f> is constant and equal to its value at the boundary.

(ft) Again, if d(f)/dn be zero at every point of the boundary of
such a region as is above described, $ will be constant throughout
the interior. For the condition d(f>/dn = 0 expresses that no lines
of motion enter or leave the region, but that they are all contained
within it. This is however, as we have seen, inconsistent with
the other conditions which the lines must conform to. Hence, as
before, there can be no motion, and </> is constant.

This theorem may be otherwise stated as follows : no con- j
tinuous irrotational motion of a liquid can take place in a 1
simply-connected region bounded entirely by fixed rigid walls.

(7) Again, let the boundary of the region considered consist
partly of surfaces S over which </> has a given constant value, and
partly of other surfaces 5) over which d<p/dn = 0. By the previous
argument, no lines of motion can pass from one point to another
of 8, and none can cross 2. Hence no such lines exist; <£ is
therefore constant as before, and equal to its value at 8.

It follows from these theorems that the irrotational motion of a
liquid in a simply-connected region is determinate when either the
value of </>, or the value of the inward normal velocity — d<j>/dn, is
prescribed at all points of the boundary, or (again) when the value
of <j) is given over part of the boundary, and the value of — d<f>/dn
over the remainder. For if fa, <£.2 be the velocity-potentials of
two motions each of which satisfies the prescribed boundary-
conditions, in any one of these cases, the function <f>1 — <f>2
satisfies the condition (a) or (ft) or (7) of the present Article,
and must therefore be constant throughout the region.

46 IRROTATIONAL MOTION. [CHAP. Ill

41. A class of cases of great importance, but not strictly in
cluded in the scope of the foregoing theorems, occurs when the
region occupied by the irrotationally moving liquid extends to
infinity, but is bounded internally by one or more closed surfaces.
We assume, for the present, that this region is simply-connected,
and that <£ is therefore single- valued.

If $ be constant over the internal boundary of the region, and
tend everywhere to the same constant value at an infinite distance
from the internal boundary, it is constant throughout the region.
For otherwise <£ would be a maximum or a minimum at some
point.

We infer, exactly as in Art. 40, that if <£ be given arbitrarily
over the internal boundary, and have a given constant value at
infinity, its value is everywhere determinate.

Of more importance in our present subject is the theorem
that, if the normal velocity be zero at every point of the internal
boundary, and if the fluid be at rest at infinity, then <f> is every
where constant. We cannot however infer this at once from the
proof of the corresponding theorem in Art. 40. It is true that we
may suppose the region limited externally by an infinitely large
surface at every point of which d<f>/dn is infinitely small ; but it is
conceivable that the integral ffd^/dn . dS, taken over a portion of
this surface, might still be finite, in which case the investigation
referred to would fail. We proceed therefore as follows.

Since the velocity tends to the limit zero at an infinite
distance from the internal boundary ($, say), it must be possible
to draw a closed surface S, completely enclosing 8, beyond which
the velocity is everywhere less than a certain value e, which
value may, by making 2 large enough, be made as small as we
please. Now in any direction from 8 let us take a point P at such
a distance beyond 2 that the solid angle which S subtends at it is
infinitely small ; and with P as centre let us describe two spheres,
one just excluding, the other just including 8. We shall prove
that the mean value of <£ over each of these spheres is, within
an infinitely small amount, the same. For if Q, Q' be points of
these spheres on a common radius PQQ', then if Q, Q' fall within
S the corresponding values of <f> may differ by a finite amount ;
but since the portion of either spherical surface which falls within
2 is an infinitely small fraction of the whole, no finite difference

41-42] REGION EXTENDING TO INFINITY. 47

in the mean values can arise from this cause. On the other hand,
when Q, Q' fall without 2, the corresponding values of <£ cannot
differ by so much as e . QQ', for e is by definition a superior limit
to the rate of variation of 0. Hence, the mean values of (/> over
the two spherical surfaces must differ by less than e . QQ'. Since
QQ' is finite, whilst e may by taking 2 large enough be made as
small as we please, the difference of the mean values may, by
taking P sufficiently distant, be made infinitely small.

Now we have seen in Art. 39, that the mean value of <f> over
the inner sphere is equal to its value at P, and that the mean
value over the outer sphere is (since M = 0) equal to a constant
quantity C. Hence, ultimately, the value of cf> at infinity tends
everywhere to the constant value C.

The same result holds even if the normal velocity be not
zero over the internal boundary; for in the theorem of Art. 39
M is divided by r, which is in our case infinite.

It follows that if dfyfdn = 0 at all points of the internal
boundary, and if the fluid be at rest at infinity, it must be every
where at rest. For no lines of motion can begin or end on the
internal boundary. Hence such lines, if they existed, must come
from an infinite distance, traverse the region occupied by the
fluid, and pass off again to infinity ; i.e. they must form infinitely
long courses between places where (/> has, within an infinitely
small amount, the same value C, which is impossible.

The theorem that, if the fluid be at rest at infinity, the motion
is determinate when the value of —d(f>/dn is given over the in
ternal boundary, follows by the same argument as in Art. 40.

Greens Theorem.

42. In treatises on Electrostatics, &c., many important pro
perties of the potential are usually proved by means of a certain
theorem due to Green. Of these the most important from our
present point of view have already been given; but as the
theorem in question leads, amongst other things, to a useful
expression for the kinetic energy in any case of irrotational
motion, some account of it will properly find a place here.

Let U, V, W be any three functions which are finite, con
tinuous, and single- valued at all points of a connected region

48 IRROTATIONAL MOTION. [CHAP. Ill

completely bounded by one or more closed surfaces 8', let SS be
an element of any one of these surfaces, and I, m, n the direction-
cosines of the normals to it drawn inwards. We shall prove in the
first place that

where the triple-integral is taken throughout the region, and the
double-integral over its boundary.

If we conceive a series of surfaces drawn so as to divide the
region into any number of separate parts, the integral

Jf(lU+mV+nW)dS .................. (2),

taken over the original boundary, is equal to the sum of the
similar integrals each taken over the whole boundary of one of
these parts. For, for every element So- of a dividing surface,
we have, in the integrals corresponding to the parts lying on
the two sides of this surface, elements (IU + mV+nW)S<r,
and (T U + m' V + n W) Scr, respectively. But the normals to
which Z, m, n, and /', m', n' refer being drawn inwards in each
case, we have I' = — I, m' = — m, n' = — n ; so that, in forming the
sum of the integrals spoken of, the elements due to the dividing
surfaces disappear, and we have left only those due to the original
boundary of the region.

Now let us suppose the dividing surfaces to consist of three
systems of planes, drawn at infinitesimal intervals, parallel to
yz, zx, xy, respectively. If #, y, z be the co-ordinates of the
centre of one of the rectangular spaces thus formed, and
&B, %, $z the lengths of its edges, the part of the integral
(2) due to the 7/^-face nearest the origin is

and that due to the opposite face is

The sum of these is —dUjdx. §x§y§z. Calculating in the same
way the parts of the integral due to the remaining pairs of faces,
we get for the final result

_/dU dV fr
\ dx dy dz

42-43] GREEN'S THEOREM. 49

Hence (1) simply expresses the fact that the surface-integral (2),
taken over the boundary of the region, is equal to the sum of the
similar integrals taken over the boundaries of the elementary
spaces of which we have supposed it built up.

The interpretation of this result when U, F, W denote the
component velocities of a continuous substance is obvious. In the
particular case of irrotational motion we obtain

(3),

where $n denotes an element of the inwardly-directed normal to
the surface S.

Again, if we put U, V, W = pu, pv, pw, respectively, we
reproduce in substance the investigation of Art. 8.

Another useful result is obtained by putting U, V, W
= u(f>, V(f), wtf), respectively, where u, v, w satisfy the relation

du dv dw _ _

dx dy dz ~
throughout the region, and make

lu + mv + nw = 0
over the boundary. We find

The function $ is here merely restricted to be finite, single- valued,
and continuous, and to have its first differential coefficients finite,
throughout the region.

43. Now let $, </>' be any two functions which, together with
their first derivatives, are finite, continuous, and single-valued
throughout the region considered ; and let us put

ir.V.F.*tf,::i¥..#*.

r dx ^ dy r dz
respectively, so that

Substituting in (1) we find

+|f+ff>**

(5)

50 IRROTATIONAL MOTION. [CHAP. Ill

By interchanging <j) and <£' we obtain

f f_i> <ty j a f f /W d$ d<t> d<t>' d</> d<t>'\ 777
\\<i> -j dS=- I I j2- j + -J^-T^+T^ - J- \dxdydz
JJ^ dn JJJ\dx dx dy dy dz dz )

-Jf!<t>'V*<f>dxdydz ..................... (6).

Equations (5) and (6) together constitute Green's theorem*.

44. If $, <p' be the velocity-potentials of two distinct modes
of irrotational motion of a liquid, so that

v^ = o, vy = o ........................ (i),

we obtain tf "*' ^ .................. (2)'

If we recall the physical interpretation of the velocity-potential,
given in Art. 19, then, regarding the motion as generated in each
case impulsively from rest, we recognize this equation as a
particular case of the dynamical theorem that

where pr, qr and pr't qr' are generalized components of impulse and
velocity, in any two possible motions of a systemf.

Again, in Art. 43 (6) let </>' = <£, and let <f> be the velocity-
potential of a liquid. We obtain

[[[ \ /cty\a (d<l>\* , fd<l>\*\j , , ff .<tyjo

III M j I +? + j I \dxdydz = — 6 ~d8 ...... (3).

JJJ \\dxj \dy) \dz>} J ] ^ dn

To interpret this we multiply both sides by ^ p. Then
on the right-hand side — d^>/dn denotes the normal velocity of
the fluid inwards, whilst p<f> is, by Art. 19, the impulsive pres
sure necessary to generate the motion. It is a proposition in
Dynamics J that the work done by an impulse is measured by the
product of the impulse into half the sum of the initial and final
velocities, resolved in the direction of the impulse, of the point to
which it is applied. Hence the right-hand side of (3), when
modified as described, expresses the work done by the system of
impulsive pressures which, applied to the surface S, would
generate the actual motion; whilst the left-hand side gives
the kinetic energy of this motion. The formula asserts that

* G. Green, Essay on Electricity and Magnetism, Nottingham, 1828, Art. 3.
Mathematical Papers (ed. Ferrers), Cambridge, 1871, p. 23.

t Thomson and Tait, Natural Philosophy, Art. 313, equation (11).
I Thomson and Tait, Natural Philosophy, Art. 308.

43-45] KINETIC ENERGY. 51

these two quantities are equal. Hence if T denote the total
kinetic energy of the liquid, we have the very important result

-j^dS (4).

If in (3), in place of </>, we write d<f)/dx, which will of course satisfy
= Q, and apply the resulting theorem to the region included within a
spherical surface of radius r having any point (#, y, z) as centre, then with the
same notation as in Art. 39, we have

d (<kt>\ 7
~ -T- (-=-*- dS
dx dn dx

1 <id [f *j [fdujo f

2 r T I I u d™ = I / u -j- dS = - I I

dr J J J J dr J J

M

Hence, writing q2 =

Since this latter expression is essentially positive, the mean value of q2, taken
over a sphere having any given point as centre, increases with the radius of
the sphere. Hence q cannot be a maximum at any point of the fluid, as was
proved otherwise in Art. 38.

Moreover, recalling the formula for the pressure in any case of irrotational
motion of a liquid, viz.

we infer that, provided the potential Q. of the external forces satisfy the
condition

V2Q = 0 ....................................... (iii),

the mean value of p over a sphere described with any point in the interior of
the fluid as centre will diminish as the radius increases. The place of least
pressure will therefore be somewhere on the boundary of the fluid. This has
a bearing on the point discussed in Art. 24.

45. In this connection we may note a remarkable theorem
discovered by Lord Kelvin*, and afterwards generalized by him
into an universal property of dynamical systems started impulsively
from rest under prescribed velocity-conditions -f-.

The irrotational motion of a liquid occupying a simply-con
nected region has less kinetic energy than any other motion
consistent with the same normal motion of the boundary.

* (W. Thomson) "On the Vis- Viva of a Liquid in Motion," Carrib. and Dub.
Math. Journ., 1849; Mathematical and Physical Papers, t. i., p. 10.7.
t Thomson and Tait, Natural Philosophy, Art. 312.

4—2

52 IRROTATIONAL MOTION. [CHAP. Ill

Let T be the kinetic energy of the irrotational motion to which
the velocity-potential <£ refers, and T^ that of another motion
given by

where, in virtue of the equation of continuity, and the prescribed
boundary-condition, we must have

duo dv0 dw0 _
dx dy dz

throughout the region, and

luQ + mv<) + nw0 = 0
over the boundary. Further let us write

T^kpfffW + vf + wfidxdydz ............ (6).

We find

Since the last integral vanishes, by Art. 42 (4), we have

T, = T+T0 .......................... (7),

which proves the theorem.

46. We shall require to know, hereafter, the form assumed by
the expression (4) for the kinetic energy when the fluid extends
to infinity and is at rest there, being limited internally by one or
more closed surfaces S. Let us suppose a large closed surface S
described so as to enclose the whole of S. The energy of the fluid
included between 8 and 2 is

where the integration in the first term extends over S, that in the
second over S. Since we have by the equation of continuity

(8) may be written

-0)^ ......... (9),

where C may be any constant, but is here supposed to be the
constant value to which (> was shewn in Art. 39 to tend at an

45-47] CYCLIC REGIONS. 53

infinite distance from 8. Now the whole region occupied by the
fluid may be supposed made up of tubes of flow, each of which
must pass either from one point of the internal boundary to
another, or from that boundary to infinity. Hence the value of
the integral

JJ dn '

taken over any surface, open or closed, finite or infinite, drawn
within the region, must be finite. Hence ultimately, when 2) is
taken infinitely large and infinitely distant all round from 8, the
second term of (9) vanishes, and we have

j)/77 I / / JL S~1\ M^P 7 r/ /-I f\\

21 = — O 11(9 — 0) -r^ttO (lv),

JJ an

where the integration extends over the internal boundary only.
If the total flux across the internal boundary be zero, we have

Jids-o,

.. dn
so that (10) becomes

simply.

On Multiply-connected Regions.

47. Before discussing the properties of irrotational motion in
multiply-connected regions we must examine more in detail the
nature and classification of such regions. In the following synopsis
of this branch of the geometry of position we recapitulate for the
sake of completeness one or two definitions already given.

We consider any connected region of space, enclosed by bound
aries. A region is 'connected' when it is possible to pass from
any one point of it to any other by an infinity of paths, each of
which lies wholly in the region.

Any two such paths, or any two circuits, which can by continu
ous variation be made to coincide without ever passing out of the
region, are said to be ' mutually reconcileable.' Any circuit which
can be contracted to a point without passing out of the region is
said to be ' reducible.' Two reconcileable paths, combined, form a
reducible circuit. If two paths or two circuits be reconcileable, it

54 IRROTATIONAL MOTION. [CHAP. Ill

must be possible to connect them by a continuous surface, which
lies wholly within the region, and of which they form the complete
boundary ; and conversely.

It is further convenient to distinguish between 'simple' and
'multiple' irreducible circuits. A 'multiple' circuit is one which
can by continuous variation be made to appear, in whole or in
part, as the repetition of another circuit a certain number of times.
A 'simple' circuit is one with which this is not possible.

A 'barrier/ or 'diaphragm,' is a surface drawn across the
region, and limited by the line or lines in which it meets the
boundary. Hence a barrier is necessarily a connected surface, and
cannot consist of two or more detached portions.

A 'simply-connected' region is one such that all paths joining
any two points of it are reconcileable, or such that all circuits
drawn within it are reducible.

A 'doubly-connected' region is one such that two irreconcileable
paths, and no more, can be drawn between any two points A, B of
it; viz. any other path joining AB is reconcileable with one of
these, or with a combination of the two taken each a certain
number of times. In other words, the region is such that one
(simple) irreducible circuit can be drawn in it, whilst all other
circuits are either reconcileable with this (repeated, if necessary),
or are reducible. As an example of a doubly-connected region we
may take that enclosed by the surface of an anchor-ring, or that
external to such a ring and extending to infinity.

Generally, a region such that n irreconcileable paths, and no
more, can be drawn between any two points of it, or such that n — 1
(simple) irreducible and irreconcileable circuits, and no more, can
be drawn in it, is said to be ' n-ply-connected.'

The shaded portion of the figure on p. 37 is a triply-con
nected space of two dimensions.

It may be shewn that the above definition of an /^-ply-connected
space is self-consistent. In such simple cases as n — 2, n = 3, this
is sufficiently evident without demonstration.

48. Let us suppose, now, that we have an ?i-ply-connected
region, with n — 1 simple independent irreducible circuits drawn
in it. It is possible to draw a barrier meeting any one of these

47-49] CYCLIC REGIONS. 55

circuits in one point only, and not meeting any of the n — 2
remaining circuits. A barrier drawn in this manner does not
destroy the continuity of the region, for the interrupted circuit
remains as a path leading round from one side to the other. The
order of connection of the region is however diminished by unity ;
for every circuit drawn in the modified region must be reconcileable
with one or more of the n — 2 circuits riot met by the barrier.

A second barrier, drawn in the same manner, will reduce the
order of connection again by one, and so on ; so that by drawing
n — I barriers we can reduce the region to a simply-connected one.

A simply-connected region is divided by a barrier into two
separate parts ; for otherwise it would be possible to pass from a
point on one side the barrier to an adjacent point on the other side
by a path lying wholly within the region, which path would in the
original region form an irreducible circuit.

Hence in an n-ply- connected region it is possible to draw u — 1
barriers, and no more, without destroying the continuity of the
region. This property is sometimes adopted as the definition of
an n-ply-connected space.

Irrotational Motion in Multiply-connected Spaces.

49. The circulation is the same in any two reconcileable
circuits AEG A, A'B'C'A' drawn in a region occupied by fluid
moving irrotationally. For the two circuits may be connected by
a continuous surface lying wholly within the region ; and if we
apply the theorem of Art. 33 to this surface, we have, remembering
the rule as to the direction of integration round the boundary,

/ (ABC A) + 1 (A'C'B'A') = 0,
or / (ABCA) = I (A'B'C'A'}.

If a circuit ABC A be reconcileable with two or more circuits
A'B'C'A', A"B"C"A", &c., combined, we can connect all these
circuits by a continuous surface which lies wholly within the
region, and of which they form the complete boundary. Hence

/ (ABCA) + 1 (A'C'B'A') + 1 (A"C"B"A") + &c. = 0,
or / (ABCA) = I (A'B'C'A') + 1 (A"B"C"A") + &c. ;
i.e. the circulation in any circuit is equal to the sum of the

56 . IRROTATIONAL MOTION. [CHAP. Ill

circulations in the several members of any set of circuits with
which it is reconcileable.

Let the order of connection of the region be ^ + 1, so that
n independent simple irreducible circuits a1} a^,...an can be
drawn in it; and let the circulations in these be tcl} K^,...Kn,
respectively. The sign of any K will of course depend on the
direction of integration round the corresponding circuit ; let the
direction in which K is estimated be called the positive direction
in the circuit. The value of the circulation in any other circuit
can now be found at once. For the given circuit is necessarily
reconcileable with some combination of the circuits alt a2,...an;
say with «j taken pl times, a2 taken p.2 times and so on, where of
course any p is negative when the corresponding circuit is taken
in the negative direction. The required circulation then is

p1/c1+pzK,i+...+pnien (1).

Since any two paths joining two points A, B of the region
together form a circuit, it follows that the values of the flow in
the two paths differ by a quantity of the form (1), where, of
course, in particular cases some or all of the p's may be zero.

50. Let us denote by — <£ the flow to a variable point P from a

fixed point A, viz.

rp
$ = — 1 (udx + vdy + wdz) (2).

J A.

So long as the path of integration from A to P is not specified,
</> is indeterminate to the extent of a quantity of the form (1).

If however n barriers be drawn in the manner explained in
Art. 48, so as to reduce the region to a simply-connected one,
and if the path of integration in (2) be restricted to lie within
the region as thus modified (i.e. it is not to cross any of the
barriers), then </> becomes a single-valued function, as in Art. 36.
It is continuous throughout the modified region, but its values
at two adjacent points on opposite sides of a barrier differ by
+ K. To derive the value of (f> when the integration is taken along
any path in the unmodified region we must subtract the quantity
(1), where any p denotes the number of times this path crosses
the corresponding barrier. A crossing in the positive direction of
the circuits interrupted by the barrier is here counted as positive,
a crossing in the opposite direction as negative.

49-51] VELOCITY-POTENTIALS IN CYCLIC REGIONS. 5*7

By displacing P through an infinitely short space parallel to
each of the co-ordinate axes in succession, we find

d6 dd> d(j)

u = — ^ir) 0 = — y- , w = --f- ;
dx dy dz

so that <j> satisfies the definition of a velocity-potential (Art. 18).
It is now however a many- valued or cyclic function ; i. e. it is not
possible to assign to every point of the original region a unique
and definite value of $, such values forming a continuous system.
On the contrary, whenever P describes an irreducible circuit, <j>
will not, in general, return to its original value, but will differ from
it by a quantity of the form (1). The quantities Klt /c2,...fcn, which
specify the amounts by which <£ decreases as P describes the several
independent circuits of the region, may be called the ' cyclic con
stants ' of </>.

It is an immediate consequence of the ' circulation-theorem' of
Art. 34 that under the conditions there presupposed the cyclic
constants do not alter with the time. The necessity for these
conditions is exemplified in the problem of Art. 30, where the
potential of the extraneous forces is itself a cyclic function.

The foregoing theory may be illustrated by the case of Art. 28 (2), where
the region (as limited by the exclusion of the origin, where the formula
would give an infinite velocity) is doubly-connected ; since we can connect
any two points A, B of it by two irre-
concileable paths passing on opposite
sides of the axis of z, e.g. ACB, ADB in
the figure. The portion of the plane zx
for which x is positive may be taken as
a barrier, and the region is thus made
simply-connected. The circulation in
any circuit meeting this barrier once

only, e.g. in ACBDA, is j^ p/r. rd0, or 2?r/i. That in any circuit not meeting
the barrier is zero. In the modified region 0 may be put equal to a single-
valued function, viz. —pdt but its value on the positive side of the barrier is
zero, that at an adjacent point on the negative side is -2w/i.

More complex illustrations of irrotational motion in multiply-connected
spaces will present themselves in the next chapter.

51. Before proceeding further we may briefly indicate a some
what different method of presenting the above theory.

Starting from the existence of a velocity-potential as the characteristic of
the class of motions which we propose to study, and adopting the second

58 IRROTATIONAL MOTION. [CHAP. Ill

definition of an n + I -ply-connected region, indicated in Art. 48, we remark
that in a simply-connected region every equipotential surface must either be a
closed surface, or else form a barrier dividing the region into two separate
parts. Hence, supposing the whole system of such surfaces drawn, we see
that if a closed curve cross any given equipotential surface once it must cross
it again, and in the opposite direction. Hence, corresponding to any element
of the curve, included between two consecutive equipotential surfaces, we have
a second element such that the flow along it, being equal to the difference
between the corresponding values of $, is equal and opposite to that along
the former ; so that the circulation in the whole circuit is zero.

If however the region be multiply-connected, an equipotential surface may
form a barrier without dividing it into two separate parts. Let as many
such surfaces be drawn as it is possible to draw without destroying the
continuity of the region. The number of these cannot, by definition, be
greater than n. Every other equipotential surface which is not closed will
be reconcileable (in an obvious sense) with one or more of these barriers. A
curve drawn from one side of a barrier round to the other, without meeting any
of the remaining barriers, will cross every equipotential surface reconcileable
with the first barrier an odd number of times, and every other equipotential
surface an even number of times. Hence the circulation in the circuit thus
formed will not vanish, and <£ will be a cyclic function.

In the method adopted above we have based the whole theory on the
equations

dw dv _ du dw _ dv du

dy dz ' dz dx ' dx dy "

and have deduced the existence and properties of the velocity -potential in the
various cases as necessary consequences of these. In fact, Arts. 35, 36, and
49, 50 may be regarded as a treatise on the integration of this system of
differential equations.

The integration of (i), when we have, on the right-hand side, instead of
zero, known functions of ^, y, 2, will be treated in Chapter vn.

52. Proceeding now, as in Art. 37, to the particular case of
an incompressible fluid, we remark that whether </> be cyclic or not,
its first derivatives d<f>jdx, dfyjdy, d^/dz, and therefore all the
higher derivatives, are essentially single-valued functions, so that
c/> will still satisfy the equation of continuity

v^ = o (i),

or the equivalent form

where the surface-integration extends over the whole boundary of
any portion of the fluid.

51-52] MULTIPLE CONNECTIVITY. 59

The theorem (a) of Art. 40, viz. that $ must be constant
throughout the interior of any region at every point of which (1)
is satisfied, if it be constant over the boundary, still holds when
the region is multiply-connected. For c/>, being constant over the
boundary, is necessarily single-valued.

The remaining theorems of Art. 40, being based on the assump
tion that the stream-lines cannot form closed curves, will require
modification. We must introduce the additional condition that
the circulation is to be zero in each circuit of the region.

Removing this restriction, we have the theorem that the
irrotational motion of a liquid occupying an n-ply-connected region
is determinate when the normal velocity at every point of the
boundary is prescribed, as well as the values of the circulations in
each of the n independent and irreducible circuits which can be
drawn in the region. For if $lt <p.2 be the (cyclic) velocity-poten
tials of two motions satisfying the above conditions, then <f> = fa — </>2
is a single-valued function which satisfies (1) at every point of
the region, and makes d(f>/d)i = 0 at every point of the boundary.
Hence by Art. 40, (f> is constant, and the motions determined by
</>! and <p.2 are identical.

The theory of multiple connectivity seems to have been first developed by
Riemarm* for spaces of two dimensions, a propos of his researches on the
theory of functions of a complex variable, in which connection also cyclic
functions, satisfying the equation

through multiply-connected regions, present themselves.

The bearing of the theory on Hydrodynamics, and the existence in certain
cases of many-valued velocity-potentials were first pointed out by von Helm-
holtzf. The subject of cyclic irrotational motion in multiply-connected regions
was afterwards taken up and fully investigated by Lord Kelvin in the paper
on vortex- motion already referred to J.

* Grundlagen flir eine allgemeine Theorie der Functional einer veranderlichen
complexen Grosse, Gottingen, 1851; Mathematische WerJce, Leipzig, 1876, p. 3;
"Lehrsatze aus der Analysis Situs," Grelle, t. liv. (1857) ; Werke, p. 84.

t Crelle, t. lv., 1858.

J See also Kirchhoff, "Ueber die Krafte welche zwei unendlich diinne starre
Einge in einer Fliissigkeit scheinbar auf einander ausiiben konnen," Crelle, t. Ixxi.
(1869); Ges.AWi.,^. 404.

60 IRROTATIONAL MOTION. [CHAP. Ill

Lord Kelvins Extension of Greens Theorem.

53. It was assumed in the proof of Green's Theorem that $
and <£' were both single-valued functions. If either be a cyclic
function, as may be the case when the region to which the inte
grations in Art. 43 refer is multiply-connected, the statement of the
theorem must be modified. Let us suppose, for instance, that <£
is cyclic ; the surface-integral on the left-hand side of Art. 43 (5),
and the second volume-integral on the right-hand side, are then
indeterminate, on account of the indeterminateness in the value of
(f> itself. To remove this indeterminateness, let the barriers neces
sary to reduce the region to a simply-connected one be drawn, as
explained in Art. 48. We may now suppose (f> to be continuous
and single-valued throughout the region thus modified; and the
equation referred to will then hold, provided the two sides of each
barrier be reckoned as part of the boundary of the region, and
therefore included in the surface-integral on the left-hand side.
Let So-j be an element of one of the barriers, /cx the cyclic constant
corresponding to that barrier, dfy'jdn the rate of variation of <£' in
the positive direction of the normal to So^. Since, in the parts
of the surface-integral due to the two sides of 8<rlt d<p'/dn is to be
taken with opposite signs, whilst the value of <j> on the positive
side exceeds that on the negative side by KI} we get finally for the
element of the integral due to 8al) the value ^d^/dn.B^.
Hence Art. 43 (5) becomes, in the altered circumstances,

- l d^ + Ki AT, + &c.

dn JJ dn

+ d$ d<l>'\ dx

dx dx dy dy dz dz )

. ............ (1);

where the surface-integrations indicated on the left-hand side
extend, the first over the original boundary of the region only,
and the rest over the several barriers. The coefficient of any K is
evidently minus the total flux across the corresponding barrier,
in a motion of which $' is the velocity-potential. The values of </>
in the first and last terms of the equation are to be assigned in
the manner indicated in Art. 50.

53-54] EXTENSION OF GREEN'S THEOREM. 61

If (/>' also be a cyclic function, having the cyclic constants
AC/, #3', &c., then Art. 43 (6) becomes in the same way

++

dx dy dy dz dz

(2).

Eqiiations (1) and (2) together constitute Lord Kelvin's extension
of Green's theorem.

54. Tf 0, $' are both velocity-potentials of a liquid, we have

V*(t> = 0, V2<£'=:0 ..................... (3),

and therefore

To obtain a physical interpretation of this theorem it is
necessary to explain in the first place a method, imagined by Lord
Kelvin, of generating any given cyclic irrotational motion of a liquid
in a multiply-connected space.

Let us suppose the fluid to be enclosed in a perfectly smooth
and flexible membrane occupying the position of the boundary.
Further, let n barriers be drawn, as in Art. 48, so as to convert the
region into a simply-connected one, and let their places be occupied
by similar membranes, infinitely thin, and destitute of inertia. The
fluid being initially at rest, let each element of the first-mentioned
membrane be suddenly moved inwards with the given (positive or
negative) normal velocity — d<f>/dn, whilst uniform impulsive pres
sures Kip, K2p,...Knp are simultaneously applied to the negative
sides of the respective barrier-membranes. The motion generated
will be characterized by the following properties. It will be
irrotational, being generated from rest; the normal velocity at
every point of the original boundary will have the prescribed
value ; the values of the impulsive pressure at two adjacent points
on opposite sides of a membrane will differ by the corresponding

62 IRROTATIONAL MOTION. [CHAP. Ill

value of Kp, and the values of the velocity-potential will therefore
differ by the corresponding value of tc ; finally, the motion on one
side of a barrier will be continuous with that on the other. To
prove the last statement we remark, first, that the velocities
normal to the barrier at two adjacent points on opposite sides of it
are the same, being each equal to the normal velocity of the
adjacent portion of the membrane. Again, if P, Q be two consecu
tive points on a barrier, and if the corresponding values of </> be on
the positive side <pp, <f)Q, and on the negative side </>'p, <j>qt we have

and therefore <f>Q — <f>p = </>'Q — </>',,,

». *., if PQ = &, d<j)/ds = dp/ds.

Hence the tangential velocities at two adjacent points on
opposite sides of the barrier also agree. If then we suppose the
barrier-membranes to be liquefied immediately after the impulse,
we obtain the irrotational motion in question.

The physical interpretation of (4), when multiplied by — p,
now follows as in Art. 44. The values of p/c are additional com
ponents of momentum, and those of — ffd(j)/dn. dcr, the fluxes
through the various apertures of the region, are the corresponding
generalized velocities.

55. If in (2) we put <f>' — <f>, and suppose <£ to be the velocity-
potential of an incompressible fluid, we find

The last member of this formula has a simple interpretation in terms
of the artificial method of generating cyclic irrotational motion just
explained. The first term has already been recognized as equal
to twice the work done by the impulsive pressure p(f) applied to
every part of the original boundary of the fluid. Again, p^ is the
impulsive pressure applied, in the positive direction, to the in
finitely thin massless membrane by which the place of the first
barrier was supposed to be occupied ; so that the expression

dA.

-^ rfo-,

-*//

54-56] ENERGY IN CYCLIC REGIONS. 63

denotes the work done by the impulsive forces applied to that
membrane ; and so on. Hence (5) expresses the fact that the
energy of the motion is equal to the work done by the whole
system of impulsive forces by which we may suppose it generated.

In applying (5) to the case where the fluid extends to
infinity and is at rest there, we may replace the first term of
the third member by

-C)dS ..................... (6),

where the integration extends over the internal boundary only.
The proof is the same as in Art. 46. When the total flux across
this boundary is zero, this reduces to

f(^d<t>

"'jJfaK

The minimum theorem of Lord Kelvin, given in Art. 45, may
now be extended as follows :

The irrotational motion of a liquid in a multiply-connected
region has less kinetic energy than any other motion consistent
with the same normal motion of the boundary and the same value
of the total flux through each of the several independent channels
of the region.

The proof is left to the reader.

Sources and Sinks.

56. The analogy with the theories of Electrostatics, the
Steady Flow of Heat, &c., may be carried further by means of the
conception of sources and sinks.

A c simple source ' is a point from which fluid is imagined to
flow out uniformly in all directions. If the total flux outwards
across a small closed surface surrounding the point be 4nrin*, then
m is called the ' strength ' of the source. A negative source is
called a ' sink.' The continued existence of a source or a sink
would postulate of course a continual creation or annihilation of
fluid at the point in question.

* The factor 4?r is introduced to keep up the analogy referred to.

64 IRROTATIONAL MOTION. [CHAP. Ill

The velocity-potential at any point P, due to a simple source,
in a liquid at rest at infinity, is

$ = m/r (1),

where r denotes the distance of P from the source. For this gives
a radial flow from the point, and if 8S, = r^vr, be an element of a
spherical surface having its centre at the surface, we have

-^- dS = 47nft,
dr

a constant, so that the equation of continuity is satisfied, and the
flux outwards has the value appropriate to the strength of the
source.

A combination of two equal and opposite sources + m', at a
distance 8s apart, where, in the limit, 8s is taken to be infinitely
small, and mf infinitely great, but so that the product m'8s is finite
and equal to p (say), is called a ' double source ' of strength /A, and
the line 8s, considered as drawn in the direction from — m' to + m',
is called its axis.

To find the velocity-potential at any point (x, y, z) due to a
double source ft situate at (#', y', z'\ and having its axis in the
direction (I, m, n), we remark that, / being any continuous function,
f(af + 18s, y' + m8s, z' + n8s) -f(x, y', z')

ultimately. Hence, putting /(a/, y', z) = m'/r, where
r = {(x - xj + (y- yj + (z - /)'}*,

ni /, d d d\ I

we find <t> = ^l- + m- + n^- ............... (2),

d d d\l

— + m-r +n-j- - ............... (3),

dx dy dz) r

where, in the latter form, S- denotes the angle which the line r,
considered as drawn from (x, y', z) to (x, y, z}, makes with the
axis (I, m, n).

We' might proceed, in a similar manner (see Art. 83), to build
up sources of higher degrees of complexity, but the above is
sufficient for our immediate purpose.

56-57] SOURCES AND SINKS. 65

Finally, we may imagine simple or double sources, instead of
existing at isolated points, to be distributed continuously over
lines, surfaces, or volumes.

57. We can now prove that any continuous acyclic irro-
tational motion of a liquid mass may be regarded as due
to a certain distribution of simple and double sources over
the boundary.

This depends on the theorem, proved in Art. 44, that if <£>, <£' be
any two functions which satisfy V2c/> = 0, V-(/>' = 0, and are finite,
continuous, and single- valued throughout any region, then

where the integration extends over the whole boundary. In the
present application, we take c/> to be the velocity-potential of the
motion in question, and put </>' = 1/r, the reciprocal of the distance
of any point of the fluid from a fixed point P.

We will first suppose that P is in the space occupied by the
fluid. Since <j>' then becomes infinite at P, it is necessary to ex
clude this point from the region to which the formula (5) applies ;
this may be done by describing a small spherical surface about P
as centre. If we now suppose 82 to refer to this surface, and 8$
to the original boundary, the formula gives

//*£©«+//**©*

At the surface 2 we have d/dn (l/r) = — 1/r2; hence if we put
82 = r^dix, and finally make r = 0, the first integral on the left-
hand becomes = — 4rir<l>P, where </>p denotes the value of <£ at P,
whilst the first integral on the right vanishes. Hence

l [/•,£(!)« ......... (7).

?r J j r dn \r J

This gives the value of <£ at any point P of the fluid in terms
of the values of <£ and dfyjdn at the boundary. Comparing with
the formula? (1) and (2) we see that the first term is the velocity-

L.

66 IRROTATIONAL MOTION. [CHAP. Ill

potential due to a surface distribution of simple sources, with a
density — 1/4-Tr . d(f>/dn per unit area, whilst the second term is the
velocity-potential of a distribution of double sources, with axes
normal to the surface, the density being 1/4-Tr . <£.

When the fluid extends to infinity and is at rest there, the
surface-integrals in (7) may, on a certain understanding, be taken
to refer to the internal boundary alone. To see this, we may take
as external boundary an infinite sphere having the point P as
centre. The corresponding part of the first integral in (7)
vanishes, whilst that of the second is equal to G, the constant
value to which, as we have seen in Art. 41, (f> tends at infinity.
It is convenient, for facility of statement, to suppose (7=0;
this is legitimate since we may always add an arbitrary con
stant to <£.

When the point P is external to the fluid, <£' is finite through
out the original region, and the formula (5) gives at once

o ds + ds ......... (8),

47rJJ r dn 4t7rJJ ^ dn\rj

where, again, in the case of a liquid extending to infinity, and at
rest there, the terms due to the infinite part of the boundary may
be omitted.

58. The distribution expressed by (7) can, further, be re
placed by one of simple sources only, or of double sources only,
over the boundary.

Let <f> be the velocity-potential of the fluid occupying a certain
region, and let (/>' now denote the velocity-potential of any possible
acyclic irrotational motion through the rest of infinite space, with
the condition that </>, or <£', as the case may be, vanishes at infinity.
Then, if the point P be internal to the first region, and therefore
external to the second, we have

47rJJ r dn 4m ]] ^ dn

(9),

o ,

4?r Jj r dn 4-TrJJ ^ dn

where $n, &n denote elements of the normal to dS, drawn inwards

57-58] SURFACE-DISTRIBUTIONS. 6 "7

to the first and second regions respectively, so that d/dn' = — d/dn.
By addition, we have

dn J 4?r J J^ ' dn

The function </>' will be determined by the surface-values of <j>'
or d(j>'/dri, which are as yet at our disposal.

Let us in the first place make (/>'= <f>. The tangential velocities
on the two sides of the boundary are then continuous, but the
normal velocities are discontinuous. To assist the ideas, we may
imagine a fluid to fill infinite space, and to be divided into two
portions by an infinitely thin vacuous sheet within which an
impulsive pressure p$> is applied, so as to generate the given
motion from rest. The last term of (10) disappears, so that

Mr8 (11)>

that is, the motion (on either side) is that due to a surface-distri
bution of simple sources, of density

1 (d$ d<f>'\*
4-7T \dn dn J

Secondly, we may suppose that dfi/dn = d<f>/dn. This gives
continuous normal velocity, but discontinuous tangential velocity,
over the original boundary. The motion may in this case be
imagined to be generated by giving the prescribed normal velocity
- d(f>/dn to every point of an infinitely thin membrane coincident in
position with the boundary. The first term of (10) now vanishes,
and we have

shewing that the motion on either side may be conceived as due
to a surface-distribution of double sources, with density

It is obvious that cyclic irrotational motion of a liquid cannot be re
produced by any arrangement of simple sources. It is easily seen, however,
that it may be represented by a certain distribution of double sources over

* This investigation was first given by Green, from the point of view of Electro
statics ; I.e. ante p. 50.

5—2

68 IRROTATIONAL MOTION. [CHAP. Ill

the boundary, together with a uniform distribution of double sources over
each of the barriers necessary to render the region occupied by the fluid
simply- Connected. In fact, with the same notation as in Art. 53, we find

//<*- »i © *+£ Hi © ^ H

where $ is the single- valued velocity-potential which obtains in the modified
region, and $' is the velocity-potential of the acyclic motion which is generated
in the external space when the proper normal velocity - d$/dn is given to each
element dS of a membrane coincident in position with the original boundary.

Another mode of representing the irrotational motion of a
liquid, whether cyclic or not, will present itself in the chapter on
Vortex Motion.

CHAPTER IV.

MOTION OF A LIQUID IN TWO DIMENSIONS.

59. IF the velocities u, v be functions of x, y only, whilst w
is zero, the motion takes place in a series of planes parallel to xy,
and is the same in each of these planes. The investigation of the
motion of a liquid under these circumstances is characterized by
certain analytical peculiarities; and the solutions of several pro
blems of great interest are readily obtained.

Since the whole motion is known when we know that in the
plane 2 = 0, we may confine our attention to that plane. When
we speak of points and lines drawn in it, we shall understand
them to represent respectively the straight lines parallel to the
axis of 2, and the cylindrical surfaces having their generating
lines parallel to the axis of 2, of which they are the traces.

By the flux across any curve we shall understand the volume
of fluid which in unit time crosses that portion of the cylindrical
surface, having the curve as base, which is included between the
planes z = 0, z = 1.

Let A, P be any two points in the plane xy. The flux across
any two lines joining AP is the same, provided they can be
reconciled without passing out of the region occupied by the
moving liquid ; for otherwise the space included between these
two lines would be gaining or losing matter. Hence if A be
fixed, and P variable, the flux across any line AP is a function
of the position of P. Let -^ be this function ; more precisely, let
^r denote the flux across AP from right to left, as regards an
observer placed on the curve, and looking along it from A in the
direction of P. Analytically, if I, m be the direction-cosines of the

70 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

normal (drawn to the left) to any element Ss of the curve, we
have

rP

= 1 (lu + mv)ds (1).

J A

If the region occupied by the liquid be aperiphractic (see p. 43),
ty is necessarily a single-valued function, but in periphractic regions
the value of ty may depend on the nature of the path AP.
For spaces of two dimensions, however, periphraxy and multiple-
connectivity become the same thing, so that the properties of ^,
when it is a many-valued function, in relation to the nature of
the region occupied by the moving liquid, may be inferred from
Art. 50, where we have discussed the same question with regard
to $. The cyclic constants of ty, when the region is peri
phractic, are the values of the flux across the closed curves
forming the several parts of the internal boundary.

A change, say from A to B, of the point from which >/r is
reckoned has merely the effect of adding a constant, viz. the flux
across a line BA, to the value of ^ ; so that we may, if we
please, regard ty as indeterminate to the extent of an additive
constant.

If P move about in such a manner that the value of ^ does
not alter, it will trace out a curve such that no fluid anywhere
crosses it, i.e. a stream-line. Hence the curves ^ = const, are the
stream-lines, and i/r is called the ' stream-function.'

If P receive an infinitesimal displacement PQ (= %) parallel
to y, the increment of A/T is the flux across PQ from right to left,
i.e. &\/r = — u. PQ, or

u — ** ........................... (2).

dy

Again, displacing P parallel to xt we find in the same way

The existence of a function ty related to u and v in this manner
might also have been inferred from the form which the equation of
continuity takes in this case, viz.

J' + ^ = () ....... (4),

dx dy

59-60] STREAM-FUNCTION. 71

which is the analytical condition that udy — vdx should be an
exact differential*.

The foregoing considerations apply whether the motion be
rotational or irrotational. The formulse for the component angular
velocities, given in Art. 31, become

f=o, 77=0, r»tiVi

V fi *Y

so that in irrotational motion we have

$H|r c£2-\/r ...

~d^+w= (>

60. In what follows we confine ourselves to the case of
irrotational motion, which is, as we have already seen, character
ized by the existence, in addition, of a velocity-potential </>,
connected with u, v by the relations

u = -^ v--^ m

dx' dy"

and, since we are considering the motion of incompressible fluids
only, satisfying the equation of continuity

dtf+~df =

The theory of the function $, and the relation between its
properties and the nature of the two-dimensional space through
which the irrotational motion holds, may be readily inferred
from the corresponding theorems in three dimensions proved in
the last chapter. The alterations, whether of enunciation or of
proof, which are requisite to adapt these to the case of two
dimensions are for the most part purely verbal.

An exception, which we will briefly examine, occurs however in the case
of the theorem of Art. 39 and of those which depend on it.

If 8s be an element of the boundary of any portion of the plane xy which
is occupied wholly by moving liquid, and if 8n be an element of the normal to
8s drawn inwards, we have, by Art. 37,

* The function ^ was first introduced in this way by Lagrange, Nouv. mem. de
VAccid. de Berlin, 1781 ; Oeuvres, t. iv., p. 720. The kinematical interpretation
is due to Rankine,

72 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

the integration extending round the whole boundary. If this boundary be a
circle, and if r, 6 be polar co-ordinates referred to the centre P of this circle
as origin, the last equation may be written

- or

1 ft*

Hence the integral ^- I

2n J o

i.e. the mean- value of $ over a circle of centre P, and radius r, is independent
of the value of r, and therefore remains unaltered when r is diminished
without limit, in which case it becomes the value of <£ at P.

If the region occupied by the fluid be periphractic, and if we apply (i) to
the space enclosed between one of the internal boundaries and a circle with
centre P and radius r surrounding this boundary, and lying wholly in the
fluid, we have

/"27T /~7/4>

.(ii);

where the integration in the first member extends over the circle only, and
£TT M denotes the flux into the region across the internal boundary. Hence

d 1 •**,, M

V;

which gives on integration

i.e. the mean value of </> over a circle with centre P and radius r is equal to
- M log r + 6'y, where C is independent of r but may vary with the position of P.
This formula holds of course only so far as the circle embraces the same
internal boundary, and lies itself wholly in the fluid.

If the region be unlimited externally, and if the circle embrace the whole
of the internal boundaries, and if further the velocity be everywhere zero at
infinity, then C is an absolute constant ; as is seen by reasoning similar to
that of Art. 41. It may then be shewn that the value of 0 at a very great
distance r from the internal boundary tends to the value - J/log r + C. In the
particular case of M=0 the limit to which <£ tends at infinity is finite; in
all other cases it is infinite, and of the opposite sign to M. We infer, as before,
that there is only one single- valued function $ which 1° satisfies the equation
(2) at every point of the plane xy external to a given system of closed curves,
2° makes the value of d<p/dn, equal to an arbitrarily given quantity at every
point of these curves, and 3° has its first differential coefficients all zero at
infinity.

If we imagine point-sources, of the type explained in Art. 56, to be distri
buted uniformly along the axis of z, it is readily found that the velocity at a
distance r from this axis will be in the direction of r, and equal to m/r, where
m is a certain constant. This arrangement constitutes what may be called a
1 line-source,' and its velocity -potential may be taken to be

0 = — m log r (iv).

60-62] KINETIC ENERGY. 73

The reader who is interested in the matter will have no difficulty in working
out a theory of two-dimensional sources and sinks, similar to that of Arts.
56—58 *.

61. The kinetic energy T of a portion of fluid bounded by a
cylindrical surface whose generating lines are parallel to the axis
of zy and by two planes perpendicular to the axis of z at unit dis
tance apart, is given by the formula

where the surface-integral is taken over the portion of the
plane xy cut off by the cylindrical surface, and the line-integral
round the boundary of this portion. Since

the formula (1) may be written

2!T=p#ety ........................ (2),

the integration being carried in the positive direction round the
boundary.

If we attempt by a process similar to that of Art. 46 to calculate the
energy in the case where the region extends to infinity, we find that its value
is infinite, except when the total flux outwards (2»rJO is zero. For if we
introduce a circle of great radius r as the external boundary of the portion
of the plane xy considered, we find that the corresponding part of the
integral on the right-hand side of (I) tends, as r increases, to the value
rrpM(Mlogr- C\ and is therefore ultimately infinite. The only exception is
when M = 0, in which case we may suppose the line-integral in (1) to extend
over the internal boundary only.

If the cylindrical part of the boundary consist of two or more
separate portions one of which embraces all the rest, the enclosed
region is multiply-connected, and the equation (1) needs a correc
tion, which may be applied exactly as in Art. 55.

62. The functions <£ and ^ are connected by the relations

d(f> _ dty d(f) _ d^r , .

doc dy ' dy dx ' '

These are the conditions that (fr + iifr, where i stands for V~l>
should be a function of the ' complex ' variable x + iy. For if

* This subject has been treated very fully by C. Neumann, Ueber das logarith-
misclie und Newton'sche Potential, Leipzig, 1877.

74 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

) ..................... (2),

we have ^(<£ + fy) = if (x + iy) = i -jt(<l> + ty) ...... (3),

whence, equating separately the real and the imaginary parts, we
obtain (1).

Hence any assumption of the form (2) gives a possible case of
irrotational motion. The curves (f> = const, are the curves of equal
velocity-potential, and the curves ty = const, are the stream-lines.
Since, by (].),

d(t> djr d(f> dty _
dx dx dy dy

we see that these two systems of curves cut one another at right
angles, as already proved. Since the relations (1) are unaltered
when we write — ty for </>, and <f> for -v/r, we may, if we choose, look
upon the curves *fr = const, as the equipotential curves, and the
curves <£ = const, as the stream-lines ; so that every assumption of
the kind indicated gives us two possible cases of irrotational motion.

For shortness, we shall through the rest of this Chapter follow
the usual notation of the Theory of Functions, and write

z = x+iy .............................. (4),

W = <f> + l'\fr ........................... (5).

At the present date the reader may be assumed to be in possession of at
all events the elements of the theory referred to*. We may, however, briefly
recall a few fundamental points which are of special importance in the hydro-
dynamical applications of the subject.

The complex variable x + iy may be represented, after Argand and Gauss,
by a vector drawn from the origin to the point (&', y). The result of adding
two complex expressions is represented by the geometric sum of the corre
sponding vectors. Regarded as a multiplying operator, a complex expression
a + ib has the effect of increasing the length of a vector in the ratio r : 1, and
of simultaneously turning it through an angle 6, where r = (a2 + 62)*, and

The fundamental property of & function of a complex variable is that it
has a definite differential coefficient with respect to that variable. If 0, \//-
denote any functions whatever of x and y, then corresponding to every value
of x-\-iy there must be one or more definite values of (f) + i\^; but the ratio
of the differential of this function to that of x-\-iy^ viz.

* See, for example, Forsyth, Ttieory of Function*, Cambridge, 1898, cc. i., ii.

62] COMPLEX VARIABLE. 75

depends in general on the ratio 8x : §y. The condition that it should be the
same for all values of this ratio is

j^j
ay dy

which is equivalent to (1) above. This property may therefore be taken, after
Kiemann, as the definition of a function of the complex variable x+iy\ viz.
such a function must have, for every assigned value of the variable, not only
a definite value or system of values, but also for each of these values a definite
differential coefficient. The advantage of this definition is that it is quite
independent of the existence of an analytical expression for the function.

Now, w being any function of st we have, corresponding to any point
P of the plane xy (which we may call the plane of the variable z), one or
more definite values of w. Let us choose any one of these, and denote it
by a point P' of which 0, >//• are the rectangular co-ordinates in a second
plane (the plane of the function w). If P trace out any curve in the plane
of z, P' will trace out a corresponding curve in the plane of w. By mapping
out the correspondence between the positions of P and P', we may exhibit
graphically all the properties of the function w.

Let now Q be a point infinitely near to P, and let Q' be the corresponding
point infinitely near to P'. We may denote PQ by §z, PQ' by 8w. The
vector P'Q' may be obtained from the vector PQ by multiplying it by the
differential coefficient dw/dz, whose value is by definition dependent only on
the position of P, and not on the direction of the element dz or PQ. The effect
of this operator dw/dz is to increase the length of PQ in some definite ratio, and
to turn it through some definite angle. Hence, in the transition from the plane
of z to that of w, all the infinitesimal vectors drawn from the point P have
their lengths altered in the same ratio, and are turned through the same angle.
Any angle in the plane of z is therefore equal to the corresponding angle in
the plane of w, and any infinitely small figure in the one plane is similar to
the corresponding figure in the other. In other words, corresponding figures
in the planes of z and w are similar in their infinitely small parts.

For instance, in the plane of w the straight lines 0 = const., ^ = const.,
where the constants have assigned to them a series of values in arithmetical
progression, the common difference being infinitesimal and the same in each
case, form two systems of straight lines at right angles, dividing the plane into
infinitely small squares. Hence in the plane xy the corresponding curves
<f> = const., \js = const., the values of the constants being assigned as before,
cut one another at right angles (as has already been proved otherwise) and
divide the plane into a series of infinitely small squares.

Conversely, if $, \//> be any two functions of #, y such that the curves
0=we, \//- = ?if, where € is infinitesimal, and m, n are any integers, divide the
plane xy into elementary squares, it is evident geometrically that

dx _ dy dx _ dy
d$=±d$' djr^+df'

If we take the upper signs, these are the conditions that x+iy should be a

76 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

function of (f> + i\js. The case of the lower signs is reduced to this by re
versing the sign of -v/r. Hence the equation (2) contains the complete solution
of the problem of orthomorphic projection from one plane to another*.

The similarity of corresponding infinitely small portions of the planes w
and z breaks down at points where the differential coefficient dwldz is zero or
infinite. Since

j-sMs .............................. »

the corresponding value of the velocity, in the hydrodynamical application, is
zero or infinite.

A 'uniform' or 'single-valued' function is one which returns to its
original value whenever the representative point completes a closed circuit
in the plane xy. All other functions are said to be ' multiform,' or ' many-
valued.' A simple case of a multiform function is that of A If we put

s=x + iy = r (cos 0 + *' sin 0),
we have z* =

Hence when P describes a closed circuit surrounding the origin, 6 increases
by 2?r, and the function £5 does not return to its former value, the sign being
reversed. A repetition of the circuit restores the original value.

A point (such as the origin in this example), at which two or more values
of the function coincide, is called a 'branch-point.' In the hydrodynamical
application ' branch-points ' cannot occur in the interior of the space occupied
by the fluid. They may however occur on the boundary, since the function
will then be uniform throughout the region considered.

Many-valued functions of another kind, which may conveniently be
distinguished as ' cyclic,' present themselves, in the Theory of Functions, as
integrals with a variable upper limit. It is easily shewn that the value of
the integral

taken round the boundary of any portion of the plane xy throughout which
/ (z), and its derivative /' (z\ are finite, is zero. This follows from the two-
dimensional form of Stokes's Theorem, proved in Art. 33, viz.

the restrictions as to the values of P, Q being as there stated. If we put
P=f (z\ Q = if (z\ the result follows, since

Hence the value of the integral (iii), taken from a fixed point A to a variable
point P, is the same for all paths which can be reconciled with one another
without crossing points for which the above conditions are violated.

* Lagrange, " Sur la construction des cartes geographiques," Nouv. mem. de
VAcad. de Berlin, 1779 ; Oeuvres, t. iv., p. 636.

62]

COMPLEX VARIABLE.

77

Points of the plane xy at which the conditions in question break down may
be isolated by drawing a small closed curve round each. The rest of the plane
is a multiply-connected region, and the value of the integral from ^1 to P
becomes a cyclic function of the position of P, as in Art. 50.

In the hydrodynamical applications, the integral (iii), considered as a
function of the upper limit, is taken to be equal to <f)+i\js. If we denote
any cyclic constant of this function by K + IH, then K denotes the circulation
in the corresponding circuit, and p. the flux across it outwards.

As a simple example we may take the logarithmic function, considered as
denned by the equation

(v).

Since z~l is infinite at the origin, this point must be isolated, e.g. by drawing
a small circle about it as centre. If we put

we have

so that the value of (v) taken round the circle is

Hence, in the simply-connected region external to the circle, the function (v)
is many-valued, the cyclic constant being - 27rt.

In the theory referred to, the exponential function is defined as the inverse
function of (v), viz. if w = logz, we have ew=z. It follows that ew is periodic,
the period being 2^'. The correspondence between the planes of z and w
is illustrated by the annexed diagram. The circle of radius unity, described
about the origin as centre, in the upper figure, corresponds over and over

again to lengths 2;r on the imaginary axis of w, whilst the inner and outer
portions of the radial line 0 = 0 correspond to a system of lines parallel to

78 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

the real axis of w, drawn on the negative and positive sides, respectively*.
The reader should examine these statements, as we shall have repeated occasion
to use this transformation.

63. We can now proceed to some applications of the foregoing-
theory.

First let us assume w = Azn,

A being real. Introducing polar co-ordinates r, 0, we have

<p = Arn cos n0, \

^ = Arnsmn0 j ' '*

The following cases may be noticed.

1°. If ?i = l, the stream-lines are a system of straight lines
parallel to xt and the equipotential curves are a similar system
parallel to y. In this case any corresponding figures in the planes
of w and z are similar, whether they be finite or infinitesimal.

2°. If n = 2, the curves (/> = const, are a system of rectangular
hyperbolas having the axes of co-ordinates as their principal axes,
and the curves -\Jr = const, are a similar system, having the co
ordinate axes as asymptotes. The lines 6 = 0, 0 — \ IT are parts of
the same stream-line ty = 0, so that we may take the positive parts
of the axes of #, y as fixed boundaries, and thus obtain the case of
a fluid in motion in the angle between two perpendicular walls.

3°. If n = — 1, we get two systems of circles touching the
axes of co-ordinates at the origin. Since now <p = A/r.cos 0, the
velocity at the origin is infinite ; we must therefore suppose the
region to which our formulas apply to be limited internally by a
closed curve.

4°. If n = — 2, each system of curves is composed of a double
system of lemniscates. The axes of the system <£ = const, coincide
with x or y ; those of the system -fy = const, bisect the angles be
tween these axes.

5°. By properly choosing the value of n we get a case of
irrotational motion in which the boundary is composed of two
rigid walls inclined at any angle a. The equation of the stream
lines being

rn sin nO = const (2),

* It should be remarked that no attempt has been made to observe the same
scale in corresponding figures, in this or in other examples, to be given later.

63-64] EXAMPLES. 79

we see that the lines 0 = 0, 6 = irjn are parts of the same stream
line. Hence if we put n = TT/«, we obtain the required solution in
the form

- ft -ft

(f> = Aracos—) ^ = Arasm — ............... (3).

The component velocities along and perpendicular to r, are

A IT a"1 ^ JJ A TT a'1 • 1*6

— A-r cos — , and A -r sin — ;
a a a a

and are therefore zero, finite, or infinite at the origin, according as
a. is less than, equal to, or greater than TT.

64. We take next some cases of cyclic functions.
1°. The assumption

W= — /JL\OgZ ........................... (1)

gives <£ = -^logr, ^ = -^0 ..................... (2).

The velocity at a distance r from the origin is fi/r'} this point
must therefore be isolated by drawing a small closed curve
round it.

If we take the radii 6 — const, as the stream-lines we get the
case of a (two-dimensional) source at the origin. (See Art. 60.)

If the circles r = const, be taken as stream-lines we get the
case of Art. 28 ; the motion is now cyclic, the circulation in any
circuit embracing the origin being 2 7r/i. C£

J ~/

2°. Let us take *

(3).

If we denote by r1} r2 the distances of any point in the plane xy
from the points (+ a, 0), and by 0lt 62, the angles which these
distances make with positive direction of the axis of #, we have

z — a = r^ty z 4- a = r2e^2,
whence </> = -//,logn/r2, ->|r = - /z (6^ - 02) ............ (4).

The curves </> = const., ty — const, form two orthogonal systems of
' coaxal ' circles.

80 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

Either of these systems may be taken as the equipotential
curves, and the other system will then form the stream-lines.
In either case the velocity at the points (+ a, 0) will be infinite.

If these points be accordingly isolated by drawing closed curves
round them, the rest of the plane xy becomes a triply-connected
region.

If the circles 01 — 02 = const, be taken as the stream-lines we
have the case of a source and a sink, of equal intensities, situate
at the points (± a, 0). If a is diminished indefinitely, whilst fia
remains finite, we reproduce the assumption of Art. 63, 3°, which
therefore corresponds to the case of a double line-source at the
origin. (See the first diagram of Art. 68.)

If, on the other hand, we take the circles r^rz — const, as the
stream-lines we get a case of cyclic motion, viz. the circulation in
any circuit embracing the first (only) of the above points is STT/J,,
that in a circuit embracing the second is — ZTT/JL ; whilst that in a
circuit embracing both is zero. This example will have additional
interest for us when we come to treat of ' Rectilinear Vortices.'

64-65] INVERSE FORMULA. 81

65. If w be a function of z, it follows at once from the defini
tion of Art. 62 that z is a function of w. The latter form of
assumption is sometimes more convenient analytically than the
former.

The relations (1) of Art. 62 are then replaced by
dxdy dx dy

d<f> dr} d^

dw dd> .dty

Also since -= - = —?~ + i — I-

dz dx dx

, dz \ \ in .v

we have

dw u — iv q \q q.
where q is the resultant velocity at (x, y). Hence if we write

and imagine the properties of the function f to be exhibited
graphically in the manner already explained, the vector drawn
from the origin to any point in the plane of f will agree
in direction with, and be in magnitude the reciprocal of, the
velocity at the corresponding point of the plane of z.

Again, since l/q is the modulus of dzjdw, i.e. of dx/d(f> + idyjd<}>,
we have

which may, by (1), be put into the equivalent forms

f

W.

\ttyy wcp u^ n^r uy>

The last formula, viz.

simply expresses the fact that corresponding elementary areas in
the planes of z and w are in the ratio of the square of the modulus
of dz/dw to unity.

I, 6

82

MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

66. The following examples of this are important.

1°. Assume z = ccoshw (1),

or x = c cosh </> cos 1

y = c sinh (/> sin -

The curves $ = const, are the ellipses

__ -|

and the curves >/r = const, are the hyperbolas

(2).
(3),

a?"

c2 cos2

c'2 sin2

= 1

these conies having the common foci (± c, 0).

Since at the foci we have 0 = 0, ty = HTT, n being some integer,
we see by (2) o£~the preceding Art. that the velocity there
is infinite. If the hyperbolas be taken as the stream-lines, the
portions of the axis of a; which lie outside the points (+ c, 0) may
be taken as rigid boundaries. We obtain in this manner the case

66]

SPECIAL CASES.

83

of a liquid flowing from one side to the other of a thin plane
partition, through an aperture of breadth 2c ; the velocity at the
edges is however infinite.

If the ellipses be taken as the stream-lines we get the case of
a liquid circulating round an elliptic cylinder, or, as an extreme
case, round a rigid lamina whose section is the line joining the foci
(± e, 0).

At an infinite distance from the origin (/> is infinite, of the
order log r, where r is the radius vector ; and the velocity is
infinitely small of the order 1/r.

2°. Let z = w + ew ................. < ......... (5),

X = (f) + 6* COS l/r, y = ty -f 6* sill i/r ............ (6).

Or

The stream-line i/r = 0 coincides with the axis of x. Again the
portion of the line y=rrr between x = — GO and x = — 1, considered

as a line bent back on itself, forms the stream-line i|r = TT ; viz. as
$ decreases from -f- oo through 0 to — oo , x increases from — oc to
— 1 and then decreases to — oo again. Similarly for the stream
line i|r = — TJ-.

6—2

84 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

Since f = — dz\dw = — 1 — e* cos i/r — ie* sin -$-,
it appears that for large negative values of <£ the velocity is
in the direction of ^-negative, and equal to unity, whilst for large
positive values it is zero.

The above formulae therefore express the motion of a liquid
flowing into a canal bounded by two thin parallel walls from an
open space. At the ends of the walls we have (f> = 0, ty = ± TT, and
therefore f = 0, i.e. the velocity is infinite. The direction of the
flow will be reversed if we change the sign of w in (5). The forms
of the stream-lines, drawn, as in all similar cases in this chapter,
for equidistant values of ty, are shewn in the figure*.

67. A very general formula for the functions <£, ^ may be
obtained as follows. It may be shewn that if a function f (z) be
finite, continuous, and single- valued, and have its first derivative
finite, at all points of a space included between two concentric
circles about the origin, its value at any point of this space can be
expanded in the form

f(z) = AQ + A1z + Ass? + ... -f B^-1 + B,z~2 + (1).

If the above conditions be satisfied at all points within a circle
having the origin as centre, we retain only the ascending series ;
if at all points without such a circle, the descending series, with
the addition of the constant A0, is sufficient. If the conditions be
fulfilled for all points of the plane xy without exception, f(z) can
be no other than a constant A0.

Putting f(z) = <t> + fy, introducing polar co-ordinates, and
writing the complex constants An> Bn> in the forms Pn + iQn,
Rn + iSn , respectively, we obtain

0 = P0+ T,rn(Pn cos n0 - Qn sin n&) + ^r~n(Encos nO+ 8nsmnO)\
^ = Q0+ 2>w(Qw cos n& + Pn sin nd) + ^r~n(Sn cosn0-Rnsmn0))

(2).

These formula? are convenient in treating problems where we
have the value of $, or of d<j>/dn, given over the circular boun
daries. This value may be expanded for each boundary in a series
of sines and cosines of multiples of 6, by Fourier's theorem. The
series thus found must be equivalent to those obtained from (2);
whence, equating separately coefficients of sin n6 and cos 116, we
obtain four systems of linear equations to determine Pn, Qn, Rn, Sn.

* This example was given by von Helmholtz, Perl. Monatsber., April 23, 1868;
Phil Mag., Nov. 1868; Ges. Abh., t. i., p. 154.

66-68] GENERAL FORMULAE. 85

68. As an example let us take the case of an infinitely long
circular cylinder of radius a moving with velocity u perpendicular
to its length, in an infinite mass of liquid which is at rest at infinity.

Let the origin be taken in the axis of the cylinder, and the
axes of x, y in a plane perpendicular to its length. Further let
the axis of x be in the direction of the velocity u. The motion,
having originated from rest, will necessarily be irrotational, and
<£ will be single-valued. Also, since fd<p/dn . ds, taken round the
section of the cylinder, is zero, ty is also single-valued (Art. 59),
so that the formulae (2) apply. Moreover, since d(f>/dn is given at
every point of the internal boundary of the fluid, viz.

— -~ = u cos 6, for r — a ..................... (3),

CLi

and since the fluid is at rest at infinity, the problem is determinate,
by Art. 41. These conditions give Pn = 0, Qn= 0, and

u cos 0 = ^ narn~l (Rn cos nO + 8n sin nO),

which can be satisfied only by making Rt = ua2, and all the other
coefficients zero. The complete solution is therefore

(f>= -- cos 0, -\/r = -- — sin 0 ............... (4).

The stream-lines ^r = const, are circles, as shewn on the next page.

The kinetic energy of the liquid is given by the formula (2) of
Art. 61, viz.

sauV P*cos2 6 d6 = m'u2 ......... (5),

Jo

if m', = 7ra2p, be the mass of fluid displaced by unit length of the
cylinder. This result shews that the whole effect of the presence
of the fluid may be represented by an addition m' to the inertia
of the cylinder. Thus, in the case of rectilinear motion, if we have
an extraneous force X acting on the cylinder, the equation of
energy gives

(imu2 + im'u-) = Xu,

or (m + no-X ..................... (6),

where m represents the mass of the cylinder.

86 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

Writing this in the form

du v , du

mdt=x-m^'

we learn that the pressure of the fluid is equivalent to a force
— m' du/dt in the direction of motion. This vanishes when u is
constant.

The above result may of course be verified by direct calculation. The
pressure is given by the formula

where we have omitted the term due to the extraneous forces (if any) acting on
the fluid, the effect of which can be found by the rules of Hydrostatics. The
term dfyjdt here expresses the rate at which 0 is increasing at a fixed point
of space, whereas the value of </> in (4) is referred to an origin which is in
motion with the velocity u. In consequence of this the value of r for any
fixed point is increasing at the rate — u cos 0, and that of 6 at the rate
U/r . sin Q. Hence we must put

d<t> du. a2 .. d(f> u sin 6 d$ dn a2 u*a2

-j-= -j- - COS 0 - U COS 0 -~ + - -£=-77 COS0+-.. COS 20.

dt dt r dr r dd dt r r2

68-69] MOTION OF A CIRCULAR CYLINDER. 87

Since, also, £2 = U2a4/r4, the pressure at any point of the cylindrical surface

The resultant force on unit length of the cylinder is evidently parallel to
the initial line 6 = 0 ; to find its amount we multiply by - add . cos 6 and
integrate with respect to & between the limits 0_ and TT. The result is _.
— m'dvi/dt, as before.

If in the above example we impress on the fluid and the
cylinder a velocity — u we have the case of a current flowing
with the general velocity u past a fixed cylindrical obstacle.
Adding to <f) and *fy the terms ur cos 6 and ur sin 6, respectively,
we get

If no extraneous forces act, and if u be constant, the resultant force
on the cylinder is zero.

69. To render the formula (1) of Art. 67 capable of repre
senting any case of irrotational motion in the space between two
concentric circles, we must add to the right-hand side the term

4 log* (1).

88 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

If A = P + iQ, the corresponding terms in $, ty are

Plogr-Qd, P0 + Q\ogr (2),

respectively. The meaning of these terms is evident from Art. 64 ;
viz. 2?rP, the cyclic constant of A/T, is the flux across the inner
(or outer) circle ; and 2?rQ, the cyclic constant of <£, is the circu
lation in any circuit embracing the origin.

For example, returning to the problem of the last Art., let us
suppose that in addition to the motion produced by the cylinder
we have an independent circulation round it, the cyclic constant
being K. The boundary-condition is then satisfied by

<£=uj 0030-^0 (3).

The effect of the cyclic motion, superposed on that due to the
cylinder, will be to augment the velocity on one side, and to
dimmish (and, it may be, to reverse) it on the other. Hence

when the cylinder moves in a straight line with constant velocity,
there will be a diminished pressure on one side, and an increased
pressure on the other, so that a constraining force must be applied
at right angles to the direction of motion.

69] CYLINDER WITH CIRCULATION. 89

The figure shews the lines of flow. At a distance from the origin they
approximate to the form of concentric circles, the disturbance due to the
cylinder becoming small in comparison with the cyclic motion. When, as in
the case represented, u>K/2ira, there is a point of zero velocity in the fluid.
The stream-line system has the same configuration in all cases, the only effect
of a change in the value of u being to alter the scale, relative to the diameter
of the cylinder.

To calculate the effect of the fluid pressures on the cylinder when moving
in any manner we write

where % is the angle which the direction of motion makes with the axis of x.
In the formula for the pressure [Art. 68 (i)] we must put, for r=a,

and ^2

The resultant force on the cylinder is found to be made up of a component

in the direction of motion, and a component

.................................... (v),

at right angles, where m.' = npa2 as before. Hence if P, Q denote the
components of the extraneous forces, if any, in the directions of the tangent
and the normal to the path, respectively, the equations of motion of the
cylinder are

(vi).

If there be no extraneous forces, u is constant, and writing
where R is the radius of curvature of the path, we find

(vii).
The path is therefore a circle, described in the direction of the cyclic motion*.

* Lord Rayleigh, " On the Irregular Flight of a Tennis Ball," Megs, of Math.,
t. vii. (1878); Greeiihill, ibid., t. ix., p. 113 (1880).

90 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

If X, y be the rectangular co-ordinates of the axis of the cylinder, the
equations (vi) are equivalent to

')x=-*py+.n
Kp± + rJ"

where Jf, Y are the components of the extraneous forces. To find the effect
of a constant force, we may put

, F=0 ........................... (ix).

The solution then is

X = a + a cos (nt + e),

g>

provided n = Kp/(m + m') .................................... (xi).

This shews that the path is a trochoid, described with a mean velocity g'/n
perpendicular to x*. It is remarkable that the cylinder has on the whole
no progressive motion in the direction of the extraneous force.

70. The formula (1) of Art. 67, as amended by the addition of
the term A log zt may readily be generalized so as to apply to any
case of irrotational motion in a region with circular boundaries,
one of which encloses all the rest. In fact, corresponding to each
internal boundary we have a series of the form

where c, = a + ib say, refers to the centre, and the coefficients
A, Alt A2} ... are in general complex quantities. The difficulty
however of determining these coefficients so as to satisfy given
boundary conditions is now so great as to render this method of
very limited application.

Indeed the determination of the irrotational motion of a liquid
subject to given boundary conditions is a problem whose exact
solution can be effected by direct processes in only a very few
cases f. Most of the cases for which we know the solution have

* Greenhill, I.e.

t A very powerful method of transformation, applicable to cases where the
boundaries of the fluid consist of fixed plane walls, has however been deve
loped by Schwarz (" Ueber einige Abbildungsaufgaben," Crelle, t. Ixx., Gesam-
melte Abhandlungen, Berlin, 1890, t. ii., p. 65), Christoffel (" Sul problema delle
temperature stazionarie e la rappresentazione di una data superficie," Annali di
Matematica, Serie n., t. i. , p. 89), and Kirchhoff (" Zur Theorie des Conden-
sators," Berl. Monatsber., March 15, 1877; Ges. Abh., p. 101). Many of the
solutions which can be thus obtained are of great interest in the mathematically
cognate subjects of Electrostatics, Heat-Conduction, &c. See for example, J. J.
Thomson, Recent Researches in Electricity and Magnetism, Oxford, 1893, c. iii.

69-71] INDIRECT METHODS. 91

been obtained by an inverse process ; viz. instead of trying to find
a solution of the equation V2<£ = 0 or V2-\Jr = 0, satisfying given
boundary conditions, we take some known solution of the differen
tial equations and enquire what boundary conditions it can be
made to satisfy. Examples of this method have already been
given in Arts. 63, 64, and we may carry it further in the following
two important cases of the general problem in two dimensions.

71. CASE I. The boundary of the fluid consists of a rigid
cylindrical surface which is in motion with velocity u in a
direction perpendicular to its length.

Let us take as axis of x the direction of this velocity u, and
let Bs be an element of the section of the surface by the plane xy.

Then at all points of this section the velocity of the fluid in
the direction of the normal, which is denoted by d-fr/ds, must be
equal to the velocity of the boundary normal to itself, or — udyjds.
Integrating along the section, we have

T/T = — My + const (1).

If we take any admissible form of i/r, this equation defines a
system of curves each of which would by its motion parallel to
x give rise to the stream-lines ^r = const. * We give a few
examples.

1°. If we choose for ^r the form — uy, (1) is satisfied
identically for all forms of the boundary. Hence the fluid
contained within a' cylinder of any shape which has a motion
of translation only may move as a solid body. If, further, the
cylindrical space occupied by the fluid be simply-connected, this is
the only kind of motion possible. This is otherwise evident from
Art. 40 ; for the motion of the fluid and the solid as one mass
evidently satisfies all the conditions, and is therefore the only
solution which the problem admits of.

2°. Let -v/r = A jr . sin 6 ; then (1) becomes

A

sin 6 = - u r sin 6 + const (2).

In this system of curves is included a circle of radius a, provided

* Cf. Kankine, "On Plane Water-Lines in Two Dimensions," Phil. Trans.,
1864, where the method is applied to obtain curves resembling the lines of ships.

92 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

A /a = - ua. Hence the motion produced in an infinite mass of
liquid by a circular cylinder moving through it with velocity u
perpendicular to its length, is given by

11 r/2

* = -— sintf (3),

which agrees with Art. 68.

3°. Let us introduce the elliptic co-ordinates f, rj, connected
with x, y by the relation

x + iy = c cosh (f + iy) (4),

or X — G cosh f cos rj,

y = c sinh f sin rj j
(cf. Art. 66), where f may be supposed to range from 0 to oo , and
rj from 0 to 2-7T. If we now put

0 + ty = (V-tf+*,) ........................ (6),

where G is some real constant, we have

i|r = - Ce~t sin rj ........................ (7),

so that (1) becomes

Ce~% sin 77 = uc sinh {• sin T; 4- const.

In this system of curves is included the ellipse whose parameter
fo is determined by

If a, b be the semi-axes of this ellipse we have

, 6 = csinhf0,

xl. r» 7.

so that (7 = - r = u6

a — 6

Hence the formula

fsin7; .................. (8)

gives the motion of an infinite mass of liquid produced by an
elliptic cylinder of semi-axes a, b, moving parallel to the greater
axis with velocity u.

That the above formulae make the velocity zero at infinity
appears from the consideration that, when f is large, S% and S?/
are of the same order as e^Sf or etBrj, so that d^r/dx, d-^rfdy are of
the order e~2^ or 1/r2, ultimately, where r denotes the distance of
any point from the axis of the cylinder.

If the motion of the cylinder were parallel to the minor axis
the formula would be

71]

TRANSLATION OF A CYLINDER.

93

The stream-lines are in each case the same for all confucal
elliptic forms of the cylinder, so that the formulae hold even when
the section reduces to the straight line joining the foci. In this
case (9) becomes

A|T = vc e~£ cos ?; (10),

which would give the motion produced by an infinitely long
lamina of breadth 2c moving ' broadside on ' in an infinite mass of

liquid. Since however this solution makes the velocity infinite at
the edges, it is subject to the practical limitation already indicated
in several instances*.

* This investigation was given in the Quart. Journ. of Math., i. xiv. (1875).
Results equivalent to (8), (9) had however been obtained, in a different manner, by
Beltrami, " Sui principii fondamentali dell' idrodinamica razionale," Mem. dell'
Accad. delle Scienze di Bologna, 1873, p. 394.

94 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

The kinetic energy of the fluid is given by

2T= p Ud^

J

o
........................ (11),

where b is the half-breadth of the cylinder perpendicular to the
direction of motion.

If the units of length and time be properly chosen we may write
£f%

whence f*^(i+

These formula) are convenient for tracing the curves 0 = const., ^ = const.,
which are figured on the preceding page.

By superposition of the results (8) and (9) we obtain, for the case of an
elliptic cylinder having a motion of translation whose components are u, V,

^= - (^T£ V e~^ (u6 sin 77 - va cos 7?) ..................... (i).

To find the motion relative to the cylinder we must add to this the expression
vy - vx — c (u sinh £ sin rj - v cosh £ cos q) ............... (ii).

For example, the stream-function for a current impinging at an angle of 45°
on a plane lamina whose edges are at x=±c is

V^= - -TO 9oc sintl £ (cos *) ~ sin ri ........................ ("*)»

where q0 is the velocity at infinity. This immediately verifies, for it makes
•^• = 0 for £ = 0, and gives

for £=oo. The stream-lines for this case are shewn in the annexed figure

(turned through 45° for convenience). This will serve to illustrate some
results to be obtained later in Chap, vi,

71-72] RELATIVE STREAM-LINES. 95

If we trace the course of the stream-line ^ = 0 from <£=+oc to <£ = - oo ,
we find that it consists in the first place of the hyperbolic arc TJ^^TT, meeting
the lamina at right angles ; it then divides into two portions, following the
faces of the lamina, which finally re-unite and are continued as the hyperbolic
arc 17 = |- IT. The points where the hyperbolic arcs abut on the lamina are
points of zero velocity, and therefore of maximum pressure. It is plain that the
fluid pressures on the lamina are equivalent to a couple tending to set it broad
side on to the stream ; and it is easily found that the moment of this couple,
per unit length, is ^irpq^c2'. Compare Art. 121.

72. CASE II. The boundary of the fluid consists of a rigid
cylindrical surface rotating with angular velocity w about an axis
parallel to its length.

Taking the origin in the axis of rotation, and the axes of x, y
in a perpendicular plane, then, with the same notation as before,
d-fy/ds will be equal to the normal component of the velocity of the
boundary, or

d^lr dr

7 = wr -j- ,

ds ds

if r denote the radius vector from the origin. Integrating we
have, at all points of the boundary,

i/r = £o> r2 + const ......................... (1 ).

If we assume any possible form of ty, this will give us the
equation of a series of curves, each of which would, by rotation
round the origin, produce the system of stream-lines determined
by t-

As examples we may take the following :
1°. If we assume

<^=Ar*co$20=.A(x°-y*) .................. (2),

the equation (1) becomes

(iw - A)x* + (Jo) + A) yz=C,

which, for any given value of A, represents a system of similar
conies. That this system may include the ellipse

a? y1 _
^~*>aa1'

we must have (Jo> — A) a2 = (\u> + A)^,

a2 - b2

4 -*••:?**.•

Hence f = k"> • (^ - f) ............... (3),

96 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

gives the motion of a liquid contained within a hollow elliptic
cylinder whose semi-axes are a, b, produced by the rotation of the
cylinder about its axis with angular velocity o>. The arrangement
of the stream-lines ty = const, is given in the figure on p. 99.

The corresponding formula for (f> is

The kinetic energy of the fluid, per unit length of the cylinder,
is given by

(

dy) j -dxdy = 1 -^- «» x ^«6 . . .(5).

This is less than if the fluid were to rotate with the boundary, as
one rigid mass, in the ratio of

/a2 -fry

U«+&v

to unity. We have here an illustration of Lord Kelvin's minimum
theorem, proved in Art. 45.

2°. Let us assume

i/r = Ar3 cos 30 = A (x3 — 3xy-\
The equation (1) of the boundary then becomes

A (x3 — 3#2/2) — \w (a? + 7/2) = C (6).

We may choose the constants so that the straight line x = a shall
form part of the boundary. The conditions for this are

Aa3 - i<wft2 = C, 3Aa + \G> = 0.
Substituting the values of A, C hence derived in (6), we have

x3 — a3 — Sxy~ -\- 3a (a? — a~ + 2/2) = 0.
Dividing out by x — a, we get

or x + 2a = + \/3 . y.

The rest of the boundary consists therefore of two straight lines
passing through the point (— 2a, 0), and inclined at angles of 30°
to the axis of x.

72] ROTATION OF A CYLINDER. 97

We have thus obtained the formulae for the motion of the
fluid contained within a vessel in the form of an equilateral prism,
when the latter is rotating with angular velocity w about an axis
parallel to its length arid passing through the centre of its section ;
viz. we have

^ = - J - ?•» cos 30, (/> = £-r3sin3<9 ............ (7),

Ct (t

where 2 /v/3a is the length of a side of the prism.

The problem of fluid motion in a rotating cylindrical case is to a certain
extent mathematically identical with that of the torsion of a uniform rod or
bar*. The above examples are mere adaptations of two of de Saint- Tenant's
solutions of the latter problem.

3°. In the case of a liquid contained in a rotating cylinder
whose section is a circular sector of radius a and angle 2a, the
axis of rotation passing through the centre, we may assume

COS 26 /?A(2«+l,7r/2a ^0

c^ + 2^+1 (a) C°S (2U + l) 2S- • -<8>'

the middle radius being taken as initial line. For this makes
ty = ^cor2 for 6= ± a, and the constants A^n+i can be determined
by Fourier's method so as to make -x/r = Jwa2 for r = a. We
find

w + 1)7r_4a - (^Fl)^ + (IT-n^+lSI

......... (9).

The conjugate expression for <£ is

sin 26 . /r\(2n+Diry2« . 7J-0

0

/r

(a

where A2n+l has the value (9).
The kinetic energy is given by

a (11),

* See Thomson and Tait, Natural Philosophy, Art. 704, et seq.

t This problem was first solved by Stokes, "On the Critical Values of the
Sums of Periodic Series," Camb. Trans., t. viii. (1847), Math. andPhys. Papers, t. i.,
p. 305. See also papers by Hicks and Greenhill, Mess, of Math., t. viii., pp. 42, 89,
and t. x., p. 83.

L. 7

98 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

where <£a denotes the value of <£ for 0 = a, the value of cUf>/dn
being zero over the circular part of the boundary.

The case of the semicircle a = JTT will be of use to us later.
We then have

l 2

2n-I
and therefore

191

-j USIA/ ^ JU _L ^ X

)aT(lT = Z 7; ~fT ^rt T — c* T^T "T ^

_ 7T2

1 "

7T

Hence*

2 x iwpw'a2 ...... (13).

This is less than if the fluid were solidified, in the ratio of "6212
to 1. See Art. 45.

4°. With the same notation of elliptic coordinates as in
Art. 71, 3°, let us assume

(j>+i^ = Cie-^+ir» ..................... (14).

Since #2 + yz — £c2 (cosh 2f + cos 2?;),

the equation (1) becomes

Ce~^ cos 2?; — Jwc2 (cosh 2f + cos 2??) = const.

This system of curves includes the ellipse whose parameter
is £0, provided

or, using the values of a, b already given,

a = Jo,(a + 6)2,

so that -^ = J&> (a + b)~ e~^ cos 2?;, "|

^ = J«B(a + 6)ae-2*sin2i7. J

At a great distance from the origin the velocity is of the
order 1/Y".

* Greenhill, I c,

72] ELLIPTIC CYLINDER. 90

The above formulae therefore give the motion of an infinite
mass of liquid, otherwise at rest, produced by the rotation of
an elliptic cylinder about its axis with angular velocity &>*. The
diagram shews the stream-lines both inside arid outside a rigid
elliptical cylindrical case rotating about its axis.

The kinetic energy of the external fluid is given by

0* ........................... (16).

It is remarkable that this is the same for all confocal elliptic
forms of the section of the cylinder.

Combining these results with those of Arts. 66, 71 we find that if an elliptic
cylinder be moving with velocities u, v parallel to the principal axes of its
cross-section, and rotating with angular velocity o>, and if (further) the fluid
be circulating irrotationally round it, the cyclic constant being K, then the
stream-function relative to the aforesaid axes is

(Mb sin TJ - va cos^+Jco (a + b}*e~^ cos 2r) + £- £.

* Quart. Journ. Math., t. xiv. (1875) ; see also Beltrami, 1. c. ante p. 93.

7—2

100 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

Discontinuous Motions.

73. We have, in the preceding pages, had several instances of
the flow of a liquid round a sharp projecting edge, and it appeared
in each case that the velocity there was infinite. This is indeed a
necessary consequence of the assumed irrotational character of the
motion, whether the fluid be incompressible or not, as may be
seen by considering the configuration of the equipotential surfaces
(which meet the boundary at right angles) in the immediate
neighbourhood.

The occurrence of infinite values of the velocity may be
avoided by supposing the edge to be slightly rounded, but even
then the velocity near the edge will much exceed that which
obtains at a distance great in comparison with the radius of
curvature.

In order that the motion of a fluid may conform to such
conditions, it is necessary that the pressure at a distance should
greatly exceed that at the edge. This excess of pressure is
demanded by the inertia of the fluid, which cannot be guided
round a sharp curve, in opposition to centrifugal force, except by
a distribution of pressure increasing with a very rapid gradient
outwards.

Hence unless the pressure at a distance be very great, the
maintenance of the motion in question would require a negative
pressure at the corner, such as fluids under ordinary conditions
are unable to sustain.

To put the matter in as definite a form as possible, let us
imagine the following case. Let us suppose that a straight tube,
whose length is large compared with the diameter, is fixed in the
middle of a large closed vessel filled with frictionless liquid, and
that this tube contains, at a distance from the ends, a sliding
plug, or piston, P, which can be moved in any required manner by
extraneous forces applied to it. The thickness of the walls of
the tube is supposed to be small in comparison with the diameter;
and the edges, at the two ends, to be rounded off, so that there are
no sharp angles. Let us further suppose that at some point of the
walls of the vessel there is a lateral tube, with a piston Q, by
means of which the pressure in the interior can be adjusted at will.

73] DISCONTINUOUS MOTIONS. 101

Everything being at rest to begin with, let a slowly increasing
velocity be communicated to the plug P, so that (for simplicity)
the motion at any instant may be regarded as approximately

steady. At first, provided a sufficient force be applied to Q, a
continuous motion of the kind indicated in the diagram on p. 83
will be produced in the fluid, there being in fact only one type of
motion consistent with the conditions of the question. As the
acceleration of the piston P proceeds, the pressure on Q may
become enormous, even with very moderate velocities of P, and if
Q be allowed to yield, an annular cavity will be formed at each end
of the tube.

The further course of the motion in such a case has not
yet been worked out from a theoretical stand-point. In actual
liquids the problem is modified by viscosity, which prevents any
slipping of the fluid immediately in contact with the tube, and
must further exercise a considerable influence on such rapid
differential motions of the fluid as are here in question.

As a matter of fact, the observed motions of fluids are often
found to deviate very widely from the types shewn in our dia
grams. In such a case as we have just described, the fluid issuing
from the mouth of the tube does not immediately spread out in
all directions, but forms, at all events for some distance, a more or
sb compact stream, bounded on all sides by fluid nearly at rest.

102 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

A familiar instance is the smoke-laden stream of gas issuing from
a chimney.

74. Leaving aside the question of the manner in which the
motion is established, von Helmholtz* and Kirchhofff have
endeavoured to construct types of steady motion of a frictionless
liquid, in two dimensions, which shall resemble more closely what
is observed in such cases as we have referred to. In the problems
to be considered, there is either a free surface or (what comes to
the same thing) a surface of discontinuity along which the moving
liquid is in contact with other fluid at rest. In either case, the
physical condition to be satisfied at this surface is that the
pressure must be the same at all points of it ; this implies, in
virtue of the steady motion, and in the absence of extraneous
forces, that the velocity must also be uniform along this surface.

The most general method we possess of treating problems of
this class is based on the properties of the function % introduced
in Art. 65. In the cases we shall discuss, the moving fluid is
supposed bounded by stream-lines which consist partly of straight
walls, and partly of lines along which the resultant velocity (q)
is constant. For convenience, we may in the first instance suppose
the units of length and time to be so adjusted that this constant
velocity is equal to unity. Then in the plane of the function
£ the lines for which q = 1 are represented by arcs of a circle
of unit radius, having the origin as centre, and the straight
walls (since the direction of the flow along each is constant) by
radial lines drawn outwards from the circumference. The points
where these lines meet the circle correspond to the points where
the bounding stream-lines change their character.

Consider, next, the function log f. In the plane of this
function the circular arcs for which q = I become transformed into
portions of the imaginary axis, and the radial lines into lines
parallel to the real axis. It remains then to frame an assumption
of the form

log?=/(M;)
such that the now rectilinear boundaries shall correspond, in the

* I. c. ante p. 24.

t "Zur Theorie freier Fliissigkeitsstrahlen," Crelle, t. Ixx. (1869), Ges. Abh.,
p. 416; see also Mechanik, cc. xxi., xxii. Considerable additions to the subject
have been recently made by Michell, "On the Theory of Free Stream Lines,'1
Phil. Trans., A., 1890.

73-75]

DISCONTINUOUS MOTIONS.

103

plane of w, to straight lines i/r = constant. There are further
conditions of correspondence between special points, one on the
boundary, and one in the interior, of each area, which render the
problem determinate. These will be specified, so far as is neces
sary, as occasion arises. The problem thus presented is a particular
case of that solved by Schwarz, in the paper already cited. His
method consists in the conformal representation of each area in
turn on a half-plane*; we shall find that, in such simple cases as
we shall have occasion to consider, this can be effected by the
successive use of transformations already studied, and figured, in
these pages.

When the correspondence between the planes of f and w has
been established, the connection between z and w is to be found,
by integration, from the relation dzjdw = — f. The arbitrary con
stant which appears in the result is due to the arbitrary position
of the origin in the plane of z.

75. We take first the case of fluid escaping from a large
vessel by a straight canal projecting in wards "f. This is the two-
dimensional form of Borda's mouthpiece, referred to in Art. 25.

-I'

A

-I'

A' X'

The figure shews the forms of the boundaries in the planes of

* See Forsyth, Theory of Functions, c. xx.

t This problem was first solved by von Helmholtz, /. c. ante p. 24.

104 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

z, f, w, and of two subsidiary variables z1} z.2*. A reference to
the diagram on p. 77 will shew that the relation

zl = xl + iy, = log f ( 1 )

transforms the boundaries in the plane of £ into the axis of #,
from (oo , 0) to the origin, the axis of yl from the origin to (0, — 2?r),
and the line y1 — — 27r from (0, - 2?r) to (oo , - 2?r), respectively.
If we now put

22 = #2 + iy* = cosh fa (2),

these boundaries become the portions of the axis of #2 for which
#2 > 1, 1 > #2 > — 3, and a?2 < — 1, respectively ; see Art. 66, 1°. It
remains to transform the figure so that the positive and negative
portions of the axis of xz shall correspond respectively to the two
bounding stream-lines, and that the point z2=Q (marked / in the
figure) shall correspond to w — — oo . All these conditions are satis
fied by the assumption

w = log*2 (3),

(see Art. 62), provided the two bounding stream -lines be taken
to be ty = 0, T/T =• — TT respectively. In other words the final
breadth of the stream (where q-=l) is taken to be equal to TT.
This is equivalent to imposing a further relation between the units
of length and time, in addition to that already adopted in Art. 74,
so that these units are now, in any given case, determinate. An
arbitrary constant might be added to (3) ; the equation, as it
stands, makes the edge A of the canal correspond to w = 0.

Eliminating zlt z.2, we get f* + f~* = 2ew, whence, finally,

? = - 1 + 2^ + 2^(^-1)* (4).

The free portion of the stream-line \/r = 0 is that for which f is complex
and therefore </> < 0. To trace its form we remark that along it we
have — dty/ds = q = 1, and therefore <£>=• — $, the arc being measured
from the edge of the canal. Also f = dx/ds + idy/ds. Hence

das/ds = - 1 + 2e~26', dy/ds = - 2e~s (1 - e~-s)* (5),

or, integrating,

x = 1 - s - e~-s, y = -\TT +e~s(l- e~-s)* + sin"1 e~s . . . (6),
the constants of integration being so chosen as to make the origin
of (as, y) coincide with the point A of the first figure. For s = oo ,

* The heavy lines represent rigid boundaries, and the fine continuous lines the
free surfaces. Corresponding points in the various figures are indicated by the
same letters.

75-76]

BOKDA'S MOUTHPIECE.

105

we have y = — JTT, which shews that, on our scale, the total breadth
of the canal is 2?r. The coefficient of contraction is therefore J,
in accordance with Borda's theory.

If we put docjds = cos 0, and therefore s = log sec %0, we get

# = sin2 10 -log sec J0, y = -%0 + ±sin0 (7),

by means of which the curve in question is easily traced.

Line of Symmetry.

76. The solution for the case of fluid issuing from a large
vessel by an aperture in a plane wall is analytically very similar.
The successive steps of the transformation, viz.

^1 = logf, #3 = cosh #! , w = log£, (1),

Z,-loff 5

A T A

-I'

] 06 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

are sufficiently illustrated by the figures. We thus get

or ? = ew + (eaw-l)* (2).

For the free stream-line starting from the edge A of the aperture
we have t|r — 0, </> < 0, whence

dxjds = e~-\ dyfds = — (1 — e~*M)
or x=l-e~s, y = (l-e-M)*-i1°gi — ?r

the origin being taken at the point A. If we put dx/ds = cos 6,
these may be written

•x = 2 sin- ^0, y = sin 6 — log tan ( J TT -f i 0) (o).

Line of Symmetry.

When 5 = x , we have &• = 1 ; and therefore, since on our scale
the final breadth of the stream is TT, the total width of the aperture
is represented by TT+ 2; i.e. the coefficient of contraction is

7T/(7T + 2), ='611.

* This example was given by Kirchhoff (I.e.'], and discussed more fully by Lord
Rayleigh, " Notes on Hydrodynamics," Phil. May., December 187G.

76-77]

VENA CONTRACTA.

107

77. The next example is of importance in the theory of the
resistance of fluids. We suppose that a steady stream impinges
directly on a fixed plane lamina, behind which is a region of dead
water bounded on each side by a surface of discontinuity.

The middle stream-line, after meeting the lamina at right
angles, branches off into two parts, which follow the lamina to the
edges, and thence the surfaces of discontinuity. Let this be the
line -\/r = 0, and let us further suppose that at the point of
divergence we have <£ = 0. The forms of the boundaries in the
planes of zt f, w are shewn in the figures. The region occupied

i

by the moving fluid corresponds to the whole of the plane of
w, which must be regarded however as bounded internally by
the two sides of the line ^ = 0, (f> < 0.

As in Art. 76, the transformations
zl = log f,
z» = cosh #

give us as boundaries the segments of the axis y.2 = 0 made by the
points a?2= + 1. The further assumption

*-- «•' (2),

converts these into segments of the negative portion of the axis
2/3 = 0, taken twice. The boundaries now correspond to those of
the plane w, except that to w = Q corresponds £3 = oo, and con
versely. The transformation is therefore completed by putting

w = zrl (3).

Hence, finally, f= ( - I)' + ( - I - 1)* (4).

1 08

MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

For i/r = 0, and 0></»-l,£is real ; this corresponds to the portion
CA of the stream-line. To find the breadth I of the lamina on
the scale of our formulae, we have, putting $ = -</>',

For the free portion A I of the stream-line, we have (f> < — 1, and
therefore, putting $ = — 1 — s,

Hence, taking the origin at the centre of the lamina,
x = ITT + 2 (1 + *)*, y = [5 (1+ «)}* - log {** + (1 + *

or, putting s — tan2 ^,

^ = ITT + 2 sec 0, y = tan 0 sec 6 - log tan (JTT + J0)

(7).

Line of Symmetry.

The excess of pressure at any point on the anterior face of the
lamina is, by Art. 24 (7),

the constant being chosen so as to make this vanish at the surface
of discontinuity. To find the resulting force on the lamina we

77-78] RESISTANCE OF A LAMINA. 109

must multiply by dx and integrate between the proper limits.
Thus since, at the face of the lamina,

we find

(9).

This result has been obtained on the supposition of special
units of length and time, or (if we choose so to regard the matter)
of a special value (unity) of the general stream-velocity, and a
special value (4 + IT) of the breadth of the lamina. It is evident
from Art. 24 (7), and from the obvious geometrical similarity of
the motion in all cases, that the resultant pressure (P0, say)
will vary directly as the square of the general velocity of the
stream, and as the breadth of the Jamina, so that for an arbitrary
velocity q0, and an arbitrary breadth I, the above result becomes

P»= ^—-pqjl (10)*,

or •{ ' "

78. If the stream be oblique to the lamina, making an angle a,
say, with its plane, the problem is altered in the manner indicated
in the figures.

/

^

The first two steps of the transformation are the same as before, viz.

* Kirchhoff, 1. c. ante p. 102 ; Lord Rayleigh, " On the Resistance of Fluids,"
Phil. Mag., Dec. 1876.

110 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

and we note that for the point I which represents the parts of the stream-line
\I/ = Q for which <£= ±oc , we now have

£= e~* (7r~a), zl = - (IT - a) ?', z.2 = - COS a.
The remaining step is then given by

+ COSa)2=--,

leading to

Along the surface of the lamina we have \^ = 0 and £ real, so that the
corresponding values of <£ range between the limits given by

( - — )—

V W

cosa= ±1.
The resultant pressure is to be found as in Art. 77 from the formula

-^

rf 1 1 — /3 cos a

If we put — - cos a= -- — ,

05 0-cosa '

the limits of /3 are ±1, and the above expression becomes

The relation between A- and /3 for any point of the lamina is given by

-^4 j (1 - 13 cos « + sin « (! -

sin

the origin being chosen so that x shall have equal and opposite values when
j3= ±1, i.e. it is taken at the centre of the lamina. The breadth is therefore,
on the scale of our formulae,

4+rrsina

(iV).

sin a

We infer from (ii) and (iv) that the resultant pressure (P0) on a lamina of
breadth I, inclined at an angle a to the general direction of a stream of
velocity £0, will be

* The solution was carried thus far by Kirchhoff (Crelle, I. c. ) ; the subsequent
discussion is taken substantially from the paper by Lord Rayleigh.

78] RESISTANCE OF A LAMINA. Ill

To find the centre of pressure we take moments about the origin. Thus

the remaining terms under the integral sign being odd functions of /3 and
therefore contributing nothing to the final result. The value of the last
integral is |TT, so that the moment

3 cos a

smda

P X t • A '

4 sin4 a

The first factor represents the total pressure ; the abscissa x of the centre of
pressure is therefore given by the second, or in terms of the breadth,

(vi).

This shews that the point in question is on the up-stream side of the
centre. As a decreases from ^n to 0, x increases from 0 to ^l. Hence if
the lamina be free to turn about an axis in its plane coincident with the
medial line, or parallel to this line at a distance of not more than ^ of the
breadth, the stable position will be that in which it is broadside on to the
stream.

In the following table, derived from Lord Rayleigh's paper, the column I
gives the excess of pressure on the anterior face, in terms of its value when
a = 0; whilst columns II and III give respectively the distances of the centre
of pressure, and of the point where the stream divides, from the middle point
of the lamina, expressed as fractions of the total breadth.

a

I

II

III

90°

1-000

•000

•000

70°

•965

•037

•232

50°

•854

•075

•402

30°

•641

•117

•483

20°

•481

•139

•496

10°

•273

•163

•500

The results contained in column I are in good agreement with some
experiments by Vince (Phil. Trans. 1798).

112 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

79. An interesting variation of the problem of Art. 77 has
been discussed by Bobyleff*. A stream is supposed to impinge
symmetrically on a bent lamina whose section consists of two
equal straight lines forming an angle.

If 2a be the angle, measured on the down-stream side, the boundaries of
the plane of £ can be transformed, so as to have the same shape as in the
Art. cited, by the assumption

f-CTs

provided C and n be determined so as to make £' = 1 when £=--e~l(*"~a\ and
(' = - l when £= e~{ (^+al . This gives

The problem is thus reduced to the former case, viz. we have

Hence for \^ = 0, and 0>$> — 1, we have, putting 0= — 0' as before,

q

The subsequent integrations are facilitated by putting q = t^\ whence

4t

/! i Frvn1
.l^-in

Thus

We have here used the formulae

P r* J -A

+9*--*+*£i+j*

where

Since q = d<f>'/ds, where 8s is an element of a stream-line, the breadth of
either half of the lamina is given by (iii), viz. it is

1+?-"+^

* Journal of the Russian Physico -Chemical Society, t. xiii. (1881) ; Wiedemann's
Bciblntter, t. vi., p. 103.

79]

BOBYLEFF S PROBLEM.

113

The definite integral which occurs in this expression can be calculated from
the formula

Tx+J*(i-i*)-i*(i-i*) (vi)>

i:

where *(£)» = d/dt.\ogU(t)9 is the function introduced and tabulated by
Gauss*

The normal pressure on either half is, by the method of Art. 77,

rifl*

•dt

sin ^nrr
2a2

^ ' ?r sin a '

The resultant pressure in the direction of the stream is therefore

4a2

Hence, for any arbitrary velocity qQ of the stream, and any breadth b of
either half of the lamina, the resultant pressure is

.(viii),

where L stands for the numerical quantity (v).

For a = £TT, we have L — 2 + 577, leading to the same result as in Art. 77(10).

In the following table, taken (with a slight modification) from Bobyleff's
paper, the second column gives the ratio P/P0 of the resultant pressure to

a

PIP,

Pjpq^bsina

P/P0 sin a

10°

•039

•199

•227

20°

•140

•359

•409

30°

•278

•489

•555

40°

•433

•593

•674

45°

•512

•637

•724

50°

•589

•677

•769

60°

•733

•745

•846

70°

•854

•800

•909

80°

•945

•844

•959

90°

1-000

•879

1-000

100°

1-016

•907

1-031

110°

•995

•931

1-059

120°

•935

•950

1-079

130°

•840

•964

1-096

135°

•780

•970

1-103

140°

•713

•975

1-109

150°

•559

•984

1-119

160°

•385

•990

1-126

170°

•197

•996

1-132

* " Disquisitiones generates circa seriem infinitam," Werke, Gottingen, 1870—77,
t. iii., p. 161.

L. 8

MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

that experienced by a plane strip of the same area. This ratio is a maximum
when a = 100°, about, the lamina being then concave on the up-stream side.
In the third column the ratio of P to the distance (26 sin a) between the
edges of the lamina is compared with ^PSo2- For values of a nearly equal
to 180°, this ratio tends to the value unity, as we should expect, since the
fluid within the acute angle is then nearly at rest, and the pressure-excess
therefore practically equal to |p^02- The last column gives the ratio of the
resultant pressure to that experienced by a plane strip of breadth 26 sin a.

80. One remark, applicable to several of the foregoing
investigations, ought not to be omitted here. It will appear at a
later stage in our subject that surfaces of discontinuity are,
as a rule, highly unstable. This instability may, however, be
mitigated by viscosity ; moreover it is possible, as urged by
Lord Rayleigh, that in any case it may not seriously affect the
character of the motion within some distance of the points on the
rigid boundary at which the surfaces in question have their
origin.

Flow in a Curved Stratum.

81. The theory developed in Arts. 59, 60, may be readily
extended to the two-dimensional motion of a curved stratum of
liquid, whose thickness is small compared with the radii of
curvature. This question has been discussed, from the point of
view of electric conduction, by Boltzmann*, Kirchhofff, T6pler§,
and others.

As in Art. 59, we take a fixed point A, and a variable point P,
on the surface defining the form of the stratum, and denote by -fy
the flux across any curve AP drawn on this surface. Then ty is a
function of the position of P, and by displacing P in any direction
through a small distance 8s, we find that the flux across the
element Bs is given by d^r/ds . 8,9. The velocity perpendicular to
this element will be ty/h&s, where h is the thickness of the
stratum, not assumed as yet to be uniform.

If, further, the motion be irrotational, we shall have in addition
a velocity-potential (/>, and the equipotential curves <£ = const, will
cut the stream-lines ty = const, at right angles.

* Wiener Sitzungsberichte, t. lii., p. 214 (1865).

t Berl. Monatsler., July 19, 1875 ; Gen. Abh., p. 56.

§ Pogg. Ann., t. clx., p. 375 (1877).

79-81] FLOW IN A CURVED STRATUM. 115

In the case of uniform thickness, to which we now proceed, it
is convenient to write ^ for -fr/h, so that the velocity perpendicular
to an element &s is now given indifferently by d-^/ds and dcfr/dn,
Sn being an element drawn at right angles to 8s in the proper
direction. The further relations are then exactly as in the plane
problem ; in particular the curves <£ = const., ty = const., drawn for
a series of values in arithmetic progression, the common difference
being infinitely small and the same in each case, will divide the
surface into elementary squares. For, by the orthogonal property,
the elementary spaces in question are rectangles, and if 8^, 8s2 be
elements of a stream-line and an equipotential line, respectively,
forming the sides of one of these rectangles, we have d^jr/ds2
= d<f>/dsl) whence Bs1 = Ss.2, since by construction S^ = 8<£.

Any problem of irrotational motion in a curved stratum (of
uniform thickness) is therefore reduced by orthomorphic projection
to the corresponding problem in piano. Thus for a spherical
surface we may use, among an infinity of other methods, that
of stereographic projection. As a simple example of this, we may
take the case of a stratum of uniform depth covering the surface of
a sphere with the exception of two circular islands (which may be
of any size and in any relative position). It is evident that the
only (two-dimensional) irrotational motion which can take place
in the doubly-connected space occupied by the fluid is one in
which the fluid circulates in opposite directions round the two
islands, the cyclic constant being the same in each case. Since
circles project into circles, the plane problem is that solved in
Art. 64, 2°, viz. the stream-lines are a system of coaxal circles with
real 'limiting points' (A, B, say), and the equipotential lines are
the orthogonal system passing through A, B. Returning to the
sphere, it follows from well-known theorems of stereographic pro
jection that the stream-lines (including the contours of the two
islands) are the circles in which the surface is cut by a system of
planes passing through a fixed line, viz. the intersection of the
tangent planes at the points corresponding to A and B, whilst
the equipotential lines are the circles in which the sphere is cut
by planes passing through these points*.

* This example is given by Kirchhoff, in the electrical interpretation, the
problem considered being the distribution of current in a uniform spherical
conducting sheet, the electrodes being situate at any two points A, B of the surface.

8—2

116 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV

In any case of transformation by orthomorphic projection,
whether the motion be irrotational or not, the velocity (dtyjdn) is
transformed in the inverse ratio of a linear element, and therefore
the kinetic energies of the portions of the fluid occupying corre
sponding areas are equal (provided, of course, the density and
the thickness be the same). In the same way the circulation
(fd^/dn.ds) in any circuit is unaltered by projection.

CHAPTER V.

IRROTATIONAL MOTION OF A LIQUID : PROBLEMS IN
THREE DIMENSIONS.

82. OF the methods available for obtaining solutions of the
equation

V24> = 0 .............................. (1),

in three dimensions, the most important is that of Spherical
Harmonics. This is especially suitable when the boundary condi
tions have relation to spherical or nearly spherical surfaces.

For a full account of this method we must refer to the special
treatises*, but as the subject is very extensive, and has been
treated from different points of view, it may be worth while to
give a slight sketch, without formal proofs, or with mere indica
tions of proofs, of such parts of it as are most important for our
present purpose.

It is easily seen that since the operator V2 is homogeneous
with respect to #, y, z> the part of </> which is of any specified
algebraic degree must satisfy (1) separately. Any such homo
geneous solution of (1) is called a 'solid harmonic' of the algebraic
degree in question. If <£n be a solid harmonic of degree n, then
if we write

(2),

* Todhunter, Functions of Laplace, &c.y Cambridge, 1875. Ferrers, Spherical
Harmonics, Cambridge, 1877. Heine, Handbiich tier Knyelfunctionen, 2nd ed.,
Berlin, 1878. Thomson and Tait, Natural Philosophy, 2nd ed., Cambridge, 1879,
t. i., pp. 171—218.

For the history of the subject see Todhunter, History of the Theories of Attrac
tion, <#c., Cambridge, 1873, t. ii.

118 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

Sn will be a function of the direction (only) in which the point
(x, y, z) lies with respect to the origin ; in other words, a function
of the position of the point in which the radius vector meets a
unit sphere described with the origin as centre. It is therefore
called a ' surface-harmonic ' of order n.

To any solid harmonic (j>n of degree n corresponds another of
degree —n—I, obtained by division by r2n+l; i.e. </> = r~zn~l^>n is
also a solution of (1). Thus, corresponding to any surface har
monic Sn, we have the two solid harmonics rnSn and r~n~l8n.

83. The most important case is when n is integral, and when
the surface-harmonic Sn is further restricted to be finite over the
unit sphere. In the form in which the theory (for this case) is
presented by Thomson and Tait, and by Maxwell*, the primary
solution of (1) is

<t>-* = A/r (3).

This represents as we have seen (Art. 56) the velocity-potential
due to a point-source at the origin. Since (1) is still satisfied
when <j) is differentiated with respect to x, y, or zt we derive a
solution

IV /4\

dx^ "" dy^ '" dz) r "

This is the velocity-potential of a double-source at the origin,
having its axis in the direction (I, m, n}. The process can be
continued, and the general type of solid harmonic obtainable in
this way is

dn 1

, d j d d d

where -^ = 18 ,- + ms -r + ns -j- ,

dris dx dy dz

ls, ms, ns being arbitrary direction-cosines.

This may be regarded as the velocity-potential of a certain
configuration of simple sources about the origin, the dimensions
of this system being small compared with r. To construct this
system we premise that from any given system of sources we may

* Electricity and Magnetism, c. ix.

82-83] SPHERICAL HARMONICS. 119

derive a system of higher order by first displacing it through a
space ^hs in the direction (18, ms, ns), and then superposing the
reversed system, supposed displaced from its original position
through a space ^hs in the opposite direction. Thus, beginning
with the case of a simple source 0 at the origin, a first application
of the above process gives us two sources 0+, 0_ equidistant from
the origin, in opposite directions. The same process applied to the
system 0+) 0_ gives us four sources 0+ + , 0_+, 0+_, 0 — at the
corners of a parallelogram. The next step gives us eight sources at
the corners of a parallelepiped, and so on. The velocity-potential,
at a distance, due to an arrangement of 2n sources obtained in
this way, will be given by (5), where A = m'hji,z...hn, m! being the
strength of the original source at 0. The formula becomes exact,
for all distances r, when hly h^...hn are diminished, and m in
creased, indefinitely, but so that A is finite.

The surface-harmonic corresponding to (5) is given by

(6),

,, , „
dh1dh.2...dhnr

and the complementary solid harmonic by

^ .................. (7).

By the method of ' inversion *,' applied to the above configura
tion of sources, it may be shewn that the solid harmonic (7) of
positive degree n may be regarded as the velocity-potential due
to a certain arrangement of 2n simple sources at infinity.

The lines drawn from the origin in the various directions
(h, ms, ns) are called the 'axes' of the solid harmonic (5) or (7),
and the points in which these lines meet the unit sphere are
called the ' poles ' of the surface harmonic 8n. The formula (5)
involves 2n + 1 arbitrary constants, viz. the angular co-ordinates
(two for each) of the n poles, and the factor A. It can be shewn
that this expression is equivalent to the most general form of
surface-harmonic which is of integral order n and finite over the
unit sphere f.

* Explained by Thomson and Tait, Natural Philosophy, Art. 515.
t Sylvester, Phil. Mag., Oct. 1876.

120 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

84. In the original investigation of Laplace*, the equation
V20 = 0 is first expressed in terms of spherical polar coordinates
r, 6, a), where

x = r cos #, y = r sin 9 cos &>, z = r sin 6 sin &>.

The simplest way of effecting the transformation is to apply
the theorem of Art. 37 (2) to the surface of a volume-element
r$0 . r sin #£&> . 8r. Thus the difference of flux across the two
faces perpendicular to r is

-..

dr \dr

Similarly for the two faces perpendicular to the meridian (&>= const.)
we find

d d() ...

and for the two faces perpendicular to a parallel of latitude
(6 = const.)

sin 6da)
Hence, by addition,

. a d ! ^d6\ d ( . ad<f)\ , 1 &<j> ,tv

sin 0 -y- (r 2 ^- + -77; I sm 0 -£ + -. — 7, -^ = 0 . . .(1).
dr \ dr) dB \ dOJ sm 0 da?

This might of course have been derived from Art. 82 (1) by the
usual method of change of independent variables.

If we now assume that <f> is homogeneous, of degree ?i, and put

we obtain

1 d / . ~dSn\ 1 d~Sn
d d0

n\ 1 d~Sn
}+ sin"* «^

which is the general differential equation of spherical surface-
harmonics. Since the product n (n + 1) is unchanged in value
when we write — n — 1 for n, it appears that

0 = r-n-i£n

will also be a solution of (1), as already stated.

* " Th^orie de 1'attraction des sph6roides et de la figure des planetes," Mem.
de VAcad. roy. des Sciences, 1782; Oeuvres Completes, Paris, 1878..., t. x., p. 341 ;
Mecanique Celeste, Livre 2mc, c. ii.

84-85] SPHERICAL HARMONICS. 121

85. In the case of symmetry about the axis of x, the term
d2Sn/dco2 disappears, and putting cos 6 = p, we get

the differential equation of 'zonal' harmonics*. This equation,
containing only terms of two different dimensions in //,, is adapted
for integration by series. We thus obtain

;i = ^{1_'H^^ + (»-2l»(» + lK» + 3)^_^J

The series which here present themselves are of the kind
called ' hypergeometric ' ; viz. if we write, after Gauss -f*,

.....

1.2.8.7.7+1.7+2
we have

Sn = AF(- £n, i + in, 1 ^) + ^^(J- - f/i, 1 + \n, f , A62)...(4).

The series (3) is of course essentially convergent when x lies between 0
and 1 ; but when ^=1 it is convergent if, and only if

y-a-/3>0.
In this case we have

F(n 8 ^ n

1 (a' ^ ^' 1)=

where n (z) is in Gauss's notation the equivalent of Euler's r (£+!)•
The degree of divergence of the series (3) when

y-a-/3<0,
as x approaches the value 1, is given by the theorem

F(o, fty, ^ = (1-^—^^(7-0, y-8, y, x)

* So called by Thomson and Tait, because the nodal lines (Sn = 0) divide the
unit sphere into parallel belts.
t I.e. ante p. 113.
£ Forsyth, Differential Equations, London, 1885, c. vi.

122 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

Since the latter series will now be convergent when # = 1, we see that

#(****)

becomes divergent as (1 — x}^~a'~^ ;

more precisely, for values of x infinitely nearly equal to unity, we have

* (to PI y,*)

ultimately.

For the critical case where y - a - 8 =0,
we may have recourse to the formula

^ (a, fty, X)-— (a+1,
which, with (ii), gives in the case supposed

y, *)-(l~.«)r? . ^(y-a, y-0,

The last factor is now convergent when x=\, so that ^(o, 8, y, a;) is
ultimately divergent as log (! — #)• More precisely we have, for values of x
near this limit,

86. Of the two series which occur in the general expression
Art. 85 (2) of a zonal harmonic, the former terminates when n is
an even, and the latter when n is an odd integer. For other
values of n both series are essentially convergent for values of /u,
between + 1, but since in each case we have 7 — a — /3 = 0, they
diverge at the limits fi=±l, becoming infinite as log(l — //r).

It follows that the terminating series corresponding to integral
values of n are the only zonal surface-harmonics which are finite
over the unit sphere. If we reverse the series we find that both
these cases (n even, and n odd,) are included in the formula

1-8-6-<2»-1>

2.4.(2n-l)(2)l-3)

* For n even this corresponds to A = ( - )£n ' — ~J - , B = 0 ; whilst for n

2i • 4 . . . 7Z-

odd we have 4 = 0, B = ( - )*<»-i> ''^ • See Heine. *• [- » PP- 12> 147-

85-86] ZONAL HARMONICS. 123

where the constant factor has been adjusted so as to make
Pn fa) = 1 for //, = !. The formula may also be written

1 rJn

p<^=^-vn ............... <2>*

The series (1) may otherwise be obtained by development of
Art. 83 (6), which in the case of the zonal harmonic assumes the
form

As particular cases of (2) we have

The function Pn(p} was first introduced into analysis by
Legendref as the coefficient of hn in the expansion of

(1 - 2ph + h*)-*.

The connection of this with our present point of view is that if
</> be the velocity-potential of a unit source on the axis of x at
a distance c from the origin, we have, on Legendre's definition,
for values of r less than c,

0 = (C3 _ 2/icr + r2)~i

Each term in this expansion must separately satisfy V20 = 0, and
therefore the coefficient Pn must be a solution of Art. 85 (1).
Since Pn is obviously finite for all values of /it, and becomes equal
to unity for /* = 1, it must be identical with (1).

For values of r greater than c, the corresponding expansion is

* The functions P1,P.2,...P7 have been tabulated by Glaisher, for values of /j.
at intervals of -01, Brit. Ass. Reports, 1879.

+ " Sur 1'attraction des spheroides homogenes," Mem. des Savans Etrangers, t. x.,
1785.

124 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

We can hence deduce expressions, which will be useful to us
later, for the velocity-potential due to a double-source of unit
strength, situate on the axis of a? at a distance c from the origin,
and having its axis pointing from the origin. This is evidently
equal to dcfr/dc, where </> has either of the above forms; so that
the required potential is, for r < c,

and for r > c,

The remaining solution of Art. 85 (1), in the case of n integral,
can be put into the more compact form*

where

This function Qn (yu,) is sometimes called the zonal harmonic ' of
the second kind.'

Thus

- 3/t) log } + £-

2 4 n

* This is equivalent to Art. 84 (4) with, for n even, A = 0, B = ( - )i» — ."' ;

1 . o. ..(/I — 1;

whilst for n odd we have ^^(-)^(«+D2- tl'"^n~1- , B = 0. See Heine, t. i.,
pp. 141, 147.

86-87] TESSERAL HARMONICS. 125

87. When we abandon the restriction as to symmetry about
the axis of x, we may suppose 8n, if a finite and single-valued
function of &>, to be expanded in a series of terms varying as cos so)
and sin sw respectively. If this expansion is to apply to the whole
sphere (i.e. from o> = 0 to o> = 2?r), we may further (by Fourier's
theorem) suppose the values of s to be integral. The differential
equation satisfied by any such term is

If we put
this takes the form

which is suitable for integration by series. We thus obtain

(n-s-2)(n-s)(n+s+l)(n+s+3) 4
1.2.3.4 M

1.2.3,4.5

_
"-"

the factor cos sa> or sin sea being for the moment omitted. In the
hypergeometric notation this may be written

J5- iw, 1 + J* + iw, f , p?)} ...... (3).

These expressions converge when p? < 1, but since in each
case we have

the series become infinite as (1 — p?}~8 at the limits //,= + 1, unless
they terminate*. The former series terminates when n— s is an
even, and the latter when it is an odd integer. By reversing the

* Lord Rayleigh, Theory of Sound, London, 1877, Art. 338.

126 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

series we can express both these finite solutions by the single
formula

----- 4

2 . 4 . (2n - 1) (2n - 3)

On comparison with Art. 86 (1) we find that

.(4).

That this is a solution of (1) may of course be verified indepen
dently.

Collecting our results we learn that a surface-harmonic which
is finite over the unit sphere is necessarily of integral order, and is
further expressible, if n denote the order, in the form

Sn = A0Pn O) + 2SS^(AS cos 50) + Bs sin sa>) Tn* (/a)... (6),

containing 2n + 1 arbitrary constants. The terms of this involving
<w are called ' tesseral ' harmonics, with the exception of the last
two, which are given by the formula

(1 — fj?y*n (An cos no) + Bn sin nco),

and are called ' sectorial ' harmonics ; the names being suggested
by the forms of the compartments into which the unit sphere is
divided by the nodal lines 8n = 0.

The formula for the tesseral harmonic of rank s may be
obtained otherwise from the general expression (6) of Art. 83
by making n — s out of the n poles of the harmonic coincide at
the point 6 = 0 of the sphere, and distributing the remaining s
poles evenly round the equatorial circle Q = \ir.

The remaining solution of (1), in the case of n integral may be
put in the form

Sn = (As cos so) + B8 sin sco) Uns (/i) ............ (7),

where Z7W- (/,) = (1 - ^^ .................. (8)*.

This is sometimes called a tesseral harmonic ' of the second kind.'

* A table of the functions Qn (/j.), Un* (/*), for various values of n and s, has been
given by Bryan, Proc. Camb. Phil. Soc., t. vi., p. 297.

87-89] CONJUGATE PROPERTY. 127

88. Two surface-harmonics S, S' are said to be ' conjugate ' when
ffSS'dv = 0 (1),

where SOT is an element of surface of the unit sphere, and the
integration extends over this sphere.

It may be shewn that any two surface-harmonics, of different
orders, which are finite over the unit sphere, are conjugate, and also
that the 2n + 1 harmonics of any given order n, of the zonal, tes-
seral, and sectorial types specified in Arts. 86, 87 are all mutually
conjugate. It will appear, later, that the conjugate property is
of great importance in the physical applications of the subject.

Since SOT = sin 6&6$a) = — S//.Sa>, we have, as particular cases of
this theorem,

f Pm(M)<^ = 0 (2),

0 (3),

and I Tm* (LL) . Tn* (/x) dp = 0 (4),

J -i

provided m, n are unequal.

For m — n, it may be shewn that

•i 9

Finally, we may quote the theorem that any arbitrary
function of the position of a point on the unit sphere can be
expanded in a series of surface-harmonics, obtained by giving n
all integral values from 0 to oo , in Art. 87 (6). The formulae (5)
and (6) are useful in determining the coefficients in this expansion.
For the analytical proof of the theorem we must refer to the
special treatises ; the physical grounds for assuming the possibility
of this and other similar expansions will appear, incidentally, in
connection with various problems.

89. As a first application of the foregoing theory let us
suppose that an arbitrary distribution of impulsive pressure is
applied to the surface of a spherical mass of fluid initially at rest.

* Ferrers, p. 86.

128 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

This is equivalent to prescribing an arbitrary value of <f> over the
surface ; the value of <£ in the interior is thence determinate,
by Art. 40. To find it, we may suppose the given surface value
to be expanded, in accordance with the theorem quoted in Art. 88,
in a series of surface-harmonics of integral order, thus

0 = S0 + S1 + flf3+...+ Sn + .................. (1).

The required value is then

<j, = Sa+r-S,+r~X2+...+r^Sn+ ............ (2),

for this satisfies V2</>=0, and assumes the prescribed form (1)
when r — a, the radius of the sphere.

The corresponding solution for the case of a prescribed value
of (f> over the surface of a spherical cavity in an infinite mass of
liquid initially at rest is evidently

a a* a3 an+l „

Combining these two results we get the case of an infinite
mass of fluid whose continuity is interrupted by an infinitely thin
vacuous stratum, of spherical form, within which an arbitrary
impulsive pressure is applied. The values (2) and (3) of <j> are of
course continuous at the stratum, but the values of the normal
velocity are discontinuous, viz. we have, for the internal fluid,

and for the external fluid

g = -2(» + !)«„/«.

The motion, whether internal or external, is therefore that due
to a distribution of simple sources with surface-density

.......................

4?r ' a

over the sphere. See Art. 58.

90. Let us next suppose that, instead of the impulsive
pressure, it is the normal velocity which is prescribed over the
spherical surface; thus

............... (l),

89-90] SPHERICAL BOUNDARY. 129

the term of zero order being necessarily absent, since we must
have

on account of the constancy of volume of the included mass.
The value of </> for the internal space is of the form

MSn + ......... (3),

for this is finite and continuous, and satisfies V2<£ = 0, and the
constants can be determined so as to make d(f>/dr assume the
given surface-value (1); viz. we have nAnan~l = \. The required
solution is therefore

The corresponding solution for the external space is found in
like manner to be

The two solutions, taken together, give the motion produced
in an infinite mass of liquid which is divided into two portions
by a thin spherical membrane, when a prescribed normal velocity is
given to every point of the membrane, subject to the condition (2).

The value of cf> changes from a£Sn/n to — aZSn/(n +1),
as we cross the membrane, so that the tangential velocity is now
discontinuous. The motion, whether inside or outside, is that
due to a double-sheet of density

See Art. 58.

The kinetic energy of the internal fluid is given by the
formula (4) of Art. 44, viz.

*r ............ (6),

the parts of the integral which involve products of surface-
harmonics of different orders disappearing in virtue of the
conjugate property of Art. 88.

L. 9

130 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

For the external fluid we have

*dK ...... (7).

91. A particular, but very important, case of the problem of
the preceding Article is that of the motion of a solid sphere in an
infinite mass of liquid which is at rest at infinity. If we take
the origin at the centre of the sphere, and the axis of x in the
direction of motion, the normal velocity at the surface is
u%/r, = u cos 0, where u is the velocity of the centre. Hence
the conditions to determine (/> are (1°) that we must have V2<£ = 0
everywhere, (2°) that the space-derivatives of $ must vanish at
infinity, and (3°) that at the surface of the sphere (r = a), we must
have

.. ..(1).

dr

The form of this suggests at once the zonal harmonic of the first
order ; we therefore assume

, d 1 , cos 6

$ = ^-T- - = -A — r.
ax r r-

The condition (1) gives — 2A/a? = u, so that the required solution
is $ = Ju£cos0 ..................... (2)*.

It appears on comparison with Art. 56 (4) that the motion of
the fluid is the same as would be produced by a double-source of
strength -Jua3, situate at the centre of the sphere. For the forms
of the stream-lines see p. 137.

To find the energy of the fluid motion we have

T

cos20 . 27ra sin d . add

(3),

if m' = f 7r/oa3. It appears, exactly as in Art. 68, that the effect of
the fluid pressure is equivalent simply to an addition to the inertia

* Stokes, " On some cases of Fluid Motion," Camb. Trans, t. viii. (1843) ;
Math, and Pliys. Papers, t. i., p. 41.

Dirichlet, " Ueber einige Falle in welchen sich die Bewegung eines festen Korpers
in einem incompressibeln fliissigen Medium theoretisch bestimmen liisst," Berl.
Monatsber., 1852.

90-91] MOTION OF A SPHERE. 131

of the solid, the amount of the increase being now half the mass
of the fluid displaced*.

Thus in the case of rectilinear motion of the sphere, if no
external forces act on the fluid, the resultant pressure is equiva
lent to a force

in the direction of motion, vanishing when u is constant. Hence
if the sphere be set in motion and left to itself, it will continue to
move in a straight line with constant velocity.

The behaviour of a solid projected in an actual fluid is of
course quite different ; a continual application of force is necessary
to maintain the motion, and if this be not supplied the solid is
gradually brought to rest. It must be remembered however, in
making this comparison, that in a ' perfect ' fluid there is no
dissipation of energy, and that if, further, the fluid be incompres
sible, the solid cannot lose its kinetic energy by transfer to the
fluid, since, as we have seen in Chapter in., the motion of the
fluid is entirely determined by that of the solid, and therefore
ceases with it.

If we wish to verify the preceding results by direct calculation from the
formula

we must remember, as in Art. 68, that the origin is in motion, and that the
values of r and 6 for a fixed point of space are therefore increasing at the
rates - u cos 6, and u sin 0/rt respectively. We thus find, for r = a,

(ii).

The last three terms are the same for surface-elements in the positions 6 and
TT- 6 ; so that, when u is constant, the pressures on the various elements of the
anterior half of the sphere are balanced by equal pressures on the correspond
ing elements of the posterior half. But when the motion of the sphere is
being accelerated there is an excess of pressure on the anterior, and a defect of
pressure on the posterior half. The reverse holds when the motion is being
retarded. The resultant effect in the direction of motion is

%7ra sin 6 . ad6 . p cos 0,
o

which is readily found to be equal to - ^npa3 du/dt, as before.

* Green, "On the Vibration of Pendulums in Fluid Media," Edin. Trans.,
1833 ; Math. Papers, p. 322. Stokes, I c.

9—2

132 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

92. The same method can be applied to find the motion
produced in a liquid contained between a solid sphere and a fixed
concentric spherical boundary, when the sphere is moving with
given velocity u.

The centre of the sphere being taken as origin, it is evident,
since the space occupied by the fluid is limited both externally
and internally, that solid harmonics of both positive and negative
degrees are admissible; they are in fact required, in order to
satisfy the boundary conditions, which are

— d$/dr = u cos 0,

for r = a, the radius of the sphere, and

d<j>jdr = 0,

for r = b, the radius of the external boundary, the axis of x being
as before in the direction of motion.

We therefore assume
and the conditions in question give

whence A = r. -u, B = J — -u (2).

t>3 — a3 bB — a3

The kinetic energy of the fluid motion is given by

Wmt-ollt^dS,

the integration extending over the inner spherical surface, since
at the outer we have d^jdr = 0. We thus obtain

where m' stands for f Trpa3, as before. It appears that the effective
addition to the inertia of the sphere is now

* Stokes, 7. c. ante p. 130.

92-93] SPHERE WITH CONCENTRIC BOUNDARY. 133

As b diminishes from oo to a, this increases continually from
to oo , in accordance with Lord Kelvin's minimum theorem (Art. 45).
In other words, the introduction of a rigid spherical partition in an
infinite mass of liquid acts as a constraint increasing the kinetic
energy for a given velocity, and so virtually increasing the inertia
of the system.

93. In all cases where the motion of a liquid takes place in a
series of planes passing through a common line, and is the same in
each such plane, there exists a stream -function analogous in some
of its properties to the two-dimensional stream-function of the
last Chapter. If in any plane through the axis of symmetry we
take two points A and P, of which A is arbitrary, but fixed, while
P is variable, then considering the annular surface generated by
any line AP, it is plain that the flux across this surface is a
function of the position of P. Denoting this function by 27n|r,
and taking the axis of x to coincide with that of symmetry, we
may say that ^r is a function of x and w, where x is the abscissa of
P, and w, = (y^ + z2)^, is its distance from the axis. The curves
i|r = const, are evidently stream-lines.

If P' be a point infinitely near to P in a meridian plane, it
follows from the above definition that the velocity normal to PP'
is equal to

27TOT.PP"

whence, taking PP' parallel first to -or and then to x,

1 e 1 e

J~ .................. (1),

dx

where u and u are the components of fluid velocity in the directions
of x and -BT respectively, the convention as to sign being similar to
that of Art. 59.

These kinematical relations may also be inferred from the
form which the equation of continuity takes under the present
circumstances. If we express that the total flux into the annular
space generated by the revolution of an elementary rectangle
is zero, we find

S«T = 0,

134 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

which shews that tzru . dx — vru . diz

is an exact differential. Denoting this by d^jr we obtain the
relations (1)*.

So far the motion has not been assumed to be irrotational ;
the condition that it should be so is

dv du
dx dtp
which leads to

The differential equation of (f> is obtained by writing

dd> dd>

u =• — -p- , v = —

dx d^

in (2), viz. it is *+* + ?. 1=0 (4).

It appears that the functions </> and i/r are not now (as they were
in Art. 62) interchangeable. They are, indeed, of different dimen
sions.

The kinetic energy of the liquid contained in any region
bounded by surfaces of revolution about the axis is given by

**•--•<«

--<•//*

(5),

Bs denoting an element of the meridian section of the bounding
surfaces, and the integration extending round the various parts of
this section, in the proper directions. Compare Art. 61.

* The stream-function for the case of symmetry about an axis was introduced
in this manner by Stokes, "On the Steady Motion of Incompressible Fluids,"
Canib. Trans., t. vii. (1842) ; Math, and Phys. Papers, t. i., p. 14. Its analytical
theory has been treated very fully by Sampson, "On Stokes' Current-Function,"
Phil. Trans. A., 1891.

93-94] SYMMETRY ABOUT AN AXIS : STREAM-FUNCTION. 135

94. The velocity-potential due to a unit source at the origin
is

* = l/r .............................. (1).

The flux through any closed curve is in this case numerically equal
to the solid angle which the curve subtends at the origin. Hence
for a circle with Ox as axis, whose radius subtends an angle 6 at 0,
we have, attending to the sign,

Omittin the constant term we have

•f = -=5r (2).

r dx

The solutions corresponding to any number of simple sources
situate at various points of the axis of x may evidently be super
posed ; thus for the double-source

, _ d 1 _ cos 6

dx r r'2 ^

i i uTT Tjf" Sin" (j

we have ^r = — .— = =

r r

And, generally, to the zonal solid harmonic of degree -?i — 1,
viz. to

(r

corresponds ^ = A - — - (6)*

dxn+l

A more general formula, applicable to harmonics of any
degree, fractional or not, may be obtained as follows. Using
spherical polar coordinates r, 0, the component velocities along
r, and perpendicular to r in the plane of the meridian, are
found by making the linear element PP' of Art. 93 coincide
successively with rSO and 8r, respectively, viz. they are

r sin 6 rdO ' r sin 6 dr

* Stefan, " Ueber die Kraftlinieii eines um erne Axe symmetrischen Feldes,"
Wied. Ann., t. xvii. (1882).

136 PROBLEMS IN THREE DIMENSIONS [CHAP. V

Hence in the case of irrotational motion we have

smM0~ dr' dr~ M '

Thus if $ = rnSn .............................. (9),

where Sn is any surface-harmonic symmetrical about the axis, we
have, putting //, = cos 6,

d* _ nrn+i S d_±- rn Q _. ^ d8»

d^~ *n' dr~ ^}d^'

The latter equation gives

which must necessarily also satisfy the former; this is readily
verified by means of Art. 85 (1).

Thus in the case of the zonal harmonic Pn, we have as
corresponding values

and * = r— 'PBGt), * =- "(l -tf ...... (12),

of which the latter must be equivalent to (5) and (6). The same
relations hold of course with regard to the zonal harmonic of the
second kind, Qn.

95. We saw in Art. 91 that the motion produced by a solid
sphere in an infinite mass of liquid was that due to a double-
source at the centre. Comparing the formula? there given with
Art. 94 (4), it appears that the stream-function due to the
sphere is

^ = -iU-sin20 ........................ (1).

The forms of the stream-lines corresponding to a number of equidistant
values of \jf are shewn on the opposite page. The stream-lines relative to the
sphere are figured in the diagram near the end of Chapter vii.

94-95]

ST11EAM-L1NES DUE TO A SPHERE.

137

Again, the stream-function due to two double-sources having
their axes oppositely directed along the axis of xt will be of the
form

where rly r.2 denote the distances of any point from the positions,
P! and P2, say, of the two sources. At the stream-surface i/r = 0
we have

138 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

i.e. the surface is a sphere in relation to which P1 and P2 are
inverse points. If 0 be the centre of this sphere, and a its radius,
we readily find

.................. (3).

This sphere may evidently be taken as a fixed boundary to the
fluid on either side, and we thus obtain the motion due to a
double-source (or say to an infinitely small sphere moving along
Ox) in presence of a fixed spherical boundary. The disturbance
of the stream-lines by the fixed sphere is that due to a double-
source of the opposite sign placed at the ' inverse ' point, the ratio
of the strengths being given by (3)*. This fictitious double-
source may be called the ' image ' of the original one.

96. Rankine employed -f- a method similar to that of Art. 71
to discover forms of solids of revolution which will by motion
parallel to their axes generate in a surrounding liquid any given
type of irrotational motion symmetrical about an axis.

The velocity of the solid being u, and 8s denoting an element
of the meridian, the normal velocity at any point of the surface is
udvr/ds, and that of the fluid in contact is given by —d^/^ds.
Equating these and integrating along the meridian, we have

•v/r = - lutzr2 + const ...................... (1).

If in this we substitute any value of ^ satisfying Art. 93 (3), we
obtain the equation of the meridian curves of a series of solids,
each of which would by its motion parallel to x give rise to
the given system of stream-lines.

In this way we may readily verify the solution already obtained
for the sphere ; thus, assuming

+ = Au*/r* ........................... (2),

we find that (1) is satisfied for r = a, provided

A = -±ua* .......................... (3),

which agrees with Art. 95 (1).

* This result was given by Stokes, " On the Resistance of a Fluid to Two Oscil
lating Spheres," Brit. Ass. Report, 1847 ; Math, and Plnjs. Papers, t. i., p. 230.

t "On the Mathematical Theory of Stream-Lines, especially those with Four
Foci and upwards," Phil. Trans. 1871, p. 267.

95-97] RANKINE'S METHOD. 139

97. The motion of a liquid bounded by two spherical surfaces
can be found by successive approximations in certain cases. For
two solid spheres moving in the line of centres the solution is
greatly facilitated by the result given at the end of Art. 95, as to
the ' image ' of a double-source in a fixed sphere.

Let A, B be the centres, and let u be the velocity of A towards B, u' that
of B towards A. Also, P being any point, let AP = rt BP = r', FAB =6,
PBA = & '. The velocity-potential will be of the form

U<£ + U'<£' (i),

where the functions 0 and <£' arc to be determined by the conditions that
V20 = 0, v2<£' = 0 (ii),

throughout the fluid ; that their space-derivatives vanish at infinity ;
and that

( ,...N

=£-0 ...... , .................... (m),

over the surface of A, whilst

over the surface of B. It is evident that <p is the value of the velocity-
potential when A moves with unit velocity towards /?, while B is at rest ; and
similarly for <$>'.

To find dj, we remark that if B were absent the motion of the fluid would
be that due to a certain double-source at A having its axis in the direction
AB. The theorem of Art. 95 shews that we may satisfy the condition of zero
normal velocity over the surface of B by introducing a double-source, viz. the
'image' of that at A in the sphere B. This image is at fflJ the inverse
point of A with respect to the sphere B ; its axis coincides with AB, and its
strength /^ is given by

where /*, =£a3, is that of the original source at A. The resultant motion due
to the two sources at A and Hl will however violate the condition to be

PROBLEMS IN THREE DIMENSIONS.

[CHAP,

satisfied at the surface of the sphere J, and in order to neutralize the normal
velocity at this surface, due to Hlt we must superpose a double-source at H%,
the image of H^ in the sphere A. This will introduce a normal velocity at the
surface of B, which may again be neutralized by adding the image of H2 in 2?,
and so on. If p,^ ju2, /*„... be the strengths of the successive images, and
fit /2»/3»'" then* distances from A, we have, if AB=c,

•(V),

and so on, the law of formation being now obvious. The images continually
diminish in intensity, and this very rapidly if the radius of either sphere is
small compared with the shortest distance between the two surfaces.

The formula for the kinetic energy is

provided

where the suffixes indicate over which sphere the integration is to be effected.
The equality of the two forms of Jf follows from Green's Theorem (Art. 44.)

The value of 0 near the surface of ^1 can be written down at once from the
results (6) and (7) of Art. 86, viz. we have

the remaining terms, involving zonal harmonics of higher orders, being omitted,
as they will disappear in the subsequent surface-integration, in virtue of the
conjugate property of Art. 88. Hence, putting d(f>/dn= — cos$, we find with
the help of (v)

afibG

a363
1+3 -o>,+3 --,

It appears that the inertia of the sphere A is in all cases increased by the
presence of a fixed sphere B. Compare Art. 92.

97] MOTION OF TWO SPHERES.

The value of N may be written down from symmetry, viz. it is

*-)

141

where

and so on.

To calculate M we require the value of 0' near the surface of the sphere A \
this is due to double-sources //, p,/, p2'> Psi-- a^ distances c, c—f^ c — /"2',
c-/3', ... from A, where

(xii),

and so on. This gives, for points near the surface of .4
</)/ = C/^i' + /*s' + /*B' + • • 0 .2^

Hence

L CT3&3 «6&6 I

^(^/)3V/3^37^2T(^^

When the ratios a/c and 6/c are both small we have

approximately.

(xv),*

* To this degree of approximation these results may also be obtained by the
method of the next Art.

142 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

If a, but not necessarily 6, is small compared with the shortest distance
between the spherical surfaces, we have

approximately. By putting c — b + k, and then making I = oo , we get the
formula for a sphere moving perpendicularly to a fixed plane wall at a
distance h, viz.

tf ........................ (xvii),

a result due to Stokes.

This also follows from (vi) and (xv), by putting b = a, u' = U, c=2k, in which
case the plane which bisects AB at right angles is evidently a plane of
symmetry, and may therefore be taken as a fixed boundary to the fluid on
either side.

98. When the spheres are moving at right angles to the line
of centres the problem is more difficult; we shall therefore content
ourselves with the first steps in the approximation, referring, for a
more complete treatment, to the papers cited below.

Let the spheres be moving with velocities v, v' in parallel directions at
right angles to A, B, and let ?*, $, o> and /, 6 ', «' be two systems of spherical
polar coordinates having their origins at A and B respectively, and their polar
axes in the directions of the velocities v, v'. As before the velocity-potential
will be of the form

with the surface conditions

and f, = 0, f=-cos«',for/=i.

If the sphere B were absent the velocity-potential due to unit velocity of
A would be

Since r cos B=r' cos 6', the value of this in the neighbourhood of B will be

approximately. The normal velocity due to this will be cancelled by the
addition of the term

97-99] MOTION OF TWO SPHERES. 143

which, in the neighbourhood of A becomes equal to

, «363

J— 6-rcos<9,

nearly. To rectify the normal velocity at the surface of A , we add the term

T a663 cos d
s -jT ~^- '

Stopping at this point, and collecting our results, we have, over the surface
of A,

and at the surface of Z?,

Hence if we denote by P, $, R the coefficients in the expression for the
kinetic energy, viz.

r r -,7j. / ™.T7,s\ \

we have P— —

/ J

The case of a sphere moving parallel to a fixed plane boundary, at a
distance A, is obtained by putting b = a, v = v', c = 2A, and halving the conse
quent value of T ; thus

This result, which was also given by Stokes, may be compared with that of
Art. 97 (xvii)*.

99. Another interesting problem is to calculate the kinetic
energy of any given irrotational motion in a cyclic space bounded
by fixed walls, as disturbed by a solid sphere moving in any
manner, it being supposed that the radius of the sphere is small

* For a fuller analytical treatment of the problem of the motion of two spheres
we refer to the following papers : W. M. Hicks, " On the Motion of Two Spheres in
a Fluid," Phil. Trans., 1880, p. 455; E. A. Herman, " Cn the Motion of Two
Spheres in Fluid," Quart. Journ. Math., t. xxii. (1887). See also C. Neumann,
Hydrodynamische Untersuchungen, Leipzig, 1883.

144 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

in comparison with the distance from it of the nearest portion of
the original boundary.

Let 0 be the velocity-potential of the motion when the sphere is absent,
and Kj, K2,,.. the circulations in the various circuits. The kinetic energy of
the original motion is therefore given by Art. 55 (5), viz.

where the integrations extend over the various barriers, drawn as in Art. 48.
If we denote by 0 + $' the velocity-potential in presence of the sphere, and
by T the energy of the actual motion, we have

the cyclic constants of $' being zero. The integration in the first term may
be confined to the surface of the sphere, since we have d<f)/dn = 0 and d(f>'/dn = 0
over the original boundary. Now, by Art. 54 (4),

so that (ii) reduces to

•» _ ~ I I ^' — (9 A j_ ^ /7^ _ „ / / A ~5?

cfoi

Let us now take the centre of the sphere as origin. Let a be the radius of
the sphere, and u, v, W the components of its velocity in the directions of the
coordinate axes ; further, let w0, v0, WQ be the component velocities of the fluid
at the position of the centre, when the sphere is absent. Hence, in the
neighbourhood of the sphere, we have, approximately

where the coefficients A, B, C are to be determined by the condition that

for r = a. This gives

Again, -2-^- !?= (% + u) + -(^o + v) + - (^0 + w) (v^)>

when r=a. Hence, substituting from (iv), (v), and (vi), in (iii), and re
membering that SJx*dS= % a2 . 4-rra2, jjyzdS=Ot &c., &c.,
we find

99-100] ELLIPSOIDAL HARMONICS. 145

The dynamical consequences of the formula (vii) will be considered more
fully in Art. 140 ; but in the meantime we may note that if the sphere be
held at rest, so that u, V, W = 0, it experiences a force tending to diminish the
energy of the system, and therefore urging it in the direction in which the
square of the (undisturbed) fluid velocity, w02 + V+wo2> most rapidly increases*.
Hence, by Art. 38, the sphere, if left to itself, cannot be in stable equilibrium
at any point in the interior of the fluid mass.

Ellipsoidal Harmonics.

100. The method of Spherical Harmonics can also be adapted
to the solution of the equation

V^ = 0 .............................. (1),

under boundary-conditions having relation to ellipsoids of revo
lution f.

Beginning with the case where the ellipsoids are prolate, we
write

y — OT cos a), z = tar sin co, ......... (2).

where -OT = k sin 6 sinh TJ = k (1 - ffi (f2 - 1)*.

The surfaces f = const., //, = const., are confocal ellipsoids, and
hyperboloids of two sheets, respectively, the common foci being the
points (+ k, 0, 0). The value of f may range from 1 to oo , whilst
//, lies between + 1. The coordinates //-, f, co form an orthogonal
system, and the values of the linear elements SsM, Ss^, £sw described
by the point (x, y, z) when /*,, f, co separately vary, are respectively,

. 7

=

I)* Bto .................. (3).

To express (1) in terms of our new variables we equate to zero
the total flux across the walls of a volume element ds^ds^ds^,
and obtain

d fdd> * * \ * d /dd> . . \ ^ d /d6 . . \ .

j -~ 08f$8m OM+ jvl j 0*»0*« }&£ + -T- 1 -T- ^S,SsA Ba> = 0,

d/A\d% J ' d£\ds{ "J d&\d8m V

* Sir W. Thomson, " On the Motion of Eigid Solids in a Liquid &c.," Phil.
Mag. , May, 1873.

t Heine, "Ueber einige Aufgaben, welche auf partielle Differentialgleichungen
fiihren," Crelle, t. xxvi., p. 185 (1843); Kugelfunktionen, t. ii., Art. 38. See also
Ferrers, Spherical Harmonics, c. vi.

L. 10

140 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

or, on substitution from (3),

This may also be written

101. If <£ be a finite function of //, and to from /z = — 1 to
//, = + 1 and from o> = 0 to w = 2?r, it may be expanded in a series
of surface harmonics of integral orders, of the types given by Art.
87 (6), where the coefficients are functions of f ; and it appears on
substitution in (4) that each term of the expansion must satisfy
the equation separately. Taking first the case of the zonal har

monic, we write

<t> = Pn(fi).Z ........................... (5),

and on substitution we find, in virtue of Art. 85 (1),

(6),

which is of the same form as the equation referred to. We thus
obtain the solutions

* = Pn(/»).Pn(?) ........................ (7),

and tf> = P»G*).Q»(?) ........................ (8),

where

rn \

i L. « i . \ • " i •-• / \ • «• i *- / if n g

1.3. ..(2/1+1) I" 2(2?z + 3)

~* ~t> A. /Si™ i O\ /O™ ~i K\ y + • • •

The solution (7) is finite when f=l, and is therefore adapted
to the space within an ellipsoid of revolution ; whilst (8) is infinite
for f=l, but vanishes for f=oo, and is appropriate to the

* Ferrers, c. v. ; Todhunter, c. vi.; Forsyth, Differential Equations, Arts. 96—99.

100-102] FORMULAE FOR OVARY ELLIPSOID. 147

external region. As particular cases of the formula (9) we
note

The definite-integral form of Qn shews that

(10),

where the accents indicate differentiations with respect to f.

The corresponding expressions for the stream-function are
readily found ; thus, from the definition of Art. 93,

<ty=s_]L_<ty <ty__±d±

dsf -& ds^' ds^~^ds^ ............... ' ''

whence

Thus, in the case of (7), we have

whence

The same result will follow of course from the second of equations
(12).

In the same way, the stream-function corresponding to (8) is

102. We can apply this to the case of an ovary ellipsoid
moving parallel to its axis in an infinite mass of liquid. The
elliptic coordinates must be chosen so that the ellipsoid in question

10—2

148 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

is a member of the confocal family, say that for which f=£0-
Comparing with Art. 100 (2) we see that if a, c be the polar and
equatorial radii, and e the eccentricity of the meridian section we
must have

The surface condition is given by Art. 96 (1), viz. we must
have

^ = -iu&2(l-^2)(f2-l) + const ............. (1),

for £= f0- Hence putting n= 1 in Art. 101 (14), and introducing
an arbitrary multiplier A, we have

-— ...... (2),

with the condition

bo ~

1 - e2 '2e 3 1
The corresponding formula for the velocity-potential is

(4).

The kinetic energy, and thence the inertia-coefficient due to
the fluid, may be readily calculated, if required, by the formula (5)
of Art. 93.

103. Leaving the case of symmetry, the solutions of V2$ = 0
when $ is a tesseral or sectorial harmonic in /j, and w are found by
a similar method to be of the types

where, as in Art. 87,

Tns (M) = (1 — ffis — /; (3),

x/x * / rt i »S \ f*

102-103] MOTION OF AN OVARY ELLIPSOID. 149

whilst (to avoid imaginaries) we write

and 0.*

It may be shewn that

As examples we may take the case of an ovary ellipsoid
moving parallel to an equatorial axis, say that of y, or rotating
about this axis.

In the former case, the surface-condition is
d(f) _ dy

dt= vd?'

for £ = f0, where v is the velocity of translation, or

This is satisfied by putting n = l, 5 = 1, in (2), viz.
the constant A being given by

In the case of rotation about Oy, if q be the angular velocity,
we must have

d<t> f doc dz\

_L — ft I f ff I

.74* *i ^ J 4- ^ J* >

for?=f0or = ^q — ./.(l-^sino, (10).

ab Vbo — */*

Putting n = 2, 5 = 1, in the formula (2) we find

^ = Af. (1 - /»«)* (?2 - 1)1 {f flog j£y - 3 - ^j sin « ... (11),

-4 being determined by comparison with (10).

150 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

104. When the ellipsoid is of the oblate or " planetary"
form, the appropriate coordinates are given by

x = k cos 6 sinh 77 = &/*f, ]

?/ = -57 cos «, z = w sin a>, \ (1).

where w = k sin 0 cosh rj = k(I - /x,2)* (f2 + 1)*. j

Here f may range from 0 to oo (or, in some applications from
— oo through 0 to + oo ), whilst //, lies between + 1. The quadrics
f = const., ft = const, are planetary ellipsoids, and hyperboloids of
revolution of one sheet, all having the common focal circle x = 0,
& = k. As limiting forms we have the ellipsoid f = 0, which
coincides with the portion of the plane x = 0 for which •cr < k, and
the hyperboloid ^ = 0 coinciding with the remaining portion of
this plane.

With the same notation as before we find

l)^co .................. (2),

so that the equation of continuity becomes, by an investigation
similar to that of Art. 100,

or

d

This is of the same form as Art. 100 (4), with if in place of f,
and the same correspondence will of course run through the
subsequent formulaB.

In the case of symmetry about the axis we have the solutions
0 = PnG*)..pn(f) ..................... (4),

and <f» = PM0*).gn(f) ..................... (5),

where

104-105] FORMULAE FOR PLANETARY ELLIPSOID. 151

and qn (?)

= (-)" [P. (?) cot- r- ^TT^P-. (?) + 3^1)^- (?) - •••}

1.3. 5. ,.(2n + l) 1s 2(2n+3)

+ " ~~2~.4(2n+3)(2n+5) ^ * ~'"| ^*'

the latter expansion being however convergent only when ?>1.
As before, the solution (4) is appropriate to the region included
within an ellipsoid of the family f = const., and (5) to the external
space.

We note that

As particular cases of the formula (7) we have

?, or) = cot- r,

The formulae for the stream-function corresponding to (4) and
(5) are

and

105. The simplest case of Art. 104 (5) is when n = 0, viz.

(^^cot-1^ ........................... (1),

where f is supposed to range from — oo to + oo . The formula
(10) of the last Art. then assumes an indeterminate form, but we
find by the method of Art. 101,

(2).

* The reader may easily adapt the demonstrations cited in Art. 101 to the
present case.

152 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

This solution represents the flow of a liquid through a circular
aperture in an infinite plane wall, viz. the aperture is the portion
of the plane yz for which or < k. The velocity at any point of the

aperture (f = 0) is

_ !_ cty _ A

~vd™~ (#--BJa)*'

since, over the aperture, kfjb = (&2 — ^2)i The velocity is therefore
infinite at the edge. Compare Art. 66, 1°.

Again, the motion due to a planetary ellipsoid (f=?o) moving
with velocity u parallel to its axis in an infinite mass of liquid is

given by

(3),

-cot-^

where A=-ku + \j~j - cot-1 U -

IbO + -1- j

Denoting the polar and equatorial radii by a and c, we have

so that the eccentricity e of the meridian section is

«=(?.'+!)-».

In terms of these quantities

(5).

The forms of the lines of motion, for equidistant values of o/r,
are shewn on the opposite page.

The most interesting case is that of the circular disk, for
which e = 1, and A = 2uc/?r. The value (3) of cf> for the two sides
of the disk becomes equal to ± Ap, or ±^.(1 — «r2/c2)*, and the
normal velocity + u. Hence the formula (4) of Art. 44 gives

=-2p [°

Jo

.................................... (6).

The effective addition to the inertia of the disk is therefore
2/7T (='6365) times the mass of a spherical portion of the fluid, of
the same radius.

105-106] MOTION OF A CIRCULAR DISK.

153

X'

X

106. The solutions of the equation Art. 104 (3) in tesseral
harmonics are

and

where
and

(3),

154 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

These functions possess the property

S'f S ,,, S/ «'_/ V+l ' 1 /K\

- -- -T ......... m-

For the motion of a planetary ellipsoid (f = f0) parallel to the
axis of y we have w = 1, s = 1, as before, and thence

(6),

with A determined by the condition

for f = f0, v denoting the velocity of the solid. This gives

^k/V?'.2^ -«**•' Si--** (7>-

In the case of the disk (f0 = 0), we have A = 0, as we should
expect.

Again, for a planetary ellipsoid rotating about the axis of y
with angular velocity q, we have, putting n = 2, s= 1,

0 = Afi(l - n*)*(p + 1)* 3f cot-1?- 3 + ^ sin a) ...... (8),

with the surface condition

(9).

For the circular disk (f0 = 0) this gives

|7r^ = -^2q ..................... (10).

At the two surfaces of the disk we have

/i2)^ sin w, = + kq (1 - yu,2)i sin &>,

and substituting in the formula

we obtain 2T= Jf^.q8 ..................... (11).

106-107] ELLIPSOID WITH UNEQUAL AXES. 155

107. In questions relating to ellipsoids with three unequal
axes we may use the method of Lamp's Functions*, or, as they
are now often called, ' Ellipsoidal Harmonics.' Without attempting
a complete account of this, we will investigate some solutions of
the equation

V2<£ = 0 .............................. (1),

in ellipsoidal coordinates, which are analogous to spherical
harmonics of the first and second orders, with a view to their
hydrodynamical applications. It is convenient to begin with the
motion of a liquid contained in an ellipsoidal envelope, which can
be treated at once by Cartesian methods.

Thus when the envelope is in motion parallel to the axis of x
with velocity u, the enclosed fluid moves as a solid, and the velocity-
potential is simply (f> — — ux.

Next let us suppose that the envelope is rotating about a
principal axis (say that of x) with angular velocity p. The
equation of the surface being

the surface condition is

x <f> y d4> z

a8 dx ~ 62 dy ~ c?

We therefore assume <£ = Ayzt which is evidently a solution of (1),
and obtain

,

—

Hence, if the centre be moving with a velocity whose com-

* See, for example, Ferrers, Spherical Harmonics, c. vi.; W. D. Niven, "On
Ellipsoidal Harmonics," Phil. Trans., 1891, A.

J56 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

ponents are u, v, w and if p, q, r be the angular velocities about
the principal axes, we have by superposition

62 - c2 c2 - a? a2 - b2

d> = —ux — vy — vrz — r, -- r p yz — - -- - qzx -- r rxii
62 + c2 c2 + a2^ a2 + 62 "

........................ (3)*

We may also include the case where the envelope is changing
its form as well as position, but so as to remain ellipsoidal. If the
axes are changing at the rates a, b, c, respectively, the general
boundary condition, Art. 10 (3), becomes

a + + ,c + + + =

a3 b3 c3 a2 dx b2 dy c2 dz

which is satisfied by

The equation (1) requires that

a b c

which is in fact the condition which must be satisfied by the
changing ellipsoidal surface in order that the enclosed volume
(^jrabc) may be constant.

108. The solutions of the corresponding problems for an
infinite mass of fluid bounded internally by an ellipsoid involve
the use of a special sj^stem of orthogonal curvilinear coordinates.

If x, y, z be functions of three parameters X, /A, v, such that the
surfaces

X = const., /ji = const., v = const (1)

are mutually orthogonal at their intersections, and if we write

* This result appears to have been obtained independently by Beltrami,
Bjerknes, and Maxwell, in 1873. See Hicks, "Report on Recent Progress in
Hydrodynamics," Brit. Ass. Rep., 1882.

t Bjerknes, " Verallgemeinerung des Problems von den Bewegungen, welche in
einer ruhenden unelastischen Fliissigkeit die Bewegung eines Ellipsoids hervor-
bringt," Gottinger Nachrichten, 1873.

107-108] ORTHOGONAL COORDINATES. Io7

the direction-cosines of the normals to the three surfaces which
pass through (a?, y; z) will be

i7 ) "27 ' fl/2^/ ' r \ /»

tt/Lt ttyL6 a/JL

, c?^c , c?y , dz
dv ' dv ' dv
respectively. It easily follows that the lengths of linear elements
drawn in the directions of these normals will be

respectively.

Hence if </> be the velocity-potential of any fluid motion, the
total flux into the rectangular space included between the six
surfaces X ± }£\, /* ± £3/i, y £ ££y will be

d ( d<j> §v &v\ ^ , d /, c^6 81; S\\ j, d fj d<f>

- - - X + -- A2 -r^ . - . -r- 6/z + --

It appears from Art. 42 (3) that the same flux is expressed by

V2<£ multiplied by the volume of the space, i.e. by

Hence

d

Equating this to zero, we obtain the general equation of continuity
in orthogonal coordinates, of which particular cases have already
been investigated in Arts. 84, 100, 104.

* The above method was given in a paper by W. Thomson, " On the Equations
of Motion of Heat referred to Curvilinear Coordinates," Camb. Math. Journ., t. iv.
(1843) ; Math, and Pfo/s. Papers, t. i., p. 25. Reference may also be made to
Jacobi, " Ueber eine particulare Losung der partiellen Differentialgleichung ...... ,"

Crelle, t. xxxvi, (1847), Gesammelte Werke, Berlin, 1881..., t. ii., p. 198.

The transformation of v"0 t° general orthogonal coordinates was first effected
by Lam6, " Sur les lois de 1'equilibre du fluide eth6r^," Journ. de VEcole Polyt.,
t. xiv., (1834). See also Lemons sur les Coordonnees Curvilignes, Paris, 1859, p. 22.

158

PROBLEMS IN THREE DIMENSIONS.

[CHAP. V

109. In the applications to which we now proceed the triple
orthogonal system consists of the confocal quadrics

a2 -i- 8 ^ 62 4- 0 c2 + 6

-1 = 0

(1),

whose properties are explained in books on Solid Geometry.
Through any given point (x, y, z) there pass three surfaces of the
system, corresponding to the three roots of (1), considered as a cubic
in 0. If (as we shall for the most part suppose) a > b > c, one of
these roots (X, say) will lie between oo and — c2, another (/JL) be
tween — c2 and — 62, and the third (v) between — 62 and — a2. The
surfaces X, /JL, v are therefore ellipsoids, hyperboloids of one sheet,
and hyperboloids of two sheets, respectively.

It follows immediately from this definition of X, p, v, that

a2 + 6 62 + 0 c2 +

identically, for all values of 6. Hence multiplying by a2 +6, and
afterwards putting # = — a2, we obtain the first of the following
equations :

(a2 + X)(a2 + ^)(a2 + z.) ,

(62-c2)(62-a2)

(c2 + X) (c2 + ft) (c2 + v)
(c2-a2)(c2-62)

.(3).

These give

dx

d\ * a2 + X ' d\ 2 62 + X
and thence, in the notation of Art. 108 (2),

(4),

X)" (c2 + X)2

(5).

If we differentiate (2) with respect to 6 and afterwards put 6 = X,
we deduce the first of the following three relations :

109-110]

CONFOCAL QUADRICS.

159

^2=4

.(6)*.

, 2 _ .

' 0,

The remaining relations of the sets (3) and (6) have been
written down from symmetry.

Substituting in Art. 108 (4), we find

4 (i. - X)

+ (X - /t) [(a2
I

............... (7)t-

110. The particular solutions of the transformed equation
V2</> = 0 which first present themselves are those in which </> is a
function of one (only) of the variables X, ^, v. Thus <£ may be a
function of X alone, provided

(a2 + X)* (62 4- X)* (c2 + X)* d<£/dX = const.,

whence
if

.................. (2),

being chosen so as

the additive constant which attaches to
to make 0 vanish for X = oo .

In this solution, which corresponds to <f> = A/r in spherical
harmonics, the equipotential surfaces are the confocal ellipsoids,
and the motion in the space external to any one of these (say that
for which X = 0) is that due to a certain arrangement of simple
sources over it. The velocity at any point is given by the formula

* It will be noticed that hlt h2, h3 are double the perpendiculars from the origin
on the tangent planes to the three quadrics X, /x, v.

t Cf. Lame, " Sur les surfaces isothermes dans les corps solides homogenes en
6quilibre de temperature," Liouville, t. ii., (1837).

160 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

At a great distance from the origin the ellipsoids X become
spheres of radius X*, and the velocity is therefore ultimately equal
to 2(7/r2, where r denotes the distance from the origin. Over any
particular equipotential surface X, the velocity varies as the
perpendicular from the centre on the tangent plane.

To find the distribution of sources over the surface X = 0 which
would produce the actual motion in the external space, we
substitute for <j> the value (1), in the formula (11) of Art. 58, and
for <£' (which refers to the internal space) the constant value

The formula referred to then gives, for the surface-density of the
required distribution,

The solution (1) may also be interpreted as representing the
motion due to a change in dimensions of the ellipsoid, such that
the ellipsoid remains similar to itself, and retains the directions of
its axes unchanged in space. If we put

a/a — 6/6 = c/c, = k, say,
the surface-condition Art. 107 (4) becomes

— d(f>/dn = JMj,
which is identical with (3), if we put C =

A particular case of (5) is where the sources are distributed
over the elliptic disk for which X = — c2, and therefore z1 = 0. This
is important in Electrostatics, but a more interesting application
from the present point of view is to the flow through an elliptic
aperture, viz. if the plane xy be occupied by a thin rigid partition
with the exception of the part included by the ellipse

we have, putting c = 0 in the previous formula,

110-111] FLOW THROUGH AN ELLIPTIC APERTURE. 161

where the upper limit is the positive root of

and the negative or the positive sign is to be taken according as the
point for which $ is required lies on the positive or the negative side
of the plane xy. The two values of (f> are continuous at the aperture,
where X = 0. As before, the velocity at a great distance is equal to
2 A /r*, nearly. For points in the aperture the velocity may be
found immediately from (6) and (7) ; thus we may put

approximately, since \ is small, whence
dt_2A ( x- .

~'

This becomes infinite, as we should expect, at the edge. The
particular case of a circular aperture has already been solved
otherwise in Art. 105.

111. We proceed to investigate the solution of V2<£ = 0, finite
at infinity, which corresponds, for the space external to the ellipsoid,
to the solution </> = x for the internal space. Following the analogy
of spherical harmonics we may assume for trial

<}>=*x .............................. (i).

which gives V2^ + --^=0 ........................ (2),

X d/X

and inquire whether this can be satisfied by making ^ equal
to some function of X only. On this supposition we shall have, by
Art. 108 (3),

and therefore, by Art. 109 (4), (6),

xdx (\
L. 11

162 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

On substitution from Art. 109 (7) the equation (2) becomes

x = - (Jf + X) (# + X) ,

which may be written

whence y= 0 - 5 — — ..(3),

the arbitrary constant which presents itself in the second integra
tion being chosen as before so as to make % vanish at infinity.

The solution contained in (1) and (3) enables us to find the
motion of a liquid, at rest at infinity, produced by the translation
of a solid ellipsoid through it, parallel to a principal axis. The
notation being as before, and the ellipsoid

being supposed in motion parallel to x with velocity u, the surface-
condition is

dct>ld\ = -udx/d\, for X = 0 .................. (5).

Let us write, for shortness,

, r00 d\ , r00 d\ 7 r d\

a« = abc / 2 . ^ s~A > ^<> = abc fiA . -i\ A ' 7o = ooe -. 0 . A
J o (a2 + X) A J o (o + X) A J0 (c- -f X) A

.............. (6),

where A = }(a2 + X) (62 + X) (c2 + X)}* ............... (7).

It will be noticed that these quantities «0, &> 70 are pure
numerics.

The conditions of our problem are now satisfied by

M~3

Pr°Vlded

that is C'=-a6°-u ........................... (9).

111-112] TRANSLATION OF AN ELLIPSOID. 163

The corresponding solution when the ellipsoid moves parallel
to y or z can be written down from symmetry, and by superposition
we derive the case where the ellipsoid has any motion of translation
whatever*.

At a great distance from the origin, the formula (8) becomes
equivalent to

which is the velocity-potential of a double source at the origin, of
strength J C, or

f abcu/(2 - «o).
Compare Art. 91.

The kinetic energy of the fluid is given by

where I is the cosine of the angle which the normal to the surface
makes with the axis of x. The latter integral is equal to the
volume of the ellipsoid, whence

' .................. (11).

The inertia-coefficient is therefore equal to the fraction «0/(2 — o^)
of the mass displaced by the solid. For the case of the sphere
(a=b = c) we find a0 = J ; this makes the fraction equal to J, in agree
ment with Art. 91. If we put b = c, we get the case of an ellipsoid
of revolution, including (for a = 0) that of a circular disk. The
identification with the results obtained by the methods of Arts.
102, 103, 105, 106 for these cases may be left to the reader.

112. We next inquire whether the equation V2$ = 0 can
be satisfied by

* This problem was first solved by Green, "Kesearches on the Vibration of
Pendulums in Fluid Media," Trans. R. 8. Edin., 1833, Math. Papers, p. 315. The
investigation is much shortened if we assume at once from the Theory of Attrac
tions that (8) is a solution of v2<£ = 0, being in fact (save as to a constant factor)
the ^-component of the attraction of a homogeneous ellipsoid on an external
point.

11—2

164 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

where % is a function of X only. This requires

V.X + I&+?^ = 0.. .(2).

* ydy z dz

Now, from Art. 109 (4), (6),

= ----

y dy z dz l \y d\ z d\) d\

X)/ 1 1 \dx

' a

On substitution in (2) we find, by Art. 109 (7),
£ log {(* + » <* + X).

Whence =

the second constant of integration being chosen as before.

For a rigid ellipsoid rotating about the axis of x with angular
velocity p, the surface-condition is

d(j>/d\ = ipzdyldX. — ^ydz\d\ ............... (4),

for X — 0. Assuming

we find that the surface-condition (4) is satisfied, provided
0 „/! , 1\ 7o-/8.

This expression dififers only by a factor from

where 0 is the gravitation-potential of a uniform solid ellipsoid at an external
point (a-, y, z). Since v2fi = 0 it easily follows that the above is also a solution of
the equation ^<p = Q.

112] ROTATION OF AN ELLIPSOID. 165

The formulae for the cases of rotation about y or z can be written
down from symmetry*.

The formula for the kinetic energy is

if (I, m, n) denote the direction-cosines of the normal to the
ellipsoid. The latter integral

= fff(y2 - z*) dxdydz = £ (68 - c2) . f Tra&c.
Hence we find

S P'P"

The two remaining types of ellipsoidal harmonic of the second order, finite
at the origin, are given by the expression

where 6 is either root of

111

0 (ii),

this being the condition that (i) should satisfy v20 = 0.

The method of obtaining the corresponding solutions for the external
space is explained in the treatise of Ferrers. These solutions would enable us
to express the motion produced in a surrounding liquid by variations in the
lengths of the axes of an ellipsoid, subject to the condition of no variation of
volume

= 0 .............................. (iii).

We have already found, in Art. 110, the solution for the case where the
ellipsoid expands (or contracts) remaining similar to itself ; so that by super
position we could obtain the case of an internal boundary changing its
position and dimensions in any manner whatever, subject only to the con
dition of remaining ellipsoidal. This extension of the results arrived at
by Green and Clebsch was first treated, though in a different manner from
that here indicated, by Bjerknest.

* The solution contained in (5) and (6) is due to Clebsch, "Ueber die Bewegung
eines Ellipsoides in einer tropfbaren Fliissigkeit," Crelle, it. Iii., liii. (1856 — 7).
t 1. c. ante p. 156.

166 PROBLEMS IN THREE DIMENSIONS. [CHAP. V

113. The investigations of this chapter relate almost entirely
to the case of spherical or ellipsoidal boundaries. It will be under
stood that solutions of the equation V20 = 0 can be carried out, on
lines more or less similar, which are appropriate to other forms of
boundary. The surface which comes next in interest, from the
point of view of the present subject, is that of an anchor-ring,
or ' torus ' ; this problem has been very ably treated, by distinct
methods, by Hicks*, and Dyson -f-. We may also refer to the
analytically remarkable problem of the spherical bowl, which has
been investigated by Basset J.

* "On Toroidal Functions," Phil. Trans., 1881.
t "On the Potential of an Anchor-Ring," Phil. Trans., 1893.
J " On the Potential of an Electrified Spherical Bowl, &c.," Proc. Lond. Math.
Soc., t. xvi. (1885).

CHAPTER VI.

ON THE MOTION OF SOLIDS THROUGH A LIQUID :
DYNAMICAL THEORY.

114. IN this Chapter it is proposed to study the very
interesting dynamical problem furnished by the motion of one
or more solids in a liquid. The development of this subject is due
mainly to Thomson and Tait* and to Kirchhofff. The cardinal
feature of the methods followed by these writers consists in this,
that the solids and the fluid are treated as forming one dynamical
system, and thus the troublesome calculation of the effect of the
fluid pressures on the surfaces of the solids is avoided.

We begin with the case of a single solid moving through an
infinite mass of liquid, and we shall suppose in the first instance
that the motion of the fluid is entirely due to that of the solid,
and is therefore irrotational and acyclic. Some special cases of
this problem have been treated incidentally in the foregoing pages,
and it appeared that the whole effect of the fluid might be
represented by an increase in the inertia of the solid. The same
result will be found to hold in general, provided we use the term
' inertia ' in a somewhat extended sense.

Under the circumstances supposed, the motion of the fluid is
characterized by the existence of a single-valued velocity-potential
</> which, besides satisfying the equation of continuity

v^ = o .............................. (i),

* Natural Philosophy, Art. 320.

t " Ueber die Bewegung eines Rotationskorpers in einer Flussigkeit," Crelle,
t. Ixxi. (1869); Ges. Abh., p. 376; Mechanik, c. xix.

168 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

fulfils the following conditions : (1°) the value of - d(f)/dn, where Sn
denotes as usual an element of the normal at any point of the
surface of the solid, drawn on the side of the fluid, must be equal
to the velocity of the surface at that point normal to itself, and
(2°) the differential coefficients d<f)/d%, dfyjdy, dfyjdz must vanish at
an infinite distance, in every direction, from the solid. The latter
condition is rendered necessary by the consideration that a finite
velocity at infinity would imply an infinite kinetic energy, which
could not be generated by finite forces acting for a finite time on
the solid. It is also the condition to which we are led by supposing
the fluid to be enclosed within a fixed vessel infinitely large and
infinitely distant, all round, from the moving body. For on this
supposition the space occupied by the fluid may be conceived as
made up of tubes of flow which begin and end on the surface of
the solid, so that the total flux across any area, finite or infinite,
drawn in the fluid must be finite, and therefore the velocity at
infinity zero.

It has been shewn in Arts. 40, 41, that under the above con
ditions the motion of the fluid is determinate.

115. In the further study of the problem it is convenient to
follow the method introduced by Euler in the dynamics of rigid
bodies, and to adopt a system of rectangular axes Ox, Oy, Oz fixed
in the body, and moving with it. If the motion of the body at
any instant be defined by the angular velocities p, q, r about, and
the translational velocities u, v, w of the origin parallel to, the
instantaneous positions of these axes, we may write, after
Kirchhoff,

qto+rxt ............ (2),

where, as will appear immediately, <f)1} (f>.2, <f)s, ^15 %2> %3 &re certain
functions of a, y, z determined solely by the configuration of
the surface of the solid, relative to the coordinate axes. In fact,
if I, m, n denote the direction- cosines of the normal, drawn
towards the fluid, at any point of this surface, the kinematical
surface-condition is

— — = I (u + qz — ry) + m(v + rx —pz) + n (w +py — qx),

114-116] SURFACE CONDITIONS. 169

whence, substituting the value (2) of <£, we find
d&

p= = ny — mz

dn

_ _: m, - j'v - = Iz - nx
dn dn

.(3).

- -= It, - -. IIUtAJ V l

dn dn

Since these functions must also satisfy (i), and have their deri
vatives zero at infinity, they are completely determinate, by
Art. 41*.

116. Now whatever the motion of the solid and fluid at any
instant, it might have been generated instantaneously from rest by
a properly adjusted impulsive 'wrench ' applied to the solid. This
wrench is in fact that which would be required to counteract the
impulsive pressures p$ on the surface, and, in addition, to generate
the actual momentum of the solid. It is called by Lord Kelvin
the ' impulse ' of the system at the moment under consideration.
It is to be noted that the impulse, as thus defined, cannot be
asserted to be equivalent to the total momentum of the system,
which is indeed in the present problem indeterminate. We
proceed to shew however that the impulse varies, in consequence
of extraneous forces acting on the solid, in exactly the same way as
the momentum of a finite dynamical system.

Let us in the first instance consider any actual motion of a
solid, from time £0 to time tlt under any given forces applied to it,
in ^ finite mass of liquid enclosed by a fixed envelope of any form.
Let us imagine the motion to have been generated from rest,
previously to the time £0, by forces (whether gradual or impulsive)
applied to the solid, and to be arrested, in like manner, by
forces applied to the solid after the time ^. Since the momentum
of the system is null both at the beginning and at the end of this
process, the time-integrals of the forces applied to the solid, to
gether with the time-integral of the pressures exerted on the fluid

* For the particular case of an ellipsoidal surface, their values may be written
down from the results of Arts. Ill, 112.

170 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

by the envelope, must form an equilibrating system. The effect of
these latter pressures may be calculated from the formula

A pressure uniform over the envelope has no resultant effect;
hence, since <f> is constant at the beginning and end, the only
effective part of the integral pressure fpdt is given by the term

-tpffdt ........................... (2).

Let us now revert to the original form of our problem, and
suppose the containing envelope to be infinitely large, and in
finitely distant in every direction from the moving solid. It is
easily seen by considering the arrangement of the tubes of flow
(Art. 37) that the fluid velocity q at a great distance r from an
origin in the neighbourhood of the solid will ultimately be, at
most*, of the order 1/r2, and the integral pressure (2) therefore of
the order 1/r4. Since the surface-elements of the envelope are of
the order r$-ar, where £OT is an elementary solid angle, the force-
and couple-resultants of the integral pressure (2) will now both
be null. The same statement therefore holds with regard to the
time-integral of the forces applied to the solid.

If we imagine the motion to have been started instantaneously
at time tQ, and to be arrested instantaneously at time tlt the result
at which we have arrived may be stated as follows :

The ' impulse ' of the motion (in Lord Kelvin's sense) at time
£j differs from the ' impulse ' at time t0 by the time-integral of the
extraneous forces acting on the solid during the interval ^ — £0f.

It will be noticed that the above reasoning is substantially
unaltered when the single solid is replaced by a group of solids,
which may moreover be flexible instead of rigid, and even
when these solids are replaced by portions of fluid moving
rotatiorially.

117. To express the above result analytically, let f, 77, f, X, //-, v
be the components of the force- and couple-constituents of the

* It is really of the order 1/r3 when, as in the case considered, the total flux
outwards is zero.

t Sir W. Thomson, I.e. ante p. 35. The form of the argument given above was
kindly suggested to the author by Mr Larmor.

116-117] IMPULSE OF THE MOTION. 171

impulse ; and let X, Y, Z, L, M, N designate in the same manner
the system of extraneous forces. The whole variation of
? t ?7> ?» \ P> v, due partly to the motion of the axes to which
these quantities are referred, and partly to the action of the
extraneous forces, is then given by the formulae

- -

-

...(!)*.

For at time t + &t the moving axes make with their positions
at time t angles whose cosines are

(1, rSt, -q&t), (-rSt, 1, ptt), (qtt, - pSt, 1),

respectively. Hence, resolving parallel to the new position of the
axis of x,

£ + gf = f + n . rSt - £ . qSt + XSt.

Again, taking moments about the new position of Ox, and re
membering that 0 has been displaced through spaces u&t, v&t,
parallel to the axes, we find

\ + &V. = \ 4- 77 . wSt - f . v8t + p . rSt - v . qSt + LSt.

These, with the similar results which can be written down from
symmetry, give the equations (1).

When no extraneous forces act, we verify at once that these
equations have the integrals

^ const.,) }

? = const. J '

which express that the magnitudes of the force- and couple-
resultants of the impulse are constant.

* Cf. Hay ward, " On a Direct Method of Estimating Velocities, Accelerations,
and all similar Quantities, with respect to Axes moveable in any manner in space."
Cainb. Trans., t. x. (1856).

172

MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

118. It remains to express £, 77, £ X, /z, v in terms of
*£, v, w, p, q, r. In the first place let T denote the kinetic energy
of the fluid, so that

2T = -p\U^-dS (1),

where the integration extends over the surface of the moving
solid. Substituting the value of <f>, from Art. 115 (2), we get

2T = Aw3 + "Bv2 + Cw2 + 2A'vw -*- 2"B'wu + 2C'uv

+ 2q (I*'u 4- M'u + N'w)

+ m"v + N"w) ........................... (2),

where the 21 coefficients A, B, C, &c. are certain constants
determined by the form and position of the surface relative to the
coordinate axes. Thus, for example,

= p II (f)2ndS = p 1 1

(3),

- mz) dS

the transformations depending on Art. 115 (3) and on a particular
case of Green's Theorem (Art. 44 (2)). These expressions for
the coefficients were given by Kirchhoff.

The actual values of the coefficients in the expression for 2T have been
found in the preceding chapter for the case of the ellipsoid, viz. we have
from Arts. Ill, 112

P-I_ (° -cr(yo

* la /w ^2N i /;^2 i .

118-119] KINETIC ENERGY. 1*73

with similar expressions for B, C, Q, R. The remaining coefficients, as will
appear presently, in this case all vanish. We note that

so that if a>b>c, then A<B<0, as might have been anticipated.

The formulae for an ellipsoid of revolution may be deduced by putting
& = c; they may also be obtained independently by the method of Arts. 101-
106. Thus for a circular disk (« = 0, b = c) we have

P=o, 5 "

The kinetic energy, Tl say, of the solid alone is given by an
expression of the form

X = m

4- 2m {p (@w — yv) + q (ju — aw) + r (av — ftu)} ...... (4).

Hence the total energy T + Tl , of the system, which we shall
denote by T, is given by an expression of the same general form as
(2), say '

2T = AU? + Btf + Cw* + ZA'vw + 2Rwu +

+ Ppa + Qq* + Rr* + ZP'qr + ZQ'rp + ZR'pq
+ 2p (Lu + Mv + Nw)

+ N"w) .............................. (5),

where the coefficients are printed in uniform type, although six of
them have of course the same values as in (4).

119. The values of the several components of the impulse in
terms of the velocities u, v, w, p, q, r can now be found by a well-
known dynamical method *. Let a system of indefinitely great forces
(X, Y, Z, L, M, N) act for an indefinitely short time r on the solid,
so as to change the impulse from (f, 77, f, X, /u,, v) to

(f + Af, 77 + AT;, f+Af, X + AX, yu + Ayu, v -f Ai/>.

* See Thomson and Tait, Natural Philosophy, Art. 313, or Maxwell, Electricity
and Magnetism, Part iv. , c. v,

174 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

The work done by the force X is

[T Xudt,
which lies between

Uil Xdi and u» \ Xdt,
Jo Ji

where #, and u2 are the greatest and least values of u during the
time r, i.e. it lies between u^ Af and «2Af. If we now introduce the
supposition that Af, A?/, Af, AX, A//,, Ai/ are infinitely small, ^ and
u2 are each equal to u, and the work done is wAf. In the same
way we may calculate the work done by the remaining forces
and couples. The total result must be equal to the increment of
the kinetic energy, whence

aw op c?

Now if the velocities be all altered in any given ratio, the
impulses will be altered in the same ratio. If then we take

Aw _ Afl _ Aw _ Ap _ A<? _ Ar _ ,
u v w p q r
it will follow that

A| = A^_A?=AX = A/A = ^ = ^
f V "" ? X " /t z/

Substituting in (11), we find

dT +
du

T

dT

v~j +
dv

dT

W-J— +

dw

dT
P^

dT
Vdq +

dT
rTr

(2),

since T is a homogeneous quadratic function. Now performing
the arbitrary variation A on the first and last members of (2), and
omitting terms which cancel by (1), we find

f Aw + y&v + f Aw + XAp + /xAg + Mr = AT,

119-120] IMPULSE IN TERMS OF KINETIC ENERGY. 175

Since the variations AM, Av, Aw, Ap, Ag, Ar are all independent,
this gives the required formula?

dT dT dT

dT

dT

(3).

It may be noted that since £17, f, . . . are linear functions of
u, v, w, ..., the latter quantities may also be expressed as linear
functions of the former, and thence T may be regarded as a homo
geneous quadratic function of f, ??, f, X, //,, ZA When expressed in
this manner we may denote it by T'. The equation (1) then
gives at once

uk. -f-

= dr

whence

dT A dT'r dT' , dT'A d.

-7- AT; + -TU A? + jT- ^X +1T~ A/*+ "^
a?; af aX tt/A a

A",

dT'

dT'

v =

dr

dT'

dv

(4),

formulaB which are in a sense reciprocal to (3).

We can utilize this last result to obtain another integral of the
equations of motion, in the case where no extraneous forces act, in
addition to those obtained in Art. 117. Thus

dt

d\ dt*
d\

which vanishes identically, by Art. 117 (1). Hence we have the
equation of energy

(5).

120. If in the formula (3) we put, in the notation of Art. 118,

it is known from the dynamics of rigid bodies that the terms in T:
represent the linear and angular momentum of the solid by itself.

176

MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

Hence the remaining terms, involving T, must represent the
system of impulsive pressures exerted by the surface of the solid on
the fluid, in the supposed instantaneous generation of the motion
from rest.

This is easily verified. For example, the ^-component of the
above system of impulsive pressures is

= Au + C'v + "B'w + lip + L'g -f Ii"r

_dT

du

•(6),

by the formulae of Arts. 115, 118. In the same way, the moment
of the impulsive pressures about Ox is

I*u

dT
dp

-f Nw + Pp + R'# + QV

.(7).

121. The equations of motion may now be written

UU U/-L

dt du

IbJ.

dv

UjJL

^ dw

+ x, i

ddT_
dt dv *

dT

dw

rdT

du

+ Y,

ddT _

dt dw

dT

dT

+z.

,

ddT
dtdp

dT

= w-5 v
dv

dT

dw

dT

+ rdq

dT
-q-j- -f L,

* dr

ddT
dtdq

dT

= u -, w
dw

dT

du

dT
+P~dr

rdT M

dp

d dT
dtdr

dT

= V -j U

du

dT

dv

dT

dT j

-P -j - + N

" dq

(1)*.

* See Kirchhoff, I.e. ante p. 167 ; also Sir W. Thomson, "Hydrokinetic Solutions
and Observations," Phil. Mag., Nov. 1871.

120-121] EQUATIONS OF MOTION. 177

If in these we write T— T + Tl5 and separate the terms due to
T and T1 respectively, we obtain expressions for the forces exerted
on the moving solid by the pressure of the surrounding fluid ; thus
the total component (X, say) of the fluid pressure parallel to x is

d dT dT dT

X = - -T: -T- + r j - Q j- (2),

dt du dv * dw

and the moment (L) of the same pressures about x is

_d_dT_ dT _ dT dT _ JT
dt dp dv dw dq dr

For example, if the solid be constrained to move with a constant
velocity (u} v, w), without rotation, we have

X = 0, Y = 0, Z = 0,

dT dT _,, dT dT dT dT

Jj = W —. V -r— , M = U -j W -j— , N = V , U —-.-

dv dw dw du au dv

where 2T = A*t2 + Bt>2 + Cw2 + 2A'vw + 2B W -f 2C'uv.

Hence the fluid pressures reduce to a couple, which moreover
vanishes if

dT dT dT

-j— : u = -j— : v = -j- :w,
du dv dw

i.e. provided the velocity (u, v, w) be in the direction of one of the
principal axes of the ellipsoid

A#2 + "By* + Cz2 + ZAfyz + VB'zx + ZG'xy = const.. . .(5).

Hence, as was first pointed out by Kirchhoff, there are, for any
solid, three mutually perpendicular directions of permanent trans
lation ; that is to say, if the solid be set in motion parallel to one
of these directions, without rotation, and left to itself, it will continue
so to move. It is evident that these directions are determined
solely by the configuration of the surface of the body. It must be
observed however that the impulse necessary to produce one of

* The forms of these expressions being known, it is not difficult to verify them
by direct calculation from the pressure-equation, Art. 21 (4). See a paper " On the
Forces experienced by a Solid moving through a Liquid," Quart. Journ. Math.,
t. xix. (1883).

L. 12

178 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

these permanent translations does not in general reduce to a single
force ; thus if the axes of coordinates be chosen, for simplicity,
parallel to the three directions in question, so that A', B' , C' = 0,
we have, corresponding to the motion u alone,

f = 4a,'i) = 0, £=0,

\ = Lu, /JL — L'u, v = L"uy
so that the impulse consists of a wrench of pitch L/A.

With the same choice of axes, the components of the couple
which is the equivalent of the fluid pressures on the solid, in the
case of a uniform translation (u, v, w), are

L = (B - C) vw, M = (C - A) wu, N = (A - B) uv. . .(6).
Hence if in the ellipsoid

A«2 + B?/2 + Cz2 = const (7),

we draw a radius-vector r in the direction of the velocity (u, v, w)
and erect the perpendicular h from the centre on the tangent
plane at the extremity of r, the plane of the couple is that
of h and r, its magnitude is proportional to sin (h, r)/h, and its
tendency is to turn the solid in the direction from h to r. Thus if
the direction of (u, v, w) differs but slightly from that of the axis of
x, the tendency of the couple is to diminish the deviation when A
is the greatest, and to increase it when A is the least, of the
three quantities A, B, C, whilst if A is intermediate to B and C
the tendency depends on the position of r relative to the circular
sections of the above ellipsoid. It appears then that of the three
permanent translations one only is thoroughly stable, viz. that
corresponding to the greatest of the three coefficients A, B, C.
For example, the only stable direction of motion of an ellipsoid
is that of its least axis; see Art. 118*.

122. The above, although the simplest, are not the only
steady motions of which the body is capable, under the action
of no external forces. The instantaneous motion of the body at
any instant consists, by a well-known theorem of Kinematics, of a

* The physical cause of this tendency of a flat-shaped body to set itself
broadside-on to the relative motion is clearly indicated in the diagram on p. 94.
A number of interesting practical illustrations are given by Thomson and Tait,
Art. 325.

121-122] STEADY MOTIONS. 179

twist about a certain screw; and the condition that this motion
should be permanent is that it should not affect the configuration
of the impulse (which is fixed in space) relatively to the body.
This requires that the axes of the screw and of the corresponding
impulsive wrench should coincide. Since the general equations
of a straight line involve four independent constants, this gives four
linear relations to be satisfied by the five ratios u : v : w : p : q : r.
There exists then for every body, under the circumstances here
considered, a singly-infinite system of possible steady motions.

Of these the next in importance to the three motions of permanent
translation are those in which the impulse reduces to a couple. The equa
tions (1) of Art. 117 are satisfied by £, 77, £=0, and X, /u, v constant, provided

*/P = P-l(l = v/r, =£, say ........................... (i).

If the axes of coordinates have the special directions referred to in the
preceding Art., the conditions £, 77, £=0 give us at once w, v, w in terms
of p, q, r, viz.

*— (Lp + L'q+ L"r)IA,\

v=-(Mp + M'q + M"r)/B\ .......................... (ii).

w= - (Np+N'q +N"r)/C )

Substituting these values in the expressions for X, p, v obtained from Art.
119 (3), we find

. <tfe dQ dQ

X = dp' ^ = ^' " = ^ ........................... (m)>

where 26 (p, q, r) = %?2 + <%2 -f Hr2 + Vfflqr + ZQ&rp + 2R'pq ......... (iv) ;

the coefficients in this expression being determined by formulae of the types

L'L" M'M" N'N"

" '~A ' ~^r ~c~

These formulae hold for any case in which the force- constituent of the impulse
is zero. Introducing the conditions (i) of steady motion, the ratios p : q : r
are to be determined from the three equations

The form of these shews that the line whose direction-ratios are p : q : r
must be parallel to one of the principal axes of the ellipsoid

0(.r, y, 2) = const ............................... (vii).

12—2

180 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

There are therefore three permanent screw-motions such that the correspond
ing impulsive wrench in each case reduces to a couple only. The axes of
these three screws are mutually at right angles, but do not in general
intersect.

It may now be shewn that in all cases where the impulse reduces to a
couple only, the motion can be completely determined. It is convenient,
retaining the same directions of the axes as before, to change the origin.
Now the origin may be transferred to any point (#, ?/, z) by writing

u + ry-qz, v+pz-rx, w + qx-py,

for u, v, w respectively. The coefficient of vr in the expression for the kinetic
energy, Art. 118 (7), becomes -J3&+M", that of wq becomes Cx + N', and so
on. Hence if we take

M" N' . N

the coefficients in the transformed expression for 2 T will satisfy the relations
M"/B=N'/C, N/C=L"/A, L'/A = M/B ................ (ix).

If we denote the values of these pairs of equal quantities by a, /3, y re
spectively, the formulae (ii) may now be written

d¥ d* d*

u=—-j-, v=--r-. w= — =- ..................... (x),

dp ' dq dr

where 2* (p, q, r) = j p2 + -^ f + -^ r2 + Zaqr + Z$rp + Zypq ...... (xi).

The motion of the body at any instant may be conceived as made up of two
parts ; viz. a motion of translation equal to that of the origin, and one of
rotation about an instantaneous axis passing through the origin. Since
£} rj, £=Q the latter part is to be determined by the equations

d\ da dv .

-drr»-i"' di=p"~r^ arf-**

which express that the vector (X, /u, v) is constant in magnitude and has a fixed
direction in space. Substituting from (iii),

d de_ de de

dt dp dq dr

d de de de

d^de_ de_ de
dt dr~q dp P dq

(xii).

These are identical in form with the equations of motion of a rigid body
about a fixed point, so that we may make use of Poinsot's well-known solution
of the latter problem. The angular motion of the body is therefore obtained
by making the ellipsoid (vii), which is fixed in the body, roll on the plane

\x -\- p.y + vz =• const. ,

122-123] IMPULSIVE COUPLE. 181

which is fixed in space, with an angular velocity proportional to the length 01
of the radius vector drawn from the origin to the point of contact /. The
representation of the actual motion is then completed by impressing on the
whole system of rolling ellipsoid and plane a velocity of translation whose
components are given by (x). This velocity is in the direction of the
normal OM to the tangent plane of the quadric

*(#» y> z)=-e3 ................................. (xiii),

at the point P where 01 meets it, and is equal to

angular velocity of body

When 01 does not meet the quadric (xiii), but the conjugate quadric obtained
by changing the sign of e, the sense of the velocity (xiv) is reversed *.

123. The problem of the integration of the equations of
motion of a solid in the general case has engaged the attention of
several mathematicians, but, as might be anticipated from the
complexity of the question, the meaning of the results is not
easily grasped.

In what follows we shall in the first place inquire what
simplifications occur in the formula for the kinetic energy, for
special classes of solids, and then proceed to investigate one or
two particular problems of considerable interest which can be
treated without difficult mathematics.

1°. If the solid has a plane of symmetry, as regards both its
form and the distribution of matter in its interior, then, taking this
plane as that of soy, it is evident that the energy of the motion is
unaltered if we reverse the signs of w, p, q, the motion being
exactly similar in the two cases. This requires that A', B', P', Q',
L, M, L', M', N" should vanish. One of the directions of perma
nent translation is then parallel to z. The three screws of Art. 122
are now pure rotations ; the axis of one of them is parallel to z ;
the axes of the other two are at right angles in the plane xy, but
do not in general intersect the first.

2°. If the body have a second plane of symmetry, at right
angles to the former one, let this be taken as the plane of zx.
We find, in the same way, that in this case the coefficients

* The substance of this Art. is taken from a paper, " On the Free Motion of a
Solid through an Infinite Mass of Liquid," Proc. Lond. Math. £oc., t. viii. (1877).
Similar results were obtained independently by Craig, " The Motion of a Solid in a
Fluid," Amer. Journ. of Math., t. ii. (1879).

182 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

C", R', N, L" also must vanish, so that the expression for 2T
assumes the form

2T = Au? + BV* + GW*

r ........................... (1).

The directions of permanent translation are now parallel to the
three axes of coordinates. The axis of x is the axis of one of the
permanent screws (now pure rotations) of Art. 122, and those of
the other two intersect it at right angles (being parallel to y and s
respectively), though not necessarily in the same point.

3°. If the body have a third plane of symmetry, viz. that of
yz, at right angles to the two former ones, we have

(2).

The axes of coordinates are in the directions of the three perma
nent translations ; they are also the axes of the three permanent
screw-motions (now pure rotations) of Art. 122.

4°. If, further, the solid be one of revolution, about %, say, the
value (1) of 2T must be unaltered when we write v, q, — w, — r for
w, r, v, q, respectively; for this is merely equivalent to turning the
axes of y, z through a right angle. Hence we must have B = C,
Q = R, M" — — N'. If we further transfer the origin to the point
denned by Art. 122 (viii) we have M" ' = N't Hence we must have

and 2T = An? + B (v2 + O

(3).

The same reduction obtains in some other cases, for example
when the solid is a right prism whose section is any regular
polygon*. This is seen at once from the consideration that, the
axis of x coinciding with the axis of the prism, it is impossible to
assign any uniquely symmetrical directions to the axes of y and z.

* See Larmor, "On Hydrokinetic Symmetry," Quart. Journ. Math., t. xx.
(1885).

123] SPECIAL FORMS OF SOLID. 183

5°. If, in the last case, the form of the solid be similarly
related to each of the coordinate planes (for example a sphere, or a
cube), the expression (3) takes the form

2T = A(<u? + v* + w2) + P(p2 + q* + r'*) ............... (4).

This again may be extended, for a like reason, to other cases,
for example any regular polyhedron. Such a body is practically
for the present purpose ' isotropic,' and its motion will be exactly
that of a sphere under similar conditions.

6°. We may next consider another class of cases. Let us
suppose that the body has a sort of skew symmetry about a certain
axis (say that of x), viz. that it is identical with itself turned
through two right angles about this axis, but has not necessarily a
plane of symmetry*. The expression for 2T must be unaltered
when we change the signs of v, w, q, r, so that the coefficients
B't <7, Q', R', M, N, L', L" must all vanish. We have then

2T = An? + Bv* + Cw2 + 2A'vw
+ Qq* + Rr* + ZP'qr

+ 2g (M'v + N'w)

+ Zr(M"v + N"w) ........................ (5).

The axis of x is one of the directions of permanent translation ;
and is also the axis of one of the three screws of Art. 122, the pitch
being — L/A. The axes of the two remaining screws intersect it
at right angles, but not in general in the same point.

7°. If, further, the body be identical with itself turned through
one right angle about the above axis, the expression (5) must be
unaltered when v, q, — w, — r are written for w, r, v, q, respectively.
This requires that B =C, A' = 0, Q = R, P'=0, M' = N",
N' — — M". If further we transfer the origin to the point chosen
in Art. 122 we must have N' = M" ', and therefore N' = 0, M" = 0.
Hence (5) reduces to

q + wr) ........................... (6).f

* A two-bladed screw-propeller of a ship is an example of a body of this kind.
t This result admits of the same kind of generalization as (3), e.g. it applies to
a body shaped like a screw-propeller with three symmetrically-disposed blades.

184 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

The form of this expression is unaltered when the axes of y, z
are turned in their own plane through any angle. The body is
therefore said to possess helicoidal symmetry about the axis of as.

8°. If the body possess the same properties of skew symmetry
about an axis intersecting the former one at right angles, we
must evidently have

-f 2L (pu + qv + rw) ..................... (7).

Any direction is now one of permanent translation, and any line
drawn through the origin is the axis of a screw of the kind con
sidered in Art. 122, of pitch -L/A. The form of (7) is unaltered
by any change in the directions of the axes of coordinates. The
solid is therefore in this case said to be ' helicoidally isotropic.'

124. For the case of a solid of revolution, or of any other form
to which the formula

.(1)

applies, the complete integration of the equations of motion was
effected by Kirchhoff* in terms of elliptic functions.

The particular case where the solid moves without rotation
about its axis, and with this axis always in one plane, admits of
very simple treatment •)•, and the results are very interesting.

If the fixed plane in question be that of xy we have p, q, w = 0,
so that the equations of motion, Art. 121 (1), reduce to

A du D D dv .

A -j- = rBv, B-j-= — rAu,

dt dt ,

Let x, y be the coordinates of the moving origin relative to
fixed axes in the plane (ocy) in which the axis of the solid moves,

* I.e. ante p. 167.

t See Thomson and Tait, Natural Philosophy, Art. 322; and Greenhill, "On
the Motion of a Cylinder through a Frictionless Liquid under no Forces," Mess, of
Hath., t. ix. (1880).

123-124] MOTION OF A SOLID OF REVOLUTION. 185

the axis of x coinciding with the line of the resultant impulse
(/, say) of the motion ; and let 6 be the angle which the line Ox
(fixed in the solid) makes with x. We have then

Au = Icoa0, Bv = -Isin0, r=6.

The first two of equations (2) merely express the fixity of the
direction of the impulse in space ; the third gives

sin<9cos<9 = 0 .................. (3).

We may suppose, without loss of generality, that A > B. If
we write 20 = S-, (3) becomes

which is the equation of motion of the common pendulum. Hence
the angular motion of the body is that of a 'quadrantal
pendulum,' i.e. a body whose motion follows the same law in
regard to a quadrant as the ordinary pendulum does in regard to
a half-circumference. When 0 has been determined from (3) and
the initial conditions, x, y are to be found from the equations

x = u cos 0 — v sin 0 = —r cos2 6 + ^ sin2 0,

= (:l~±]sin0cos0 = -?6

the latter of which gives

y = |0 (6),

as is otherwise obvious, the additive constant being zero since the
axis of x is taken to be coincident with, and not merely parallel
to, the line of the impulse /.

Let us first suppose that the body makes complete revolutions,
in which case the first integral of (3) is of the form

02 = o>2 (1 - &2 sin2 0) (7),

A-B I*
where fc2 = -r-D?y • — 2 W«

Hence, reckoning t from the position 0 = 0, we have

186

MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

124]

MOTION OF A SOLID OF REVOLUTION.

187

in the usual notation of elliptic integrals. If we eliminate t
between (5) and (9), and then integrate with respect to 0,
we find

-^F*ft

(10),

the origin of x being taken to correspond to the position 0 = 0.
The path can then be traced, in any particular case, by means of
Legendre's Tables. See the curve marked I in the figure.

If, on the other hand, the solid does not make a complete
revolution, but oscillates through an angle a on each side of the
position 6 = 0, the proper form of the first integral of (3) is

frs.rffl-E^ ; (11),

\ sin2 aj

where
If we put
this gives

whence

sur a =

ABQ ft)2

A-B'
sin 6 — sin a sin -

(12).

sin2 a

siu a

(13).

Transforming to ^r as independent variable, in (5), and integrating,
we find

x = -g- sin a . .F(sin a, ^r) - -y^cosec a . #(sin a, yfr

Qco
y = —j- cos >|r

The path of the point 0 is here a sinuous curve crossing the line
of the impulse at intervals of time equal to a half-period of the
angular motion. This is illustrated by the curves III and IV of the
figure.

There remains a critical case between the two preceding, where
the solid just makes a half-revolution, 6 having as asymptotic

188 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

limits the two values + JTT. This case may be obtained by putting
k = 1 in (7), or a = ^TT in (11) ; and we find

6 = co cos 6 ................................. (15),

(16),

(17).

0(0

y= *-COS0

See the curve II of the figure*.

It is to be observed that the above investigation is not restricted
to the case of a solid of revolution ; it applies equally well to the
case of a body with two perpendicular planes of symmetry, moving
parallel to one of these planes, provided the origin be properly
chosen. If the plane in question be that of a?y, then on transferring
the origin to the point (M"/B, 0, 0) the last term in the formula
(1) of Art. 123 disappears, and the equations of motion take the
form (2) above. On the other hand, if the motion be parallel to
zx we must transfer the origin to the point (— N'/C, 0, 0).

The results of this Article, with the accompanying diagrams,
serve to exemplify the statements made near the end of Art. 121.
Thus the curve IV illustrates, with exaggerated amplitude, the
case of a slightly disturbed stable steady motion parallel to an
axis of permanent translation. The case of a slightly disturbed
unstable steady motion would be represented by a curve con
tiguous to II, on one side or the other, according to the nature of
the disturbance.

125. The mere question of the stability of the motion of a
body parallel to an axis of symmetry may of course be more simply
treated by approximate methods. Thus, in the case of a body

* In order to bring out the peculiar features of the motion, the curves have
been drawn for the somewhat extreme case of A = 5B. In the case of an infinitely
thin disk, without inertia of its own, we should have A/B = cc; the curves would
then have cusps where they meet the axis of y. It appears from (5) that x has
always the same sign, so that loops cannot occur in any case.

In the various cases figured the body is projected always with the same impulse,
but with different degrees of rotation. In the curve I, the maximum angular
velocity is */2 times what it is in the critical case II ; whilst the curves III and
IV represent oscillations of amplitude 45° and 18° respectively.

124-125] STABILITY. 189

with three planes of symmetry, as in Art. 123, 3°, slightly dis
turbed from a state of steady motion parallel to x, we find,
writing u = u0 + u' ', and assuming u't v, w, p, q, r to be all small,

...(1).

D d2v A(A-B) A

Hence B ~j- + A —^ — - u<?v = 0,

with a similar equation for r, and

with a similar equation for q. The motion is therefore stable only
when A is the greatest of the three quantities A, B, C.

It is evident from ordinary Dynamics that the stability of a
body moving parallel to an axis of symmetry will be increased, or
its instability (as the case may be) will be diminished, by
communicating to it a rotation about this axis. This question
has been examined by Greenhill*.

Thus in the case of a solid of revolution slightly disturbed from a state of
motion in which u and p are constant, while the remaining velocities are
zero, if we neglect squares and products of small quantities, the first and
fourth of equations (1) of Art. 121 give

du/dt=0, dp/dt = 0,
whence u- = u0, p=pQ ........................... (i),

say, where UQ, p0 are constants. The remaining equations then take, on
substitution from Art. 123 (3), the forms

"Fluid Motion between Confocal Elliptic Cylinders, &c.," Quart. Journ.
Math., t. xvi. (1879).

190 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

If we assume that v, w, q, r vary as e?A*, and eliminate their ratios, we find

O ......... (iv).

The condition that the roots of this should be real is that

should be positive. This is always satisfied when A>B, and can be satisfied
in any case by giving a sufficiently great value to pQ.

This example illustrates the steadiness of flight which is given to an
elongated projectile by rifling.

126. In the investigation of Art. 122 the term 'steady' was
used to characterize modes of motion in which the ' instantaneous
screw ' preserved a constant relation to the moving solid. In the
case of a solid of revolution, however, we may conveniently use the
term in a somewhat wider sense, extending it to motions in which
the velocities of translation and rotation are constant in magnitude,
and make constant angles with the axis of symmetry and with
each other, although their relation to particles of the solid not on
the axis may continually vary.

The conditions to be satisfied in this case are most easily obtained from
the equations of motion of Art. 121, which become, on substitution from
Art. 123 (3),

...... (i).

It appears that p is in any case constant, and that q2 + r2 will also be constant

provided

vfa=w/r, =k, say .................... . ............ (ii).

This makes du/dt = 0, and

const.

It follows that k will also be constant; and it only remains to satisfy the
equations

125-127] MOTION OF A HELICOID. 191

which will be consistent provided

whence ulp=kBPI{AQ-WB(A-B}} ........................ (iii).

Hence there are an infinite number of possible modes of steady motion, of the
kind above defined. In each of these the instantaneous axis of rotation and
the direction of translation of the origin are in one plane with the axis of the
solid. It is easily seen that the origin describes a helix about the resultant
axis of the impulse.

These results are due to Kirchhoff.

127. The only case of a body possessing helicoidal property,
where simple results can be obtained, is that of the 'isotropic
helicoid' defined by Art. 123 (7). Let 0 be the centre of the
body, and let us take as axes of coordinates at any instant, a line
Ox, parallel to the axis of the impulse, a line Oy drawn outwards
from this axis, and a line Oz perpendicular to the plane of the
two former. If / and G denote the force- and couple-constituents
of the impulse, we have

Au + Lp = f = /, Pp + La = \=G,

Aw + Lr = £= 0, Pr + Lw = v = I
where OT denotes the distance of 0 from the axis of the impulse.

Since AP — L2 4= 0, the second and fifth of these equations
shew that 0 = 0, q = 0. Hence w is constant throughout the
motion, and the remaining quantities are constant ; in particular

u = (IF-GL)l(AP-L*\\
w = -^ILI(AP-D) } '

The origin 0 therefore describes a helix about the axis of the
impulse, of pitch

G/I-P/L.

This example is due to Lord Kelvin*.

* I.e. ante p. 176. It is there pointed out that a solid of the kind here in
question may be constructed by attaching vanes to a sphere, at the middle points of
twelve quadrantal arcs drawn so as to divide the surface into octants. The vanes
are to be perpendicular to the surface, and are to be inclined at angles of 45° to the
respective arcs.

For some further investigations in this field see a paper by Miss Fawcett, " On
the Motion of Solids in a Liquid," Quart. Journ. Math., t. xxvi. (1893).

192 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

128. Before leaving this part of the subject we remark that
the preceding theory applies, with obvious modifications, to the
acyclic motion of a liquid occupying a cavity in a moving solid. If
the origin be taken at the centre of inertia of the liquid, the
formula for the kinetic energy of the fluid motion is of the type

2T = m (u* + v* + w2)

+ Pp- + Qg2 + Rr2 + VP'qr + 2Q'rp + 2H'pq ...... (1).

For the kinetic energy is equal to that of the whole fluid mass
(m), supposed concentrated at the centre of mass and moving
with this point, together with the kinetic energy of the motion
relative to the centre of mass. The latter part of the energy is
easily proved by the method of Arts. 115, 118 to be a homo
geneous quadratic function of p, q, r.

Hence the fluid may be replaced by a solid of the same
mass, having the same centre of inertia, provided the principal
axes and moments of inertia be properly assigned.

The values of the coefficients in (1), for the case of an ellipsoidal cavity,
may be calculated from Art. 107. Thus, if the axes of #, y, z coincide with
the principal axes of the ellipsoid, we find

P'=0, Q' = 0, R' = 0.

Case of a Perforated Solid.

129. If the moving solid have one or more apertures or per
forations, so that the space external to it is multiply-connected,
the fluid may have a motion independent of that of the solid, viz.
a cyclic motion in which the circulations in the several irreducible
circuits which can be drawn through the apertures may have any
given constant values. We will briefly indicate how the foregoing
methods may be adapted to this case.

Let K, K, #",... be the circulations in the various circuits, and
let So-, So-', So-",... be elements of the corresponding barriers,
drawn as in Art. 48. Further, let I, m, n denote direction-cosines
of the normal, drawn towards the fluid, at any point of the surface
of the solid, or drawn on the positive side at any point of a barrier.
The velocity-potential is then of the form

128-129] PERFORATED SOLID. 193

where

</> = ufa + vfa + tufa+pxi + qx* , n ,

The functions fa, fa, fa, %1; %2, ^3 are determined by the same
conditions as in Art. 115. To determine co, we have the condi
tions : (1°) that it must satisfy V2o> = 0 at all points of the fluid ;
(2°) that its derivatives must vanish at infinity; (3°) that dco/dn=Q
at the surface of the solid ; and (4°) that w must be a cyclic function,
diminishing by unity whenever the point to which it refers com
pletes a circuit cutting the first barrier once only in the positive
direction, and recovering its original value whenever the point
completes a circuit not cutting this barrier. It appears from
Art. 52 that these conditions determine o> save as to an additive
constant. In like manner the remaining functions to', a>", ... are
determined.

By the formula (5) of Art. 55, twice the kinetic energy of the
fluid is equal to

*«•'- ...... (2).

Since the cyclic constants of 0 are zero, we have, by Art. 54 (4),

which vanishes, since dfa/dn = 0 at the surface of the solid.
Hence (2) reduces to

- ...... (3).

Substituting the values of (/>, fa from (1), we find that the kinetic
energy of the fluid is equal to

T + # .............................. (4),

where T is a homogeneous quadratic function of u, v, w, p, q, r of
the form defined by Art. 118 (2), (3), and

2K = (K, K) K2 + (V, K) K2 + . . . + 2 (K, K) KK + ......... (5),

L. 13

194 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

where, for example,

day'

The identity of the two forms of (K, K') follows from Art. 54 (4).
Hence the total energy of fluid and solid is given by

T=® + K) ........................ (7),

where ® is a homogeneous quadratic function of u, v, w, p, q, r
of the same form as Art. 118 (5), and K is defined by (5) and
(6) above.

130. The 'impulse' of the motion now consists partly of
impulsive forces applied to the solid, and partly of impulsive
pressures pK,pK,pic"... applied uniformly (as explained in Art. 54)
over the several membranes which are supposed for a moment to
occupy the positions of the barriers. Let us denote by fl3 %, f1}
Xls fa, Vi the components of the extraneous impulse applied to the
solid. Expressing that the ^-component of the momentum of the
solid is equal to the similar component of the total impulse acting
on it, we have

(1).

where, as before, Tx denotes the kinetic energy of the solid, and T
that part of the energy of the fluid which is independent of the
cyclic motion. Again, considering the angular momentum of the
solid about the axis of a?,

= X, - p jj (</> + </,0) (ny - mz) dS

129-130] COMPONENTS OF IMPULSE. 195

Hence, since ffi, = T + T15 we have

. d® [[ dfaja , [[ ,d<h ,~

f i = -7- - PK oj y- dS - OK CD , T- dS - . . . ,
du JJ dn JJ dn

By virtue of Lord Kelvin's extension of Green's theorem, al
ready referred to, these may be written in the alternative forms

Adding to these the terms due to the impulsive pressures
applied to the barriers, we have, finally, for the components of the
total impulse of the motion,

d1®

ME, d®

x= ,— + x0, y^ — ~j — i-^o. v~~j

op c?g dr

where, for example,

It is evident that the constants f0, 770, ?o> X0, MOJ ^o are the
components of the impulse of the cyclic fluid motion which
remains when the solid is, by forces applied to it alone, brought
to rest.

By the argument of Art. 116, the total impulse is subject to
the same laws as the momentum of a finite dynamical system.
Hence the equations of motion of the solid are obtained by substi
tuting from (5) in the equations (1) of Art. 117*.

* This conclusion may be verified by direct calculation from the pressure-
formula of Art. 21 ; see Bryan, " Hydrodynamical Proof of the Equations of Motion
of a Perforated Solid, ...... ," Phil. May., May, 1893.

13—2

196 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

131. As a simple example we may take the case of an annular
solid of revolution. If the axis of x coincide with that of the ring,
we see by reasoning of the same kind as in Art. 123, 4° that if
the situation of the origin on this axis be properly chosen we
may write

_l_ (if lA 1/-2 /I \

-r \K, KJK \±j.

Hence f = J.w + £0> y = Bv, %=Bw,]

Substituting in the equations of Art. 117, we find dp/dt=Q, or
p = const., as is obviously the case. Let us suppose that the ring
is slightly disturbed from a state of motion in which v, w, p, q, r
are zero, i.e. a state of steady motion parallel to the axis. In
the beginning of the disturbed motion v, w, p, q, r will be small
quantities whose products we may neglect. The first of the
equations referred to then gives du/dt = 0, or u = const., and the
remaining equations become

...(3).

Eliminating r, we find

Exactly the same equation is satisfied by w. It is therefore
necessary and sufficient for stability that the coefficient of v on the
right-hand side of (4) should be negative ; and the time of a small
oscillation, in the case of disturbed stable motion, is

11 (5).

We may also notice another case of steady motion of the ring, viz. where
the impulse reduces to a couple about a diameter. It is easily seen that the
equations of motion are satisfied by £, 17, £, X, p = 0, and v constant ; in which
case

u=—£Q/A, r = const.

* Sir W. Thomson, 1. c. ante p. 176.

131-132] MOTION OF A RING. 197

The ring then rotates about an axis in the plane yz parallel to that of 2, at a
distance u/r from it.

For further investigations on the motion of a ring we refer to papers by
Basset*, who has discussed in detail various cases where the axis moves in
one plane, and Miss Fawcettf.

Equations of Motion in Generalized Coordinates.

132. When we have more than one moving solid, or when the
fluid is bounded, wholly or in part, by fixed walls, we may have
recourse to Lagrange's method of ' generalized coordinates.' This
was first applied to hydrodynamical problems by Thomson and
Tait§.

In any dynamical system whatever, if f, rj, % be the Cartesian
coordinates at time t of any particle ra, and X, Y, Z be the com
ponents of the total force acting on it, we have of course

m% = X, 77177= F, m% = Z .................. (1).

Now let f-f- Af, 77 + AT;, f + Af be the coordinates of the same
particle in any arbitrary motion of the system differing infinitely
little from the actual motion, and let us form the equation

= S (Jr A? + FAT; + Z&£) ...... (2),

where the summation 2 embraces all the particles of the system.
This follows at once from the equations (1), and includes these, on
account of the arbitrary character of the variations Af, A?;, Af.
Its chief advantages, however, consist in the extensive elimination
of internal forces which, by imposing suitable restrictions on the
values of A£, A?;, Af we are able to effect, and in the facilities
which it affords for transformation of coordinates.

If we multiply (2) by $t and integrate between the limits t0
and tit then since

* "On the Motion of a Ring in an Infinite Liquid," Proc. Camb. Phil. Soc.,
t. vi. (1887).

t 1. c. ante p. 191.

§ Natural Philosophy (1st ed.), Oxford, 1867, Art. 331.

108 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

we find

p&M (j Af + rj AT; + £A f)l '' - f ' 2m (f Af + ^Ar) + ?A£) eft

» f 2 (XA£ + FAT; + £A?) (ft.
./*

If we put, as usual,

£s) ..................... (3),

this may be written

T + 2 (ZAf + FAr; +

(4).

If we now introduce the condition that in the varied motion
the initial and final positions (at times £0 and ^) shall be respec
tively the same for each particle as in the actual motion, the
quantities Af, AT;, Af vanish at both limits, and the above
equation reduces to

= 0 ...... (5).

This formula is especially valuable in the case of a system
whose freedom is limited more or less by constraints. If
the variations Af, AT;, Af be such as are consistent with these
constraints, some of the internal forces of the system disappear as
a rule from the sum

for example, all the internal reactions between the particles of a
rigid body, and (as we shall prove presently) the mutual pressures
between the elements of an incompressible perfect fluid.

In the case of a * conservative system,' we have

(6),

where V is the potential energy, and the equation (5) takes the
form

(7)*.

* Sir W. E, Hamilton, " Oil a General Method in Dynamics," Phil. Trans.
1834, 1835.

132-133] GENERALIZED COORDINATES.

133. In the systems ordinarily considered in books on Dyna
mics, the position of every particle at any instant is completely
determined by the values of certain independent variables or
'generalized coordinates' ft, ft, •••, so that

The kinetic energy can then be expressed as a homogeneous
quadratic function of the 'generalized velocity-components'
ft, ft,..., thus

2T=Anq1* + A22&+... + 2Al2qlq2 + ............ (9),

where, for example,

v

An = 2m T- + - + >

Wft/ Uft/ )

^ + ##l

-^ -, T^ 7 7 f >

dql dq2 dql dqz)

The quantities Allt Aw,..., AM,... are called the 'inertia-coeffi
cients ' of the system ; they are, of course, in general functions of
the coordinates ft, ^2, .......

Again, we have

2X*Af+FAi7 + £A?)=Q1Agr1 + e2A0a+ ...... (11),

where, for example,

(12).

% dft dq

The quantities Qlt Q2)... are called the 'generalised components of
force.' In the case of a conservative system we have

If X', F', Z' be the components of impulsive force by which the actual
motion of the particle m could be produced instantaneously from rest, we
have of course

m|=Jr, w/7=r', mC = Z' ........................ (i),

and therefore

200 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

Now, from (8) and (10),

by (9). Again

S(X'A£+rtoi + Z'&Q = Qlt*ql + Qj*qs + ............... (iv),

where, for example,

l i

It is evident, on comparison with (12), that $/, $2',..« are the time-
integrals of Qlt Q2,... taken over the infinitely short duration of the impulse,
in other words they are the generalized components of the impulse. Equating
the right-hand sides of (iii) and (v) we have, on account of the independence
of the variations A^, A<?2v?

dT dT

aft<

The quantities

dT dT

are therefore called the ' generalized components of momentum ' of the system,
they are usually denoted by the symbols pl, p2,.... Since T is, by (9), a
homogeneous quadratic function of qlt q2,..., it follows that

In terms of the generalized coordinates q1} q.2)... the equation
(5) becomes

[tl{&T+Q1bql + Q,bq9 + ...}dt = 0 ......... (14),

Jto

where

A^ dT ^ dT A. dT A dT . ,1C.

^T=^r A<7!+ ^.-Ag2+ ... + -7- Ag^ T-Aft-f ... (15).
dq, dq, dq, dq2

Hence, by a partial integration, and remembering that, by hypo
thesis, Ag1? A^2,... all vanish at the limits t0, tl} we find

dT dT \ . ddT dT

......... (16).

Since the values of A^, A^,... within the limits of integration

133-134] LAGRANUE'S EQUATIONS. 201

are still arbitrary, their coefficients must separately vanish. We
thus obtain Lagrange's equations

dT

dt

(17)*

134. Proceeding now to the hydrodynamical problem, let
<?i» %••• be a system of generalised coordinates which serve to
specify the configuration of the solids. We will suppose, for the
present, that the motion of the fluid is entirely due to that of the
solids, and is therefore irrotational and acyclic.

In this case the velocity-potential at any instant will be of the
form

(1),

where <j>lt <f>2)... are determined in a manner analogous to that of
Art. 115. The formula for the kinetic energy of the fluid is then

... + 2A]2g1g2 + ......... (2),

where, for example,

the integrations extending over the instantaneous positions of the
bounding surfaces of the fluid. The identity of the two forms of
A12 follows from Green's Theorem. The coefficients An, A12 ,. . . will,
of course, be in general functions of the coordinates qlt q.2)....

* The above sketch is introduced with the view of rendering more intelligible
the hydrodynamical investigations which follow. Lagrange's proof, directly from
the variational equation of Art. 132 (2), is reproduced in most treatises on
Dynamics. Another proof, by direct transformation of coordinates, not involving
the method of ' variations,' was given in the first instance by Hamilton, Phil. Trans.
1835, p. 96 ; the same method was employed by Jacobi, Vorlesungen iiber Dynamik
(ed. Clebsch), Berlin, 1864,'p. 64, Werke, Supplementband, p. 64; by.Bertrand in the
notes to his edition of the Mecanique Analytique, Paris, 1853 ; and more recently by
Thomson and Tait, Natural Philosophy, (2nd ed.) Art. 318.

202 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

If we add to (2) twice the kinetic energy, Tl, of the solids
themselves, we get an expression of the same form, with altered
coefficients, say

2T=Anq1* + Az,&+... + 2Al.2q1q,+ (4).

It remains to shew that the equations of motion of the solids
can be obtained by substituting this value of T in the Lagrangian
equations, Art. 133 (17). We cannot assume this without further
consideration, for the positions of the various particles of the
fluid are evidently not determined by the instantaneous values
</i> q-2)--' of the coordinates of the solids.

Going back to the general formula

/;

dt

1*i

(5),

let us suppose that in the varied motion, to which the symbol A
refers, the solids undergo no change of size or shape, and that -the
fluid remains incompressible, and has, at the boundaries, the same
displacement in the direction of the normal as the solids with
which it is in contact. It is known that under these conditions
the terms due to the internal forces of the solids will disappear
from the sum

The terms due to the mutual pressures of the fluid elements are
equivalent to

or

jjp (JA£ + m^ + »A{) dS +fffp (^f +

where the former integral extends over the bounding surfaces,
1, 7ii, n denoting the direction-cosines of the normal, drawn towards
the fluid. The volume-integral vanishes by the condition of
incompressibility,

| |

dx dy dz

The surface-integral vanishes at a fixed boundary, where

134] APPLICATION TO HYDRODYNAMICS. 20o

and in the case of a moving solid it is cancelled by the terms due
to the pressure exerted by the fluid on the solid. Hence the
symbols X, Y, Z may be taken to refer only to the extraneous
forces acting on the system, and we may write

......... (6),

where Qlt Q2,... now denote the generalized components of ex
traneous force.

We have still to consider the right-hand side of (5). Let us
suppose that in the arbitrarily varied motion the initial and final
positions of the solids are respectively the same as in the actual
motion. For every particle of the solids we shall then have

Af=0, A77 = 0, A? = 0,

at both limits, but the same will not hold as a rule with regard to
the particles of the fluid. The corresponding part of the sum

will however vanish ; viz. we have

^^ S I 7 t— */ I 7 fc-

v ay dz

= pff(t> (l^ + m&Tj + jiAJ) dflf

of which the second term vanishes by the condition of incom-
pressibility, and the first term vanishes at the limits £0 and tl}
since we then have, by hypothesis,

at the surfaces of the solids. Hence, under the above conditions,
the right-hand side of (5) vanishes, and therefore

....}«ft«d ............ (7).

The varied motion of the fluid has still a high degree of
generality. We will now farther limit it by supposing that
whilst the solids are, by suitable forces applied to them, made to
execute an arbitrary motion, subject to the conditions that Ag1}

204 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

Ag2,...=0 for t = tQ and t = tl} the fluid is left to take its own
course in consequence of this. The varied motion of the fluid
will now be irrotational, and therefore T+AT will be the same
function of the varied coordinates q + Ag, and the varied velocities
q + Ag, that T is of q and q. Hence we may write, in (7),

dT . . dT . dT . dT . /o\*.

The derivation of the Lagrangian equations then follows
exactly as before.

It is a simple consequence of Lagrange's equations, thus established for
the present case, that the generalized components of the impulse by which
the actual motion at any instant could be generated instantaneously from
rest are

dT dT

¥1' ¥2""'

If we put 7T=T + T1, we infer that the terms

dT dT
W d&~

must represent the impulsive pressures which would be exerted by the solids
on the fluid in contact with them.

This may be verified as follows. If A£, Ar/, A£ denote arbitrary variations
subject only to the condition of incompressibility, and to the condition that
the fluid is to remain in contact with the solids, it is found as above that,
considering the fluid only,

(i).
Now by the kinematical condition to be satisfied at the surface, we have

jA£ + mAi7 + wA£=- jkA?1_ JbAg,2_ ............... (ii),

\AjJlt (.If ll

and therefore

^ dn

'2 + ---)A2'i
dT

by (1), (2), (3) above. This proves the statement.

With the help of equation (iii) the reader may easily construct a proof of
Lagrange's equations, for the present case, analogous to that usually given in
text-books of Dynamics.

* This investigation is amplified from Kirchhoff, I.e. ante p. 167.

134-135] MOTION OF A SPHERE. 205

135. As a first application of the foregoing theory we may
take an example given by Thomson and Tait, where a sphere is
supposed to move in a liquid which is limited only by an infinite
plane wall.

Taking, for brevity, the case where the centre moves in a
plane perpendicular to that of the wall, let us specify its position
at time t by rectangular coordinates x, y in this plane, of which y
denotes the distance from the wall. We have

(1),

where A and B are functions of y only, it being plain that the
term xy cannot occur, since the energy must remain unaltered
when the sign of x is reversed. The values of A, B can be
written down from the results of Arts. 97, 98, viz. if in denote the
mass of the sphere, and a its radius, we have

•(2),

U '

approximately, if y be great in comparison with a.
The equations of motion give
d

(3),

where X, Y are the components of extraneous force, supposed to
act on the sphere in a line through the centre.

If there be no extraneous force, and if the sphere be projected
in a direction normal to the wall, we have x = 0, and

Eif1 = const ............................ (4).

Since B diminishes as y increases, the sphere experiences an
acceleration from the wall.

Again, if the sphere be constrained to move in a line parallel to
the wall, we have y = 0, and the necessary constraining force is

Since dA/dy is negative, the sphere appears to be attracted by the

206 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

wall. The reason of this is easily seen by reducing the problem to
one of steady motion. The fluid velocity will evidently be greater,
and the pressure, therefore, will be less, on the side of the sphere
next the wall than on the further side ; see Art. 24.

The above investigation will also apply to the case of two
spheres projected in an unlimited mass of fluid, in such a way
that the plane y = 0 is a plane of symmetry as regards the motion.

136. Let us next take the case of two spheres moving in the
line of centres.

The kinematical part of this problem has been treated in Art. 97. If we
now denote by #, y the distances of the centres A, B from some fixed origin 0
in the line joining them, we have

* .............................. (i),

where the coefficients Z, J/, N are functions of c, —y - x. Hence the equations
of motion are

. . dJ\r .

dr

where Jf, Y are the forces acting on the spheres along the line of centres. If
the radii «, b are both small compared with c, we have, by Art. 97 (xv),
keeping only the most important terms,

...(iii)

approximately, where m, m' are the masses of the two spheres. Hence to
this order of approximation

dL dM a3&3 dN

-jf — 0. -7— = — OTTO — T~ , — y- = 0.

dc dc c4 ' dc

If each sphere be constrained to move with constant velocity, the force
which must be applied to A to maintain its motion is

dM . dM .

This tends towards B, and depends only on the velocity of B. The spheres
therefore appear to repel one another ; and it is to be noticed that the apparent
forces are not equal and opposite unless x= ±y.

Again, if each sphere make small periodic oscillations about a mean
position, the period being the same for each, the mean values of the first terms
in (ii) will be zero, and the spheres therefore will appear to act on one another
with forces equal to

135-137] MUTUAL INFLUENCE OF TWO SPHERES. 207

where \£y] denotes the mean value of xy. If #, y differ in phase by less than a
quarter- period, this force is one of repulsion, if by more than a quarter- period
it is one of attraction.

Next, let B perform small periodic oscillations, while A is held at rest. The
mean force which must be applied to A to prevent it from moving is

where [y2] denotes the mean square of the velocity of B. To the above order of
approximation dNjdc is zero, but ori reference to Art. 97 (xv) we find that
the most important term in it is - 127rpa3&°/cr, so that the force exerted on
A is attractive, and equal to

This result comes under a general principle enunciated by Lord Kelvin.
If we have two bodies immersed in a fluid, one of which (A) performs small
vibrations while the other (B) is held at rest, the fluid velocity at the surface
of B will on the whole be greater on the side nearer A than on that which is
more remote. Hence the average pressure on the former side will be less
than that on the latter, so that B will experience on the whole an attraction
towards A. As practical illustrations of this principle we may cite the
apparent attraction of a delicately-suspended card by a vibrating tuning-fork,
and other similar phenomena studied experimentally by Guthrie* and
explained in the above manner by Lord Kelvin f.

Modification of Lagranges Equations in the case of
Cyclic Motion.

137. We return to the investigation of Art. 134, with the
view of adapting it to the case where the fluid has cyclic irrota-
tional motion through channels in the moving solids, or (it may be)
in an enclosing vessel, independently of the motion of the solids
themselves.

If K, K, K", ..., be the circulations in the various independent
circuits which can be drawn in the space occupied by the fluid, the
velocity-potential will now be of the form

0 + £o,

where £=&&+&&+ ........................ (1),

* "On Approach caused by Vibration," Proc. Roy. Soc., t. xix., (1869); Phil.
Mag. , Nov. 1870.

t Reprint of Papers on Electrostatics, &c., Art. 741. For references to further
investigations, both experimental and theoretical, by Bjerknes and others on the
mutual influence of oscillating spheres in a fluid, see Hicks, "Beport on Eecent
Researches in Hydrodynamics," Brit. Ass. Rep., 1882, pp. 52...; Winkelmann,
Handbuch der Physik, Breslau, 1891..., t. i., p. 435.

208 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

the functions <j>19 fa, ..., being determined by the same conditions
as in Art. 134, and

</>0 = KO) + re V + ........................ (2),

&), &)', ... , being cyclic velocity-potentials determined as in Art. 129.
Let us imagine barrier-surfaces to be drawn across the several
channels. In the case of channels in a containing vessel we shall
suppose these ideal surfaces to be fixed in space, and in the case of
channels in a moving solid we shall suppose them to be fixed
relatively to the solid. Let us denote by #0, £0', •-., the portions
of the fluxes across these barriers which are due to the cyclic motion
alone, and which would therefore remain if the solids were held at
rest in their instantaneous positions, so that, for example,

*--//£

*=-« <*— '^'-

where So-, Scr', ... are elements of the several barriers. The total
fluxes across the respective barriers will be denoted by % + %0>
%' + %o'> . . . , so that %, %', . . . would be the surface-integrals of the
normal velocity of the fluid relative to the barriers, if the motion of
the fluid were entirely due to that of the solids, and therefore
acyclic.

The expression of Art. 55 for twice the kinetic energy of the
fluid becomes, in our present notation,

This reduces, exactly as in Art. 129, to the sum of two homogene
ous quadratic functions of q1} q%, ... , and of K, K, ..., respectively*.
Thus the kinetic energy of the fluid is equal to

T + # .............................. (5),

with 2T = An^2 + A22£22 + . . . + 2A12g1g2 + ......... (6),

and 2K = (rc, K)^ + (K') K) ^ + . . . + 2 (*, K')KK' '+ ...... (7),

where, for example,

An example of this reduction is furnished by the calculation of Art. 99.

137] CYCLIC MOTION. 209

and («,«) = - ,? AT,

dn"" }

It is evident that K is the energy of the cyclic motion which
remains when the solids are maintained at rest in the configuration
(qly q,, ...)• We note that, by (3), (7), and (9),

dK dK

.................. (11).

If we add to (5) the kinetic energy of the solids themselves, we
obtain for the total kinetic energy of the system an expression of
the form

T=1S, + K ........................... (12),

where 2® = Auq,2 + A?>q<? + ... + ZA^q, + ......... (13),

the coefficients being in general functions of qlt q2) .......

To obtain the equations of motion we have recourse as before
to the formula

2 Xb+ FAT + £A dt

......... (14).

The only new feature is in the treatment of the expression on
the right-hand side. By the usual method of partial integration
we find

dxdydz

+ p« f |(JAf + mAri + »Af ) dff+pK I |(

............... (15),

where /, m, ?? are the direction-cosines of the normal to any

element 8S of a bounding surface, drawn towards the fluid, or

L. 14

210 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

(as the case may be) of the normal to a barrier, drawn in the
direction in which the circulation is estimated.

Let us now suppose that the slightly varied motion, to which
A refers, is arbitrary as regards the solids, except only that the
initial and final configurations are to be the same as in the actual
motion, whilst the fluid is free to take its own course in accordance
with the motion of the solids. On this supposition we shall have,
both at time t0 and at time tlt

for the fluid in contact with an element SS of the surface of a
solid, and, at the barriers,

The right-hand side of (14) therefore reduces, under the present

suppositions, to

r ~\tt

L*4 (x + xo) + PIC' A (xf + XQ') + .. .J ,

and the equation may be put into the form

{ A T - p* A (X + %o) - PK' A (xf + ft') - ...

+ QiA?1 + QaA?a+...}d* = 0 ...... (16)*,

which now takes the place of Art. 134 (7).

Since the variation A does not affect the cyclic constants
K, K, ..., we have by (11),

and therefore, by (12),

...}<fc = 0 ...... (17).

It is easily seen that x, x, ... are linear functions of qlf q.2, ...,
say

%= MI + O-&+ -..}

^/ = «1/g1 + aL/g,+ ... I .................. (18),

where the coefficients are in general functions of qlt q.2, If we write

* Cf. Larmor, " On the Direct Application of the Principle of Least Action to
the Dynamics of Solid and Fluid Systems," Proc. Lond. Math. Soc., t. xv. (1884).

137-138] MODIFICATION OF LAGRANGE's EQUATIONS. 211

for shortness, & = p/c^ + p/e'a/ + . . . \

h/«V+... } (19),

J

the formula (17) becomes

(20).

Selecting the terms in A^, A^ from the expression to be
integrated, we have

i 7 2 y- •••

! ^ % ""dft dql

Hence by a partial integration, remembering that A^, A^2, ...
vanish at both limits, and equating to zero the coefficients of
Aga, A</o,... which remain under the integral sign, we obtain
the first of the following symmetrical system of equations :

d d& d& dK

T* ^-^ + (1,2)^., + (1,3)^+... + j-

dt dql dql dql

d d& d& dK

d df& d1® . dK

It d ~d+ (3' 1} «' + (3' 2) «' + ' ' ' +

We have here introduced the notation

(r}S) = d/s-d^ (22),

dqr dqs

and it is important to notice that (r, s) = — (s, r).

138. The foregoing investigation has been adopted as leading
directly, and in conformity with our previous work, to the desired
result ; but it may be worth while to give another treatment of
the question, which will bring out more fully the connection with
the theory of ' gyrostatic ' systems, and the method of ' ignoration
of coordinates.'

* These equations were first given in a paper by Sir W. Thomson, "On the
Motion of Rigid Solids in a Liquid circulating irrotationally through perforations
in them or in a Fixed Solid," Phil. Mag. , May 1873. See also C. Neumann, Hydro-
dynamische Untersuchungen (1883).

14—2

212 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

It will be necessary to modify, to some extent, our previous
notation. Let us now denote by %, %, %", . . . the total fluxes
relative to the several barriers of the region, which we shall as
before regard as ideal surfaces fixed relatively to the solid surfaces
on which they abut ; and let %, %', ^", ... be the time-integrals
of these fluxes, reckoned each from an arbitrary epoch. We
shall shew, in the first place, that the Lagrangian equations
(Art. 133 (1*7)) will still hold in the case of cyclic motion, provided
these quantities ^, ^', %' ', . . . are treated as additional generalized
coordinates of the system.

Let qlt q2) ... be, as before, the system of generalized coordi
nates which specify the positions of the moving solids. The
motion of the fluid at any instant is completely determined by
the values of the velocities q}) q.2)..., and of the fluxes X> X> •-> as
above defined. For if there were two types of irrotational motion
consistent with these values, then, in the motion which is the
difference of these, the bounding surfaces, and therefore also the
barriers, would be at rest, and the flux across each barrier would
be zero. The formula (5) of Art. 55 shews that the kinetic energy
of such a motion would be zero, and the velocity therefore every
where null.

It follows that the velocity-potential (4>, say) of the fluid
motion can be expressed in the form

. .-+%«+%V+ ............ (1),

where </>1? for example, is the velocity-potential of the motion
corresponding to

which we have just seen to be determinate.

The kinetic energy of the fluid is given by the expression

Substituting the value of <I> from (1), and adding the energy of
motion of the solids, we see that the total kinetic energy of the
system (T, say) is a homogeneous quadratic function of the quan
tities qlt fa, ...,%,%,..., with coefficients which are functions of
qlf q,, ..., only.

138-139] INTRODUCTION OP FLUX-COORDINATES. 213

We now recur to the formula (4) of Art. 132. The variations
Af, A?;, A f being subject to the condition of incompressibility, the
part of the sum

Sm(|Af + iJAi|-f£AS) (3)

which is due to the fluid is, in the present notation,

M

m AT; + 7iA f) dS

where the surface-integral extends over the bounding surfaces
of the fluid, and the symbols K, K', ... denote as usual the cyclic
constants of the actual motion. We will now suppose that
the varied motion of the solids is subject to the condition that
A</15 Ag2, ... = 0, at both limits (£0 and ^), that the varied motion
of the fluid is irrotational and consistent with the motion of the
solids, and that the (varied) circulations are adjusted so as to make
A%, AX', . . . also vanish at the limits. Under these circumstances
the right-hand side of the formula cited is zero, and we have

{AT+ 2 (ZAf + FAT; + ZAJ)} = 0 ............ (5).

If we assume that the extraneous forces do on the whole no
work when the boundary of the fluid is at rest, whatever relative
displacements be given to the parts of the fluid, the generalized
components of force corresponding to the coordinates %, %',... will
be zero, and the formula may be written

ri

J to

•(6)<

where

— A* —A- —A dT&( dl 'A* — A"

dq± dq.z " dq: dq2 dx dx

+ (7).

139. If we now follow out the process indicated at the end of
Art. 133, we arrive at the equations of motion for the present
case, in the forms

d_dT__dT_Q d_dT _dT _
dtdqi dq1 dtdq2 dq.2

d^dT _ d_dT^
dtdx dtdx' '""

214 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

Equations of this type present themselves in various problems
of ordinary Dynamics, e.g. in questions relating to gyrostats, where
the coordinates %,%',• • • , whose absolute values do not affect the
kinetic and potential energies of the system, are the angular
coordinates of the gyrostats relative to their frames. The general
theory of systems of this kind has been treated independently by
Routh* and by Thomson and Tait-)-. It may be put, briefly, as
follows.

We obtain from (1), by integration,

(2),

where, in the language of the general theory, C, C', ... are the
constant momenta corresponding to the coordinates %, %', ____

In the hydrodynamical problem, they are equal to p«, prc' ', . . . , as
will be shewn later, but we retain for the present the more
general notation.
Let us write

S = T-Cx-ffx'- ..................... (3).

The equations (2) when written in full, determine %, %',••. as
linear functions of C, C' , ... and <j,, q.2)...', and by substitution in
(3) we can express ® as a quadratic function of q1} q2, ...,C,G' .....

On this supposition we have, performing the arbitrary variation A
on both sides of (3), and omitting terms which cancel, by (2),
d® d® d®

where, for brevity, only one term of each kind is exhibited.

Hence

d® = dT d® = dT_

dq\ dq± ' dq.2 dq.2 '

d® dT d® dT

.(5).

d®= . d® = .,
dO~ X' dCf X>'

* On the Stability of a given State of Motion (Adams Prize Essay), London, 1877.
t Natural Philosophy, 2nd edition, Art. 319 (1879).

See also von Helmholtz, "Principien der Statik monocyclischer Systeme," Crelle,
t. xcvii. (1884).

139] IGNORATION OF COORDINATES. 215

Hence the equations (1) now take the form
d d® d®

__ (6)*,

dt dq.2

from which the velocities ^, ^'> ... corresponding to the 'ignored'
coordinates %, %',••• have been eliminated.

In the particular case where

0=0, C' = 0,...,

these equations are of the ordinary Lagrangian form, ® being now
equal to T, with the velocities ^, ;£', ... eliminated by means of the
relations

dT d'f

dx 0> *r

so that © is now a homogeneous quadratic function of qlt q.2,....
Cf. Art. 134 (4).

In the general case we proceed as follows. If we substitute in
(3) from the last line of (5) we obtain

Now, remembering the composition of ®, we may write, for a
moment

<H)=:<H)2)0 + <H)1)1+@0)2 ..................... (8),

where ®2j0 is a homogeneous quadratic function of qL, <£>,...,
without C, C',...j <H)lfl is a bilinear function of these two sets
of quantities; and ©0)2 is a homogeneous quadratic function
of C, C',..., without qL> q.,, .... Substituting in (7), we find

T=ea>0-®0>3 ........................ (9),

or, to return to our previous notation,

T=1& + K ........................... (10),

where *& and K are homogeneous quadratic functions of qlt q.,, ...

* Eouth, I, c.

216

MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

and of C, C',..., respectively. Hence (8) may be written in the
form

where j3lf /32, ... are linear functions of C, C', ..., say

.(12).

The meaning of the coefficients a1) «2> • • •, ai> «s'> •»•>••• aPPears fr°m
the last line of (5), viz. we have

dK

dK

.(13).

Compare Art. 137 (18).

If we now substitute from (11) in the equations (6) we obtain
the general equations of motion of a ' gyrostatic system/ in the form

d
dt

d
dt dq

d^d®
dt dq3

dql

T*^-^+X2,l)*

dq3

(I,2)g2+(l,

+ (2,
(3,2)<72

dK

dK
dq2

dK

where

dq.

(14)*,

.(15).

The equations (21) of Art. 137 are a particular case of these. To complete
the identification it remains to shew that, in the hydrodynamical application,

C=PK, C' = PK',..,, (i).

For this purpose we may imagine that in the instantaneous generation of
the actual motion from rest, the positions of the various barriers are
occupied for a moment by membranes to which uniform impulsive pressures
pK, pKj . . . are applied as in Art. 54, whilst impulsive forces are simultaneously
applied to the respective solids, whose force and couple resultants are
equal and opposite to those of the pressures f. In this way we obtain a
system of generalized components of impulsive force, corresponding to the

* These equations were obtained, in a different manner, by Thomson and Tait,
?. c. ante p. 214.

t Sir W. Thomson, I c. ante p. 211.

139-140] MOVING SPHERE IN CYCLIC REGION. 217

coordinates ^, ^', ..., viz. the virtual moment of this system is zero for any
infinitely small displacements of the solids, so long as x, /, ... do not vary.
We may imagine, for example, that the impulses are communicated to the
membranes by some mechanism attached to the solids and reacting on these*.
Denoting these components by X, X', ..., and considering an arbitrary variation
of x, x', ... only, we easily find, by an adaptation of the method employed near
the end of Art. 134, that

dT dT

'

whence the results (i) follow.

The same thing may be proved otherwise as follows. From the equations
(2) and (1) of Art. 138, we find

dT /d$ da> d& da , d® dco

da>

since v2<w = 0. The conditions by which o> is determined are that it is the
value of * when

i.e. o) is the velocity-potential of a motion in which the boundaries, and
therefore also the barriers, are fixed, whilst

Hence the right-hand side of (iii) reduces to px, as was to be proved.

140. A simple application of the equations (21) of Art. 137
is to the case of a sphere moving through a liquid which circulates
irrotationally through apertures in a fixed solid.

If the radius (a, say) of the sphere be small compared with its least
distance from the fixed boundary, then C? the kinetic energy of the system
when the motion of the fluid is acyclic, is given by Art. 91, viz.

2C=m(X-2+?/2+i2) .............................. (i),

where m now denotes the mass of the sphere together with half that of the
fluid displaced by it, and #, y, s are the Cartesian coordinates of the centre.
And by the investigation of Art. 99, or more simply by a direct calculation,
we have, for the energy of the cyclic motion by itself,

2/i= const. -27rpa3(w2 + ?/2 + w2) ..................... (ii).

Again the coefficients alf a.,, a3 of Art. 137 (18) denote the fluxes across
the first barrier, when the sphere moves with unit velocity parallel to x, y, z,
respectively. If we denote by O the flux across this barrier due to a unit
simple-source at (#, y, z\ then remembering the equivalence of a moving
sphere to a double-source (Art. 91), we have

a^fyPdQldx, a2 = ±a*da/dy, a3 = %a?d£l/dz ............... (iii),

* Burton, Phil. Mag., May, 1893.

218 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

so that the quantities denoted by (2, 3), (3, 1), (1, 2) in Art. 137 (21) vanish
identically. The equations therefore reduce in the present case to

.._ dW .._ dW_ .. dW

dx ' dv ' * dz

where W=7rpa?(t<;* + v2 + iv2) .............................. (v),

and A', F, Z are the components of extraneous force applied to the sphere.

By an easy generalization it is seen that the equations (iv) must apply to
any case where the liquid is in steady (irrotational) motion except in so far as
it is disturbed by the motion of the small sphere. It is not difficult, moreover,
to establish the equations by direct calculation of the pressures exerted on the
sphere by the fluid.

When Jf, F, Z=0, the sphere tends to move towards places where the
undisturbed velocity of the fluid is greatest.

For example, in the case of cyclic motion round a tixed circular cylinder
(Arts. 28, 64), the fluid velocity varies inversely as the distance from the axis.
The sphere will therefore move as if under the action of a force towards this
axis varying inversely as the cube of the distance. The projection of its path
on a plane perpendicular to the axis will therefore be a Cotes' spiral*.

141. If in the equations (21) of Art. 137 we put (ji=0,
^2=0,..., we obtain the generalized components of force which are
required in order to maintain the solids at rest, viz.

o - dK a -dK m

yi~%' ys-dfc'"" '

We are not dependent, of course, for this result, on the
somewhat intricate investigation which precedes. If the solids
be guided from rest in the configuration (ql} q2,...) to rest in the
configuration (ql-\-^.qlt q.2 + Aq2,...), the work done on them
is ultimately equal to

which must therefore be equal to the increment &.K of the kinetic
energy. This gives at once the equations (1).

The forces representing the pressures of the fluid on the
solids (at rest) are obtained by reversing the sign in (1), viz.
they are

_ dK _dK
dq, ' dq.2 ' '

The solids tend therefore to move so that the kinetic energy
of the cyclic motion diminishes.

* Sir W. Thomson, I.e. ante p. 211.

140-141] PRESSURES ON SOLIDS AT REST. 219

It appears from Art. 137 (10), that under the present circum
stances the fluxes through the respective apertures are given by

dK . dK

By solving these equations, the circulations K, #',... can be ex
pressed as linear functions of ^0, ;£„', ....

If these values of K, K, ...be substituted in K we obtain
a homogeneous quadratic function of ^0, ^', — When so ex
pressed, the kinetic energy of the cyclic motion may be denoted
by T0. We have then, exactly as in Art. 119,

T,+ K=2K = p*x> + pic'x;+ ..................... (4),

so that if, for the moment, the symbol A be used to indicate
a perfectly general variation of these functions, we have

dT() A dT0 . dTn , dT0

,- . ° AXO + , . A£0' + ... + • y— Aft +
d%o dxo dq, dq2

dK A dK . dK A dK A

+ A* + ---A*' + ...+ i-Aft + ,- A^ + ...
d/c d/c dqt dq.,

Omitting terms which cancel by (3), and equating coefficients
of the variations A^0, A^(/, ..., A^, A^,..., which form an inde
pendent system, we find

and

Hence the generalized components (2) of the pressures exerted by
the fluid on the solids when held at rest may also be expressed in
the forms

dT0 dT0 f .

dqi> 1fc>-

It will be shewn in Art. 152 that the energy K of the cyclic fluid
motion is proportional to the energy of a system of electric current-sheets
coincident with the surfaces of the fixed solids, the current-lines being
orthogonal to the stream-lines of the fluid.

220 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI

The electromagnetic forces between conductors carrying these currents
are proportional* to the expressions (2) with the signs reversed. Hence in
the hydrodynamical problem the forces on the solids are opposite to those
which obtain in the electrical analogue. In the particular case where the
fixed solids reduce to infinitely thin cores, round which the fluid circulates,
the current-sheets in question are practically equivalent to a system of
electric currents flowing in the cores, regarded as wires, with strengths *, *', ...
respectively. For example, two thin circular rings, having a common axis,
will repel or attract one another according as the fluid circulates in the
same or in opposite directions through themf. This might have been
foreseen of course from the principle of Art. 24.

Another interesting case is that of a number of open tubes, so narrow as
not sensibly to impede the motion of the fluid outside them. If flow be
established through the tubes, then as regards the external space the extremi
ties will act as sources and sinks. The energy due to any distribution of
positive or negative sources mlt m2, ... is given, so far as it depends on the
relative configuration of these, by the integral

taken over a number of small closed surfaces surrounding mlt w2, ... respec
tively. If 01? <£2, ... be the velocity-potentials due to the several sources, the
part of this expression which is due to the simultaneous presence of m1} m2 is

which is by Green's Theorem equal to

. dd>9 Try .....

d>, / 2dS (m).

1 dn

Since the surface-integral of d<f>2/dn is zero over each of the closed surfaces
except that surrounding w2, we may ultimately confine the integration to the
latter, and so obtain

Since the value of (f)1 at m2 is mjr12, where r12 denotes the distance between
wij and ?n2, we obtain, for the part of the kinetic energy which varies with the
relative positions of the sources, the expression

* Maxwell, Electricity and Magnetism, Art. 573.

t The theorem of this paragraph was given by Kirchhofi, 1. c. ante p. 59. See
also Sir W. Thomson, "On the Forces experienced by Solids immersed m a Moving
Liquid," Proc. R. S. Edin., 1870; Reprint, Art. xli. ; and Boltzmann, " Ueber die
Druckkrafte welche auf Hinge wirksam sind die in bewegte Fliissigkeit tauchen,"
Crelle, t. Ixxiii. (1871).

141] MUTUAL ACTION OF SOURCES. 221

The quantities mx, w2, ... are in the present problem equal to l/4?r times
the fluxes £0, xo'j ••• across the sections of the respective tubes, so that (v)
corresponds to the form T0 of the kinetic energy. The force apparently ex
erted by wij on w2, tending to increase r12, is therefore, by (8),

d

Hence two sources of like sign attract, and two of unlike sign repel, with
forces varying inversely as the square of the distance*. This result, again,
is easily seen to be in accordance with general principles. It also follows,
independently, from the electric analogy, the tubes corresponding to Ampere's
' solenoids.'

We here take leave of this somewhat difficult part of our
subject. To avoid the suspicion of vagueness which sometimes
attaches to the use of 'generalized coordinates/ an attempt has
been made in this Chapter to make the treatment as definite as
possible, even at some sacrifice of generality in the results. There
can be no doubt, for example, that with proper interpretations the
equations of Art. 137 will apply to the case of flexible bodies
surrounded by an irrotationally moving fluid, and even to cases of
isolated vortices (see Chap, vn.), but the justification of such
applications belongs rather to general Dynamics-)-.

* Sir W. Thomson, Reprint, Art. xli.

t For further investigations bearing on the subject of this Chapter see J. Purser,
"On the Applicability of Lagrange's Equations in certain Cases of Fluid Motion,"
Phil. Mag., Nov. 1878 ; Larmor, I.e. ante p. 210 ; Basset, Hydrodynamics, Cam
bridge, 1888, c. viii.

CHAPTER VII.

VORTEX MOTION.

142. OUR investigations have thus far been confined for the
most part to the case of irrotational motion. We now proceed to
the study of rotational or ' vortex ' motion. This subject was first
investigated by von Helmholtz*; other and simpler proofs of some
of his theorems were afterwards given by Lord Kelvin in the paper
on vortex motion already cited in Chapter ill.

We shall, throughout this Chapter, use the symbols f, 77, f to
denote, as in Chap, in., the components of the instantaneous
angular velocity of a fluid element, viz.

dw dv\ du dw . dv du\ .

A line drawn from point to point so that its direction is every
where that of the instantaneous axis of rotation of the fluid is
called a ' vortex-line/ The differential equations of the system of
vortex -lines are

dx _dy _dz ,,..

~~f^ — — ~~£T ........................ \ )'

£ V ?

If through every point of a small closed curve we draw the
corresponding vortex-line, \ve obtain a tube, which we call a

* "Ueber Integrate der hydrodynamischen Gleichungen welche den Wirbel-
bewegungen entsprechen," Crelle, t. lv. (1858); Ges. Alh., t. i., p. 101.

142] VORTEX-FILAMENTS. 223

' vortex-tube.' The fluid contained within such a tube constitutes
what is called a ' vortex-filament/ or simply a ' vortex.'

Let ABC, A'B'C' be any two circuits drawn on the surface of a
vortex-tube and embracing it, and let AA' be a connecting line
also drawn on the surface. Let us apply the theorem of Art. 33 to
the circuit ABCAA'C'B'A'A and the part of the surface of the

tube bounded by it. Since 1% + mi] + n% is zero at every point of
this surface, the line-integral

j(ndx + vdy -f wdz),

taken round the circuit, must vanish; i.e. in the notation of
Art. 32

/ (ABC A) + I(AA) + I(A'C'BA) + I(A'A} = Q,

which reduces to

= I(AB'C'A).

Hence the circulation is the same in all circuits embracing the
same vortex-tube.

Again, it appears from Art. 32 that the circulation round the
boundary of any cross- section of the tube, made normal to its
length, is 2&>cr, where co, = (f 2 + rf + f2)*, is the angular velocity of
the fluid, and a the infinitely small area of the section.

Combining these results we see that the product of the angular
velocity into the cross- section is the same at all points of a vortex.
This product is conveniently termed the ' strength ' of the vortex.

The foregoing proof is due to Lord Kelvin ; the theorem itself was first
given by von Helmholtz, as a deduction from the relation

224 VORTEX MOTION. [CHAP. VII

which follows at once from the values of £, ?/, £ given by (1). In fact, writing
in Art. 42 (1), & 77, £ for U, V, W, respectively, we find

S}(l£ + mq+nQdS=0 (ii),

where the integration extends over any closed surface lying wholly in the
fluid. Applying this to the closed surface formed by two cross-sections of a
vortex-tube and the portion of the tube intercepted between them, we find
o)1o-1 = a)2o-2, where o)l5 a>2 denote the angular velocities at the sections a-lt <r2,
respectively.

Lord Kelvin's proof shews that the theorem is true even when £, 17, £ are
discontinuous (in which case there may be an abrupt bend at some point of a
vortex), provided only that u, v, w are continuous.

An important consequence of the above theorem is that a
vortex-line cannot begin or end at any point in the interior of
the fluid. Any vortex-lines which exist must either form closed
curves, or else traverse the fluid, beginning and ending on its
boundaries. Compare Art. 37.

The theorem of Art. 33 (4) may now be enunciated as follows :
The circulation in any circuit is equal to twice the sum of the
strengths of all the vortices which it embraces.

143. It was proved in Art. 34 that, in a perfect fluid whose
density is either uniform or a function of the pressure only, and
which is subject to extraneous forces having a single-valued
potential, the circulation in any circuit moving with the fluid is
constant.

Applying this theorem to a circuit embracing a vortex-tube we
find that the strength of any vortex is constant.

If we take at any instant a surface composed wholly of vortex-
lines, the circulation in any circuit drawn on it is zero, by Art. 33,
for we have 1% + mr) + n£ = 0 at every point of the surface. The
preceding article shews that if the surface be now supposed to
move with the fluid, the circulation will always be zero in any
circuit drawn on it, and therefore the surface will always consist
of vortex-lines. Again, considering two such surfaces, it is plain
that their intersection must always be a vortex-line, whence
we derive the theorem that the vortex-lines move with the
fluid.

This remarkable theorem was first given by von Helmholtz for
the case of liquids ; the preceding proof, by Lord Kelvin, shews
that it holds for all fluids subject to the conditions above stated.

142-143] PERSISTENCE OF VORTICES. 225

One or two independent proofs of the theorem may be briefly indicated.

Of these perhaps the most conclusive is based upon a slight generalization
of some equations given originally by Cauchy in his great memoir on Waves*,
and employed by him to demonstrate Lagrange's velocity-potential theorem.

The equations (2) of Art. 15, yield, on elimination of the function x by
cross-differentiation,

du dy _ du dx dv dy dv d}/ dw dz dw dz _ dwQ dvQ
~ + ~ + ~~~=~

(where u, v, w have been written in place of dx/dt, dy/dt, dzfdt^ respectively),
with two symmetrical equations. If in these equations we replace the
differential coefficients of u, v, w with respect to a, b, c, by their values in
terms of differential coefficients of the same quantities with respect to ,v, y, z,
we obtain

d(a, 6) d(a, 6)

If we multiply these by dxfda, dxjdb, dxldc, in order, and add, then, taking
account of the Lagrangian equation of continuity (Art. 14(1)) we deduce the
first of the following three symmetrical equations :

=
P Po da Po db PQ do '

1L = &>dy + riQ dy + tv<fy_

p p0 da p0 db PQ dc '

p po da PQ db pQ dc

In the particular case of an incompressible fluid (p = p0) these differ only in
the use of the notation £, 77, f from the equations given by Cauchy. They
shew at once that if the initial values £0, ?70, £0 of the component rotations
vanish for any particle of the fluid, then £, 77, £ are always zero for that
particle. This constitutes in fact Cauchy's proof of Lagrange's theorem.

To interpret (ii) in the general case, let us take at time £ = 0 a linear
element coincident with a vortex-line, say

where e is infinitesimal. If we suppose this element to move with the fluid,
the equations (ii) shew that its projections on the coordinate axes at any other
time will be given by

* I. c. ante p. 18.
L. 15

226

VORTEX MOTION.

[CHAP. VII

i.e. the element will still form part of a vortex-line, and its length (&, say) will
vary as eo/p, where o> is the resultant angular velocity. But if o- be the cross-
section of a vortex-filament having 8s as axis, the product pa-ds is constant with
regard to the time. Hence the strength coo- of the vortex is constant*.

The proof given originally by von Helmholtz depends on a system of three
equations which, when generalized so as to apply to any fluid in which p is a
function of p only, become

Dt ...

D (£\ £ dw T) dw £ dw
Dt \p) p dx p dy p dz

These may be obtained as follows. The dynamical equations of Art. 6
may be written, when a force-potential Q, exists, in the forms

provided

where q* = uz + v2 + w2. From the second and third of these we obtain, elimina
ting x by cross-differentiation,

_,,,^^,_.<fa. **• -'*

dt dy dz

Remembering the relation

dz

(vi),

and the equation of continuity

Dp fdu dv dw
Dt \dx dy

we easily deduce the first of equations (iii).

To interpret these equations we take, at time t, a linear element whose
projections on the coordinate axes are

&r=e£/p, 8y=(r)/p, 8z = f£/p (viii)>

where 6 is infinitesimal. If this element be supposed to move with the fluid,

* See Nanson, Mess, of Math. t. iii., p. 120 (1874); Kirchhoff, Mechanik, Leipzig.
1876..., c. xv.; Stokes, Math, and Phys. Papers, t. ii., p. 47 (1883).
t Nanson, I. c.

143-144] VON HELMHOLTZ' THEOREM. 227

the rate at which dx is increasing is equal to the difference of the values of u
at the two ends, whence

_ % du rj du £ du
Dt p dx p dy p dz '

It follows, by (iii), that

Von Helmholtz concludes that if the relations (viii) hold at time t, they will
hold at time t-\- 8t, and so on, continually. The inference is, however, not quite
rigorous ; it is in fact open to the criticisms which Sir G. Stokes* has directed
against various defective proofs of Lagrange's velocity-potential theorem f.

By way of establishing a connection with Lord Kelvin's investigation we
may notice that the equations (i) express that the circulation is constant in
each of three infinitely small circuits initially perpendicular, respectively, to
the three coordinate axes. Taking, for example, the circuit which initially
bounded the rectangle 86 Sc, and denoting by A, B, C the areas of its pro
jections at time t on the coordinate planes, we have

d(b,c) d(b,c)

so that the first of the equations referred to is equivalent to

(x)J.

144. It is easily seen by the same kind of argument as in
Art. 41 that no irrotational motion is possible in an incompressible
fluid filling infinite space, and subject to the condition that the
velocity vanishes at infinity. This leads at once to the following
theorem :

The motion of a fluid which fills infinite space, and is at
rest at infinity, is determinate when we know the values of the

* I. c. ante p. 18.

t It may be mentioned that, in the case of an incompressible fluid, equations some
what similar to (iii) had been established by Lagrange, Miscell. Taur., t. ii. (1760),
Oeuvres, t. i., p. 442. The author is indebted for this reference, and for the above
criticism of von Helmholtz' investigation, to Mr Larmor. Equations equivalent to
those given by Lagrange were obtained independently by Stokes, I. c., and made the
basis of a rigorous proof of tbe velocity-potential theorem.

$ Nanson, Mess, of Math., t. vii., p. 182 (1878). A similar interpretation of
von Helmholtz' equations was given by the author of this work in the Mess, of
Math., t. vii., p. 41 (1877).

Finally we may note that another proof of Lagrange's theorem, based on ele
mentary dynamical principles, without special reference to the hydrokinetic equa
tions, was indicated by Stokes (Camb. Trans., t. viii.; Math. andPhys. Papers, t. i.,
p. 113), and carried out by Lord Kelvin, in his paper on Vortex Motion.

15—2

228 VORTEX MOTION. [CHAP. VII

expansion (6, say) and of the component angular velocities f, 77, f,
at all points of the region.

For, if possible, let there be two sets of values, uly vlt wl}
and u2, v.2, w2, of the component velocities, each satisfying

the equations

du dv dw fi

1 — h i — V—T- —" ........................ (1),

dx dy dz

dw dv .. du dw dv du

dy~Tz=^> dz~Tx = l7]> dx'dy-^ ......... (2)>

throughout infinite space, and vanishing at infinity. The quantities

will satisfy (1) and (2) with 0, £, 77, £ each put = 0, and will vanish
at infinity. Hence, in virtue of the result above stated, they will
everywhere vanish, and there is only one possible motion satisfying
the given conditions.

In the same way we can shew that the motion of a fluid occupying any
limited simply-connected region is determinate when we know the values of
the expansion, and of the component rotations, at every point of the region,
and the value of the normal velocity at every point of the boundary. In the
case of a multiply-connected region we must add to the above data the values
of the circulations in the several independent circuits of the region.

145. If, in the case of infinite space, the quantities 6, f, 77, f
all vanish beyond some finite distance of the origin, the complete
determination of u, v, w in terms of them can be effected as
follows*.

The component velocities (ult vl} wlt say) due to the ex
pansion can be written down at once from Art. 56 (1), it being
evident that the expansion 6' in an element bx'&y'Sz' is equivalent
to a simple source of strength l/4?r . 0'Sw'Sy'Bz'. We thus obtain

where <D =-dx'dy'dz' .................. (2),

* The investigation which follows is substantially that given by von Helmholtz.
The kinematical problem in question was first solved, in a slightly different
manner, by Stokes, "On the Dynamical Theory of Diffraction," Camb. Trans.,
t. ix. (1849), Math, and Phys. Papers, t. ii., pp. 254....

144-145] KINEMATICAL THEORY. 229

r denoting the distance between the point (#', y' ', z) at which the
volume-element of the integral is situate and the point (x, y, z)
at which the values of ul,vl, w^ are required, viz.

r = a -*

and the integration including all parts of space at which 6' differs
from zero.

To verify this result, we notice that the above values ofult vlf wl
make

^+^1+M=_V2$=

dx ay dz
by the theory of Attractions, and also vanish at infinity.

To find the velocities (u.2, v.2, w2) say) due to the vortices,
we assume

_dll dG _dF dH _dG dF

U'2~~d^~dz' V'2~~dz~ dx> W*~dx~dy""

and seek to determine F, G, H so as to satisfy the required
conditions. In the first place, these formula) make

du2 d/Vz dw2 _ Q
da dy dz

and so do not interfere with the result contained in (1). Also, they
give

2£ = ^2 _ —2 = .

dy dz dx\dx dy dz

Hence our problem will be solved if we can find three functions
F, G, H satisfying

dFdG^dU .....................

dx dy dz

and V2^=-2f, V"G = -2r), V'2H = -2£ ......... (5).

These latter equations are satisfied by making F, G, H equal to the
potentials of distributions of matter whose volume-densities at the
point (x} y, z) are f/2?r, 77/2-77-, f/2-Tr, respectively ; thus

•(6),

230 VORTEX MOTION. [CHAP. VII

where the accents attached to £, 77, f are used to distinguish the
values of these quantities at the point (xf, y', z'\ and

r={(x- #')2 + (y - y'f + (z- z'J}*,

as before. The integrations are to include, of course, all places at
which f , 77, f differ from zero.

It remains to shew that the above values of F, G, H really
satisfy (4), Since djdx .r~l = — d/dx' . r~l, we have

dF dG dH 1 ffff d , d ., d \ 1 , , , , , ,
j- + -j- + -j- = ~ a r I ? V-> + V j-/ + ? j~> - dxdy dz'
dx dy dz Sirj/Jv ** ^2/ dz J r

— 7T~ I I (1%' + m??/ + ?l?') \~ cT

ZTT J J r ZTT

(7),

by the usual method of partial integration. The volume-integral
vanishes, by Art. 142 (i), and the surface-integral also vanishes,
since lj~ + 77177 + nf = 0 at the bounding surfaces of the vortices.
Hence the formulae (3) and (6) lead to the prescribed values of
f, 77, f, and give a zero velocity at infinity.

The complete solution of our problem is now obtained by
superposition of the results contained in the formula (1) and (3),

viz. we have

d$> dH dG

u = -^j— + ^j j- 1

dx dy dz

d3> dF dH

d<$> dG dF

w = - -r + -= -j-

dz dx dy

where <l>; F, G, H have the values given in (2) and (6).

When the region occupied by the fluid is not unlimited, but is bounded (in
whole or in part) by surfaces at which the normal velocity is given, and when
further (in the case of a cyclic region) the value of the circulation in each of
the independent circuits of the region is prescribed, the problem may by a
similar analysis be reduced to one of irrotational motion, of the kind con
sidered in Chap, in., and there proved to be determinate. This may be left
to the reader, with the remark that if the vortices traverse the region,
beginning and ending on the boundary, it is convenient to imagine them
continued beyond it, or along its surface, in such a manner that they form
re-entrant filaments, and to make the integrals (6) refer to the complete system
of vortices thus obtained. On this understanding the condition (4) will still
be satisfied.

145-147] ELECTRO-MAGNETIC ANALOGY. 231

146. There is a remarkable analogy between the analytical
relations above developed and those which obtain in the theory of
Electro-magnetism. If, in the equations (1) and (2) of Art. 144,
we write

a, y8, 7, p, p} q, r

for U, V, W, 0/47T, £/27T, T;/27r, J/27T,

respectively, we obtain

da d/3 dy \

IT + ~T~ + ~J- = 47TP,

dx dy dz I

dy dj3 da dy d@ da {" '

~T~ ~ ~T = ^Trp, ~J -T~ — ^q, ~J ~J~ = ^7rr

dy dz dz dx dx dy J

which are the fundamental relations of the subject referred to ;
viz. a, /:?, 7 are the components of magnetic force, p, q, r those of
electric current, and p is the volume-density of the imaginary
magnetic matter by which any magnetization present in the field
may be represented. Hence, if we disregard constant factors, the
vortex-filaments correspond to electric circuits, the strengths of
the vortices to the strengths of the currents in these circuits,
sources and sinks to positive and negative magnetic poles, and,
finally, fluid velocity to magnetic force f.

The analogy will of course extend to all results deduced from
the fundamental relations ; thus, in equations (8) of the preceding
Art., <l> corresponds to the magnetic potential and F, G, H to the
components of ' electro-magnetic momentum.'

147. To interpret the result contained in Art. 145 (8), we
may calculate the values of u, v, w due to an isolated re-entrant
vortex-filament situate in an infinite mass of incompressible fluid
which is at rest at infinity.

Since 6 = 0, we shall have <3> = 0. Again, to calculate the
values of F, G, H, we may replace the volume-element So/Sy'S*'
by v'Ss', where &s' is an element of the length of the filament, and
<r' its cross-section. Also, we have

c., , dx' , , dy' .,, , dz
* =(a ds" "=<a ds" ?=W d?'

* Cf. Maxwell, Electricity and Magnetism, Art. 607.

f This analogy was first pointed out by von Helmholtz ; it has been extensively
Utilized by Lord Kelvin in his papers on Electrostatics and Magnetism.

232 VORTEX MOTION. [CHAP. VII

where o>' is the angular velocity of the fluid. Hence the formal*
(6) of Art. 145 become

m' dot m' dy m dz

~

where m', = a>V ', measures the strength of the vortex, and the
integrals are to be taken along the whole length of the filament.

Hence, by Art. 145 (8), we have

m (t

u = o~ I
27rJ \

d\ d 1

j — .dz--j--.
dy r dz r

with similar results for v, w. We thus find

_ m ffdy' z — z' dz' x — x'\ ds

.(2)*.

m ((dz' x — x dx y — y'^

^ — If «7 «/

27rJ\ds' r ds' r

m [/ dx' y — y' dy' z — z'\ ds'

nn ^-^- I I ^L ~L~ ^_ *^ I

27rJ\ds' r ds' r ) r~ '

If &u, Av, A^6' denote the parts of these expressions which corre
spond to the element Ss' of the filament, it appears that the
resultant of Aw, Av, At6> is a velocity perpendicular to the plane
containing the direction of the vortex- line at (x, y' , z'} and the
line r, and that its sense is that in which the point (x, y, z) would
be carried if it were attached to a rigid body rotating with the
fluid element at (x , y', z'). For the magnitude of the resultant
we have

''""""'"' (3),

where % is the angle which r makes with the vortex-line at (#', y', z).

With the change of symbols indicated in the preceding Art.
this result becomes identical with the law of action of an electric
current on a magnetic pole*)*.

* These are equivalent to the forms obtained by Stokes, 1. c. ante p. 228.

t Ampere, Theorie matheniatique des plienomenes electro-dynamiques, Paris, 1826.

147-148] MOTION DUE TO AN ISOLATED VORTEX. 233

Velocity-Potential due to a Vortex.

148. At points external to the vortices there exists of course a
velocity-potential, whose value may be obtained as follows. Taking
for shortness the case of a single re-entrant vortex, it was found
in the preceding Art. that, in the case of an incompressible fluid,

m' [i rf 1 7 , d 1 , A

= o~ I \ j-'-'dy — 7 -,-.dz } (1).

27rJ \dz r J dy r v '

dy'

By Stokes' Theorem (Art. 33 (5)) we can replace a line-integral ex
tending round a closed curve by a surface-integral taken over any
surface bounded by that curve ; viz. we have, with a slight change
of notation,

fdR dQ\ fdP dR\ fdQ dP\] ,

T-/-T^ +m j-/- j^l + »( T%-^r-,)td&.

\dy' dz) \dz doc) \dx' dy')}

If we put

we find

^ 1 p d I
— •- tf——-

dzr r' dy' r*

dR_dQ=_/d* _^1\1_^11
dy' dz ~ (dy* dz">) r ~ d^ rf '

dP _dR_ d* !_

dz' dx' ~ dx'dy' r' '

dQ_dP= _fc _!

dx dy' dx'dz r"

so that (1) may be written

inf [[/, d d d \ d 1 70(

u = x— 1 1 U -r, + m -J-, +n -j-f -r-> - ao .
27rJJ\ d» c^/ dz J dx r

Hence, and by similar reasoning, we have, since

f . r~l = - dldx . r~\

dd> dd> dd> /ON

u = - -f, -y = - y^, w = --^ , ............... (2),

dx dy dz

where

m' f/Yj rf rf rf\ 1 7ry- /ox

0 = er U TT + ^ J"7+ * :r> J - <*» ............ w>

27rJJV rf^ rf/ dW r*

Here ^, m, ?i denote the direction-cosines of the normal to the
element 8$' of any surface bounded by the vortex-filament.

234 VORTEX MOTION. [CHAP. VII

The formula (3) may be otherwise written

where ^ denotes the angle between r and the normal (I, m, n).
Since cos OSS'/'r" measures the elementary solid angle subtended
by 8$' at (x, y, z)t we see that the velocity-potential at any point,
due to a single re-entrant vortex, is equal to the product of m'/Zir
into the solid angle which any surface bounded by the vortex
subtends at that point.

Since this solid angle changes by 4?r when the point in
question describes a circuit embracing the vortex, we verify that
the value of <f> given by (4) is cyclic, the cyclic constant being
twice the strength of the vortex. Cf. Art. 142.

Comparing (4) with Art. 56 (4) we see that a vortex is, in
a sense, equivalent to a uniform distribution of double sources over
any surface bounded by it. The axes of the double sources must be
supposed to be everywhere normal to the surface, and the density
of the distribution to be equal to the strength of the vortex
divided by 2?r. It is here assumed that the relation between
the positive direction of the normal and the positive direction
of the axis of the vortex-filament is of the 'right-handed' type.
See Art. 32.

Conversely, it may be shewn that any distribution of double sources over
a closed surface, the axes being directed along the normals, may be replaced
by a system of closed vortex-filaments lying in the surface*. The same thing
will appear independently from the investigation of the next Art.

Vortex- Sheets.

149. We have so far assumed w, v, w to be continuous. We
will now shew how cases where surfaces present themselves at
which these quantities are discontinuous may be brought within
the scope of our theorems.

The case of a surface where the normal velocity is discon
tinuous has already been treated in Art. 58. If u, v, w denote the
component velocities on one side, and u', v', w' those on the other,

* Cf. Maxwell, Electricity and Magnetism, Arts. 485, 652.

148-149] VORTEX-SHEETS. 235

it was found that the circumstances could be represented by
imagining a distribution of simple-sources, with surface density

— {I (uf — u) + m (v — v) + n (wf — w)},
47T

where I, m, n denote the direction-cosines of the normal drawn
towards the side to which the accents refer.

Let us next consider the case where the tangential velocity
(only) is discontinuous, so that

I (uf — u) + m (v — v) + w (w' — w) = Q (1).

We will suppose that the lines of relative motion, which are
defined by the differential equations

dx _ _dy_ = dz .

u' — u v' — v w' — w '

are traced on the surface, and that the system of orthogonal
trajectories to these lines is also drawn. Let PQ, P'Q' be linear
elements drawn close to the surface, on the two sides, parallel to
a line of the system (2), and let PP' and QQ' be normal to the
surface and infinitely small in comparison with PQ or P'Q,'.
The circulation in the circuit P'Q'QP will then be equal to
(q — q) PQ, where q, q' denote the absolute velocities on the two
sides. This is the same as if the position of the surface were
occupied by an infinitely thin stratum of vortices, the orthogonal
trajectories above-mentioned being the vortex-lines, and the
angular velocity w and the (variable) thickness &n of the stratum
being connected by the relation 2o> . PQ . Sn = (q — q) PQ, or

co$n = %(q'-q) (3).

The same result follows from a consideration of the discontinuities which
occur in the values of u, v, w as determined by the formulae (3) and (6) of
Art. 145, when we apply these to the case of a stratum of thickness dn
within which £, 77, £ are infinite, but so that £§w, r)8n, £8n are finite*.

It was shewn in Arts. 144, 145 that any continuous motion of
a fluid filling infinite space, and at rest at infinity, may be
regarded as due to a proper arrangement of sources and vortices
distributed with finite density. We have now seen how by
considerations of continuity we can pass to the case where the
sources and vortices are distributed with infinite volume-density,

* Helmholtz, 1. c. ante p. 222.

236 VORTEX MOTION. [CHAP. VII

but finite surface-density, over surfaces. In particular, we may
take the case where the infinite fluid in question is incompressible.
and is divided into two portions by a closed surface over which the
normal velocity is continuous, but the tangential velocity dis
continuous, as in Art. 58 (12). This is equivalent to a vortex-
sheet; and we infer that every continuous irrotational motion,
whether cyclic or not, of an incompressible substance occupying
any region whatever, may be regarded as due to a certain distri
bution of vortices over the boundaries which separate it from the
rest of infinite space. In the case of a region extending to
infinity, the distribution is confined to the finite portion of the
boundary, provided the fluid be at rest at infinity.

This theorem is complementary to the results obtained in
Art. 58.

The foregoing conclusions may be illustrated by means of the results of
Art. 90. Thus when a normal velocity Sn was prescribed over the sphere
r = a, the values of the velocity-potential for the internal and external space
were found to be

respectively. Hence if §e be the angle which any linear element drawn on
the surface subtends at the centre, the relative velocity estimated in the
direction of this element will be

_

n(n~+I) ~df '

The resultant relative velocity is therefore tangential to the surface, and
perpendicular to the contour lines (Sn = const.) of the surface-harmonic Snt
which are therefore the vortex-lines.

For example, if we have a thin spherical shell filled with and surrounded
by liquid, moving as in Art. 91 parallel to the axis of x, the motion of the
fluid, whether internal or external, will be that due to a system of vortices
arranged in parallel circles on the sphere ; the strength of an elementary
vortex being proportional to the projection, on the axis of x, of the breadth
of the corresponding strip of the surface*.

Impulse and Energy of a Vortex-System.

150. The following investigations relate to the case of a
vortex-system of finite dimensions in an incompressible fluid
which fills infinite space and is at rest at infinity.

* The same statements hold also for an ellipsoidal shell moving parallel to one of
its principal axesv See Art. 111.

149-150] IMPULSE OF A VORTEX-SYSTEM. 237

If X', Y, Z' be components of a distribution of impulsive
force which would generate the actual motion (u, v, w) instan
taneously from rest, we have by Art. 12 (1)

Tr, 1 div T7, 1 d^r „, 1 dix

X' ;- = w, Y — -T~ = V> z — j =w CO.

p doc p dy p dz

where ts is the impulsive pressure. The problem of finding X', Y', Z',
TV in terms of u, v, w, so as to satisfy these three equations, is clearly
indeterminate ; but a sufficient solution for our purpose may be
obtained as follows.

Let us imagine a simply-connected surface 8 to be drawn
enclosing all the vortices. Over this surface, and through
the external space, let us put

™ = P<t> (2),

where <£ is the velocity-potential of the vortex-system, determined
as in Art. 148. Inside S let us take as the value of w any
single-valued function which is finite and continuous, is equal to
(2) at 8, and also satisfies the equation

f = P? -(3).

dn r dn

at $, where &n denotes as usual an element of the normal. It
follows from these conditions, which can evidently be satisfied in an
infinite number of ways, that the space-derivatives d^jdx, dTxjdy,
d^fjdz will be continuous at the surface S. The values of X , Y', Z'
are now given by the formula (1); they vanish at the surface S,
and at all external points.

The force- and couple-equivalents of the distribution X ', F', Z'
constitute the ' impulse ' of the vortex-system. We are at present
concerned only with the instantaneous state of the system, but it
is of interest to recall that, when no extraneous forces act, this
impulse is, by the argument of Art. 116, constant in every respect.

Now, considering the matter inclosed within the surface S, we
find, resolving parallel to x,

MpX'dxdydz = pffludxdydz - pfjl^dS (4),

if I, m, n be the direction-cosines of the inwardly-directed normal
to any element &S of the surface. Let us first take the case of a
single vortex-filament of infinitely small section. The fluid

238 VORTEX MOTION. [CHAP. VII

velocity being everywhere finite and continuous, the parts of the
volume-integral on the right-hand side of (4) which are due to the
substance of the vortex itself may be neglected in comparison
with those due to the remainder of the space included within S.
Hence we may write

fjjudxdydz = - ffjj£ dxdydz = jjl^dS + 2m' fjldS' . . , (5),

where c/> has the value given by Art. 147 (4), m denoting the
strength of the vortex (so that 2m' is the cyclic constant of c/>), and
$S' an element of any surface bounded by it. Substituting in
(4), we infer that the components of the impulse parallel to the
coordinate axes are

Zm'pffmdS', Zm'pJfndS' ............... (6).

Again, taking moments about Ox,
5S5p(yZ'-zY') dxdydz

= pfff(yw — zv) dxdydz — pff(ny — mz) (f>dS ......... (7).

For the same reason as before, we may substitute, for the volume-
integral on the right-hand side,

(ny - mz) <j>dS + 2m'ff(ny - mz) dS' (8).

Hence, and by symmetry, we find, for the moments of the impulse
about the coordinate axes,

2m'pff(ny-mz)d8', 2mpff(lz-nx)dS', 2m'ptf(mx - ly) dS' . . .(9).

The surface-integrals contained in (6) and (9) may be replaced
by line-integrals taken along the vortex. In the case of (6) it is
obvious that the coefficients of m'p are double the projections on
the coordinate axes of any area bounded by the vortex, so that the
components in question take the forms

, r/ ,dz' ,dy'\ 7 , , [( ,dx' , dz'

™ (y ^r,-z -r/ <&* > m \\z ^-,~x ^r,
JV ds ds J j\ ds ds

150-151] COMPONENTS OF IMPULSE. 239

For the similar transformation of (9) we must have recourse
to Stokes' Theorem ; we obtain without difficulty the forms

+ *>) ds',

............ (11).

From (10) and (11) we can derive by superposition the com
ponents of the force- and couple- resultants of any finite system of
vortices. Denoting these by P, Q, R, and Z, If, N, respectively,
we find, putting

, dx M , dy' . , dz' ..

~J ' — £ > w TJ ' = rl » w J ' ~ b >

ds ds ds

and replacing the volume-element a'Ss' by

P = p$jj(y% — zrj) docdydz, L = /o///(2/2 + ^2) f dxdydz,
Q = pfff(zg - a£) da?dyrf«, Jlf = p///(^2 + a?) 77 dajrfy^, [-...(12)*,
-R = P/J/C^7/ ~ ^ f ) docdydz, N = pfff(a? + f) £ dxdydz
where the accents have been dropped, as no longer necessary.

151. Let us next consider the energy of the vortex-system.
It is easily proved that under the circumstances presupposed, and
in the absence of extraneous forces, this energy will be constant.
For if T be the energy of the fluid bounded by any closed surface
S, we have, putting F = 0 in Art. 11 (5),

DT

_ = ff(lu + mv + nw) pdS .................. (1).

If the surface S enclose all the vortices, we may put

and it easily follows from Art. 148 (4) that at a great distance R
from the vortices p will be finite, and lu + mv + nw of the order
R~3, whilst when the surface 8 is taken wholly at infinity,

* These expressions were given by J. J. Thomson, On the Motion of Vortex
(Adams Prize Essay), London, 1883, pp. 5, 6.

240 VORTEX MOTION. [CHAP. VII

the elements BS ultimately vary as R2. Hence, ultimately, the
right-hand side of (1) vanishes, and we have

T = const ............................... (3).

152. We proceed to investigate one or two important kine-
matical expressions for T, still confining ourselves, for simplicity,
to the case where the fluid (supposed incompressible) extends to
infinity, and is at rest there, all the vortices being within a finite
distance of the origin.

The first of these is indicated by the electro-magnetic analogy
pointed out in Art. 146. Since 6 = 0, and therefore O = 0, we have

+ tf + w2) dxdydz

{[[< fdH dG\ (dF dH\ /dG dF\ ,
= pHI\u( -= -- • — - ) + vi-j -- ,— + w i-j ---- ,- dxdydz,
JjJ\ \dy dz) \dz dx) \dx dy )

by Art. 145 (3). The last member may be replaced by the sum of
a surface integral

pff{F(miv - nv) + G (nu - ho) + H (Iv - nm)} dS,
and a volume integral

dw dv\ „ fdu dw\ „ (dv du\] 7 7 7

--- i-)+0l-3- — -y }^H (,--^-}\ dxdydz.
dy dz) \dz dx) \dx dy)}

At points of the infinitely distant boundary, F, G, H are ultimately
of the order R~2, and u, v, w of the order R~s, so that the surface-
integral vanishes, and we have

T=PJff(FS + €h,.+ HQd*dydt .................. (1),

or, substituting the values of F, G, H from Art. 145 (6),

T = 1- fjjfjf*? + 7' + K' dxdydz dx'dy'dz ...(2),

where each volume-integration extends over the whole space
occupied by the vortices.

A slightly different form may be given to this expression as
follows. Regarding the vortex-system as made up of filaments,
let &s, 8s' be elements of length of any two filaments, a, a'
the corresponding cross-sections, and &>, &>' the corresponding
angular velocities. The elements of volume may be taken to be

151-152] KINETIC ENERGY. 241

o-Ss and a'Ss', respectively, so that the expression following the
integral signs in (2) is equivalent to

cos e . , ,£ ,
- . ftxros . ft) a- os ,
r

where e is the angle between §s and Ss'. If we put wa — m,
MO-' = m' , so that in and m' denote the strengths of the two
elementary vortices, we have

T = ^mm'jjC°^dsds' ............... (3),

where the double integral is to be taken along the axes of the
filaments, and the summation embraces every pair of such
filaments which are present.

The factor of p/Tr in (3) is identical with the expression for the
energy of a system of electric currents flowing along conductors
coincident in position with the vortex-filaments, with strengths
m, m',... respectively*. The above investigation is in fact merely
an inversion of the argument given in treatises on Electro-
magnetism, whereby it is proved that

= (a2 + & + 72) dxdydz,

i, i' denoting the strengths of the currents in the linear conductors
whose elements are denoted by 8s, 8s', and a, /3, 7 the components
of magnetic force at any point of the field.

The theorem of this Art. is purely kinematical, and rests solely
on the assumption that the functions u, v, w satisfy the equation
of continuity,

du dv dw «

dx dy dz

throughout infinite space, and vanish at infinity. It can therefore
by an easy generalization be extended to the case considered in
Art. 141, where a liquid is supposed to circulate irrotationally
through apertures in fixed solids, the values of u, v, w being now
taken to be zero at all points of space not occupied by the fluid.
The investigation of Art. 149 shews that the distribution of velocity
thus obtained may be regarded as due to a system of vortex-sheets
coincident with the surfaces of the solids. The energy of this
system will be given by an obvious adaptation of the formula (3)
above, and will therefore be proportional to that of the correspond-

* See Maxwell, Electricity and Magnetism, Arts. 524, 637.
L. 16

242 VOKTEX MOTION. [CHAP. VII

ing system of electric current-sheets. This proves a statement
made by anticipation in Art. 141.

153. Under the circumstances stated at the beginning of
Art. 152, we have another useful expression for T\ viz.

T= 2p /// {u (y% -zrj)+v Of -xQ+w (ay - y% )} dxdydz. . .(4).

To verify this, we take the right-hand member, and transform it
by the process already so often employed, omitting the surface-
integrals for the same reason as in the preceding Art. The first
of the three terms gives

dv du\ du d

du\ fdu dw\\ , , ,
T~ ~<M ~7 --- r~ r dxdydz
dyj \dz dxj)

Transforming the remaining terms in the same way, adding, and
making use of the equation of continuity, we obtain

+ yv + ZW dxdydz,

or, finally, on again transforming the last three terms,
iP JJ/(^2 + v2 + w2) dxdydz.

In the case of a finite region the surface-integrals must be retained. This
involves the addition to the right-hand side of (4) of the term

P JJ {(lu + mv + nw) (xu +2/v + zw] -^(te+my+ nz) <?2} dS,
where q2 = u2 + v2 + w2. This simplifies in the case of & fixed boundary*.

The value of the expression (4) must be unaltered by any displacement of
the origin of coordinates. Hence we must have

J7J 0>£- /Mty) dxdydz=Q,\

JJJ («*-*£) dxdydz = Q, I ........................... (i).

J/J (^17 - i>£) dxdydz = Q }

These equations, which may easily be verified by partial integration, follow
also from the consideration that the components of the impulse parallel to the
coordinate axes must be constant. Thus, taking first the case of a fluid
enclosed in a fixed envelope of finite size, we have, in the notation of Art. 150,

P = pMudxdydz-pSSl<l>dS ........................ (ii),

whence

l dS ...... (iii),

Of. J. J. Thomson, I.e.

z -pjfl

152-154] KINETIC ENERGY. 243

by Art. 143 (iv). The first and third terms of this cancel, since at the
envelope we have ^ = d$ldt. Hence for any re-entrant system of vortices
enclosed in a fixed vessel, we have

with two similar equations. If now the containing vessel be supposed
infinitely large, and infinitely distant from the vortices, it follows from the
argument of Art. 116 that P is constant. This gives the first of equations (i).

Conversely from (i), established otherwise, we could infer the constancy of
the components P, Q, R of the impulse*.

Rectilinear Vortices.

154. When the motion is in two dimensions xy we have w = 0,
whilst u, v are functions of x, y, only. Hence f = 0, j] — 0, so that
the vortex-lines are straight lines parallel to z. The theory then
takes a very simple form.

The formulae (8) of Art. 145 are now replaced by

dx dy> ' dy dx ..... '

the functions </>, -\Jr being subject to the equations

V^-0, V1^ = 2f ..................... (2),

where V^ = d?/da? + d?/dy*,

and to the proper boundary-conditions.

In the case of an incompressible fluid, to which we will now
confine ourselves, we have

, ~r ..................... ,

dy dx

where ^ is the stream-function of Art. 59. It is known from the
theory of Attractions that the solution of

V1V=2f .............................. (4),

where f is a given function of x, y> is

tf + fy ............... (5),

Cf. J. J. Thomson, Motion of Vortex Rings, p. 5.

16—2

244 VORTEX MOTION. [CHAP. VII

where f denotes the value of f at the point (x't y'), and r now
stands for

The ' complementary function ' -<fr0 may be any solution of

V,2V. = 0 ........................... (6);

it enables us to satisfy the boundary-conditions.

In the case of an unlimited mass of liquid, at rest at infinity,
we have ^0 = const. The formulae (3) and (5) then give

(7).

x — x , ,

«-

Hence a vortex-filament whose coordinates are a?', y' and whose
strength is ??^' contributes to the motion at (x, y} a velocity whose
components are

m' y — yf , m' x — x'

-- . XffJ- , and — . — - .

TT r2 TT r2

This velocity is perpendicular to the line joining the points (x, y),
(#', y'\ and its amount is m'/irr.

Let us calculate the integrals ffugdxdy, and ffv£dacdy, where
the integrations include all portions of the plane xy for which f
does not vanish. We have

gf. dxdy dx'dy',

where each double integration includes the sections of all the
vortices. Now, corresponding to any term

K'' dxdydxdy1

of this result, we have another term

and these two terms neutralize one another. Hence

0 ........................... (8),

154-155] RECTILINEAR VORTICES. 245

and, by the same reasoning,

ffv{da;dy = 0 ........................... (9).

If as before we denote the strength of a vortex by m, these results

may be written

2wM = 0, 2rav=0 ........................ (10).

We have seen above that the strength of each vortex is constant
with regard to the time. Hence (10) express that the point whose
coordinates are

rp - _ /•?/ - _ \

VO — ^ j (j — ^> j

2m Zra

is fixed throughout the motion. This point, which coincides with
the centre of inertia of a film of matter distributed over the plane
xy with the surface-density f, may be called the ' centre ' of the
system of vortices, and the straight line parallel to z of which it
is the projection may be called the 'axis' of the system.

155. Some interesting examples are furnished by the case of
one or more isolated vortices of infinitely small section. Thus :

1°. Let us suppose that we have only one vortex-filament
present, and that the rotation f has the same sign throughout its
infinitely small section. Its centre, as just defined, will lie either
within the substance of the filament, or at all events infinitely
close to it. Since this centre remains at rest, the filament as a
whole will be stationary, though its parts may experience relative
motions, and its centre will not necessarily lie always in the same
element of fluid. Any particle at a finite distance r from the
centre of the filament will describe a circle about the latter as
axis, with constant velocity m/7rr. The region external to the
filament is doubly-connected ; and the circulation in any (simple)
circuit embracing the filament is 2m. The irrotational motion of
the fluid external to the filament is the same as in Art. 28 (2).

2°. Next suppose that we have two vortices, of strengths mlt
m3, respectively. Let A, B be their centres, 0 the centre of the
system. The motion of each filament as a whole is entirely due
to the other, and is therefore always perpendicular to AB. Hence
the two filaments remain always at the same distance from one
another, and rotate with constant angular velocity about 0, which
is fixed. This angular velocity is easily found; we have only to

246 VOKTEX MOTION. [CHAP. VII

divide the velocity of A (say), viz. m,/(7r . AB), by the distance AO,
where

A0= * AB,

m.

and so obtain

7T.AB*

for the angular velocity required.

If m1, w2 be of the same sign, i.e. if the directions of rotation
in the two filaments be the same, 0 lies between A and B\ but
if the rotations be of opposite signs, 0 lies in AB, or BA,
produced.

If m1 = — w2, 0 is at infinity; in this case it is easily seen that
A, B move with constant velocity m^vr . AB) perpendicular to AB,
which remains fixed in direction. The motion at a distance from
the filaments is given at any instant by the formulas of Art. 64, 2°.

Such a combination of two equal and opposite rectilinear vortices
may be called a ' vortex-pair.' It is the two-dimensional analogue
of a circular vortex-ring (Art. 162), and exhibits many of the
characteristic properties of the latter.

The motion at all points of the plane bisecting AB at right
angles is in this latter case tangential to that plane. We may
therefore suppose the plane to form a fixed rigid boundary of the
fluid in either side of it, and so obtain the solution of the case
where we have a single rectilinear vortex in the neighbourhood of
a fixed plane wall to which it is parallel. The filament moves
parallel to the plane with the velocity m/27rd, where d is the
distance of the vortex from the wall.

The stream-lines due to a vortex-pair, at distances from the vortices great
in comparison with the linear dimensions of the cross-sections, form a system
of coaxal circles, as shewn in the diagram
on p. 80.

We can hence derive the solution of the
case where we have a single vortex-filament
in a mass of fluid which is bounded, either
internally or externally, by a fixed circular
cylinder. Thus, in the figure, let EPD be
the section of the cylinder, A the position of

the vortex (supposed in this case external), and let B be the 'image' of A
with respect to the circle EPD, viz. C being the centre, let

155] SPECIAL CASES. 247

where c is the radius of the circle. If P be any point on the circle, we have

AP AE _AD_
~BP~BE~^D~ ;

so that the circle occupies the position of a stream-line due to a pair of
vortices, whose strengths are equal and opposite in sign, situated at J, E in
an unlimited mass of fluid. Since the motion of the vortex A would then be
perpendicular to AB, it is plain that all the conditions of the problem will be
satisfied if we suppose A to describe a circle about the axis of the cylinder
with the constant velocity

m tn . CA

where m denotes the strength of A.

In the same way a single vortex of strength m, situated inside a fixed
circular cylinder, say at B, would describe a circle with constant velocity

m.CB

It is to be noticed, however*, that in the case of the external vortex the
motion is not completely determinate unless, in addition to the strength
m of the vortex, the value of the circulation in a circuit embracing the
cylinder (but not the vortex) is prescribed. In the above solution, this
circulation is that due to the vortex-image at B and is -2m. This may
be annulled by the superposition of an additional vortex + m at (7, in which
case we have, for the velocity of ,4,

m . CA m me2

7T (CA2 - C2} 7T.CA 7T.CA (CA2 - C2) '

For a prescribed circulation K we must add to this the term */2ir . CA.

3°. If we have four parallel rectilinear vortices whose centres
form a rectangle ABB' A', the strengths being m for the vortices
A', B, and — m for the vortices A, B', it is evident that the
centres will always form a rectangle. Further, the various rota
tions having the directions indicated in the figure, we see that

* See F. A. Tarleton, "On a Problem in Vortex Motion," Proc. R. L A.,
December 12, 1892.

248 VORTEX MOTION. [CHAP. VII

the effect of the presence of the pair A, A' on B, B' is to separate
them, and at the same time to diminish their velocity perpen
dicular to the line joining them. The planes which bisect AB,
A A' at right angles may (either or both) be taken as fixed rigid
boundaries. We thus get the case where a pair of vortices, of
equal and opposite strengths, move towards (or from) a plane
wall, or where a single vortex moves in the angle between two
perpendicular walls.

If x, y be the coordinates of the vortex B' relative to the planes of
symmetry, we readily find

where r2 = j?+y2. By division we obtain the differential equation of the
path, viz.

dx dy

-^ + -4=0,
*

whence a2 (x2 + y2} —

a being an arbitrary constant, or, transforming to polar coordinates,

r= a/sin 20 .................................... (ii).

Also since 4$f-y£« m/2r,

the vortex moves as if under a centre of force at the origin. This force is
repulsive, and its law is that of the inverse cube*.

156. When, as in the case of a vortex-pair, or a system of
vortex-pairs, the algebraic sum of the strengths of all the vortices
is zero, we may work out a theory of the ' impulse/ in two di
mensions, analogous to that given in Arts. 116, 149 for the
case of a finite vortex-system. The detailed examination of this
must be left to the reader. If P, Q denote the components of the
impulse parallel to x and y, and N its moment about Oz, all
reckoned per unit depth of the fluid parallel to zt it will be found
that

Q=- pSJx^dxdy, )

+ y*)Sdxdy 1 ........ '

* See Greenhill, " On plane vortex-motion,'' Quart. Journ. Math., t. xv. (1877),
where some other interesting cases of motion of rectilinear vortex-filaments are
discussed.

The literature of special problems in this part of the subject is somewhat
extensive; for references see Hicks, Brit. Ass. Ecp. 1882, pp. 41...; Love, "On
Kecent English Eesearches in Vortex Motion," Math. Ann., t. xxx., p. 326 (1887) ;
Winkelmann, Handbuch der Physik, t. i., pp. 446-451.

155-156] IMPULSE AND ENERGY. 249

For instance, in the case of a single vortex-pair, the strengths of
the two vortices being ± m, and their distance apart c, the impulse
is 2wc, in a line bisecting c at right angles.

The constancy of the impulse gives

= const., 2wy = const.,

(2).
y-) = const.

It may also be shewn that the energy of the motion in the
present case is given by

When %m is not zero, the energy and the moment of the
impulse are both infinite, as may be easily verified in the case of
a single rectilinear vortex.

The theory of a system of isolated rectilinear vortices has been put in a
very elegant form by Kirchhoff*.

Denoting the positions of the centres of the respective vortices by
(*'i> yi)> (XM 2/2)1 ••• and their strengths by mlt m,2, ..., it is evident from
Art. 154 that we may write

dxl _ dW d(/l _ d

dx. dW dy9 dW

m<> — ji — j-- , m2 — FT = -j — j

dt tfym " dt dx«

where W= - 2m1m2 log r12

if r12 denote the distance between the vortices mlt m2.

Since T7 depends only on the relative configuration of the vortices, its
value is unaltered when xlt #2,... are increased by the same amount, whence
3d W 'jdxi = 0, and, in the same way, ^,dWjdyl = Q. This gives the first two of
equations (2), but the proof is not now limited to the case of 2m = 0. The
argument is in fact substantially the same as in Art. 154.

»

Again, we obtain from (i)

/ dx dii\ ( dW dW\

2m (x -=- t -- = - 2 jc -- -11 --s-

-=- + t/ -f- ) = - 2 (jc --T -11 --s-
dt J dt) \ dy J dx

or if we introduce polar coordinates (rlt ^j), (r2, ^2), ... for the several vortices,

* Mechanik. c. xx.

250 VORTEX MOTION. [CHAP. VII

Since W is unaltered by a rotation of the axes of coordinates in their own
plane about the origin, we have 2dW/d6 = Q, whence

2mr2 = const (iv),

which agrees with the third of equations (2), but is free from the restriction
there understood.

An additional integral of (i) is obtained as follows. We have

dy dx\_~( dW dW\
V dt y dt)~^\^ dx :,y~dy)^

^de dW

or 2mr2-r = 2r ^- (v).

dt dr

Now if every r be increased in the ratio 1 + f 3 where e is infinitesimal, the
increment of W is equal to 2er . d W/dr. The new configuration of the
vortex-system is geometrically similar to the former one, so that the mutual
distances r12 are altered in the same ratio 1+e, and therefore, from (ii), the
increment of W is err"1 . 2%?ft2. Hence

»d6 1

157. The results of Art. 155 are independent of the form of
the sections of the vortices, so long as the dimensions of these
sections are small compared with the mutual distances of the
vortices themselves. The simplest case is of course when the
sections are circular, and it is of interest to inquire whether this
form is stable. This question has been examined by Lord Kelvin*.

Let us suppose, as in Art. 28, that the space within a circle r = a, having
the centre as origin, is occupied by fluid having a uniform rotation f, and that
this is surrounded by fluid moving irrotationally. If the motion be continuous
at this circle we have, for r<.a

while for r>a,

(ii).

To examine the effect of a slight irrotational disturbance, we assume, for
r<a,

and, for r>a, \ (*")>

a a"

& r i"8

where s is integral, and o- is to be determined. The constant A must have
the same value in these two expressions, since the radial component of the

* Sir W. Thomson, " On the Vibrations of a Columnar Vortex," Phil. Mag.,
Sept. 1880.

156-158] STABILITY OF A CYLINDRICAL VORTEX. 251

velocity, d^IrdO, must be continuous at the boundary of the vortex, for which
r = a, approximately. Assuming for the equation to this boundary

r=a + acos(s0 — vt} (iv),

we have still to express that the tangential component (d^/dr) of the velocity
is continuous. This gives

£r + s - cos (s0 - at] = £ — - s - cos (s6 - crt).

Ctf T Cb

Substituting from (iv), and neglecting the square of a, we find

fa = — sA I a (v).

So far the work is purely kinematical; the dynamical theorem that the
vortex-lines move with the fluid shews that the normal velocity of a
particle on the boundary must be equal to that of the boundary itself.
This condition gives

dr d\fr d\lr dr

~dt = ~ rde ~ ~dr ~rdO>

where r has the value (iv), or

A so. . .,

cra = s — + £«. — (vi).

ct ct

Eliminating the ratio A /a between (v) and (vi) we find

<r = («-l)f (vii).

Hence the disturbance represented by the plane harmonic in (iii) consists
of a system of corrugations travelling round the circumference of the vortex
with an angular velocity

<r/*= (*-!)/*.£ (viii).

This is the angular velocity in space; relative to the previously rotating
fluid the angular velocity is

the direction being opposite to that of the rotation.

When s = 2, the disturbed section is an ellipse which rotates about its
centre with angular velocity |f.

The transverse and longitudinal oscillations of an isolated rectilinear
vortex-filament have also been discussed by Lord Kelvin in the paper cited.

158. The particular case of an elliptic disturbance can be
solved without approximation as follows*.

Let us suppose that the space within the ellipse

* Kirchhoff, Mechanik, c. xx., p. 261; Basset, Hydrodynamics, Cambridge,
1888, t. ii., p. 41.

252 VORTEX MOTION. [CHAP. VII

is occupied by liquid having a uniform rotation £, whilst the surround
ing fluid is moving irrotationally. It will appear that the conditions
of the problem can all be satisfied if we imagine the elliptic boundary to
rotate without change of shape with a constant angular velocity (n, say), to
be determined.

The formula for the external space can be at once written down from
Art. 72, 4°; viz. we have

^ = %n(a + b)2e~^cos2r] + (ab£ ..................... (ii),

where £, ^ now denote the elliptic coordinates of Art. 71, 3°, and the cyclic
constant K has been put = 27j-«&£, in conformity with Art. 142.

The value of >//• for the internal space has to satisfy

..

with the boundary-condition

ux

These conditions are both fulfilled by

provided A + B = I , J

I (vi).

It remains to express that there is no tangential slipping at the boundary
of the vortex; i.e. that the values of d^/dg obtained from (ii) and (v)
coincide. Putting x = c cosh £ cos 77, y = c sinh £ sin »;, where c = (a2 — 62)2, diffe
rentiating, and equating coefficients of cos 2?/, we obtain the additional condition

- \n (a -f- b)2 e~~^ = £c2 (A- B} cosh £ sinh £,
which is equivalent to

since, at points of the ellipse (i), cosh £ = a/c, sinh
Combined with (vi) this gives

When a = 6, this agrees with our former approximate result.

The component velocities x, y of a particle of the vortex relative to the
principal axes of the ellipse are given by

whence we find -= — ?&?, f «»-.., ...(x).

a b b a

Integrating, we find

x=ka cos (nt -f e), y = kb sin (nt -f ( } (xi),

158-159] ELLIPTIC VORTEX. 253

where k, e are arbitrary constants, so that the relative paths of the particles are
ellipses similar to the section of the vortex, described according to the harmonic
law. If of, y' be the coordinates relative to axes fixed in space, we find

of =#cos nt-y&\\\ nt = -= (a + 6) cos (2nt + f ) + - (a - b) cos e, \

L.(xii).

k k

y' = x sin nt +y cos nt = -(a + b) sin (2nt -f e) - ^ (« - &) sin f j

The absolute paths are therefore circles described with angular velocity 2n*.

159. It was pointed out in Art. 81 that the motion of an
incompressible fluid in a curved stratum of small but uniform
thickness is completely defined by a stream -function ^r, so that
any kinematical problem of this kind may be transformed by
projection into one relating to a plane stratum. If, further, the
projection be ' orthomorphic,' the kinetic energy of corresponding
portions of liquid, and the circulations in corresponding circuits,
are the same in the two motions. The latter statement shews that
vortices transform into vortices of equal strengths. It follows at
once from Art. 142 that in the case of a closed simply-connected
surface the algebraic sum of the strengths of all the vortices
present is zero.

Let us apply this to motion in a spherical stratum. The
simplest case is that of a pair of isolated vortices situate at
antipodal points ; the stream-lines are then parallel small circles,
the velocity varying inversely as the radius of the circle. For
a vortex-pair situate at any two points A, B, the stream-lines are
coaxal circles as in Art. 81. It is easily found by the method of
stereographic projection that the velocity at any point P is the
resultant of two velocities m/ira . cot \0^ and m/tra . cot \0.2, per
pendicular respectively to the great-circle arcs AP, BP, where
0j, #2 denote the lengths of these arcs, a the radius of the sphere,
and ± ra the strengths of the vortices. The centre •(• (see Art. 154)

* For further researches in this connection see Hill, ' ' On the Motion of Fluid
part of which is moving rotationally and part irrotationally," Phil. Trans., 1884;
and Love, " On the Stability of certain Vortex Motions," Proc. Lond. Math. Soc.,
t. xxv., p. 18 (1893).

t To prevent possible misconception it may be remarked that the centres of
corresponding vortices are not necessarily corresponding points. The paths of these
centres are therefore not in general projective.

The problem of transformation in piano has been treated by Aouth, "Some
Applications of Conjugate Functions," Proc. Lond. Math. Soc., t. xii., p. 73 (1881).

254 VORTEX MOTION. [CHAP. VII

of either vortex moves perpendicular to AB with a velocity

m/7ra.cot^AB. The two vortices therefore describe parallel

and equal small circles, remaining at a constant distance from
each other.

Circular Vortices.

160. Let us next take the case where all the vortices present
in the liquid (supposed unlimited as before) are circular, having
the axis of a? as a common axis. Let CT denote the distance of any
point P from this axis, ^ the angle which TX makes with the plane
xij, v the velocity in the direction of OT, and co the angular
velocity of the fluid at P. It is evident that uy v, co are functions
of x, va only, and that the axis of the rotation o> is perpendicular
to xix. We have then

y = -or cos S-, z = OT sin S-, }

v = vcos$, w=vsm*b, > ............... (I).

f = 0, T) = — a) sin ^, f = a) cos ^ J

The impulse of the vortex-system now reduces to a force along
Ox. Substituting from (1) in the first formula of Art. 150 (12)
we find

............ (2),

where the integration is to extend over the sections of all the
vortices. If we denote by m the strength to&cSor of an elementary
vortex-filament whose coordinates are #, tzr, this may be written

P = 27rp2m*r2 = 2-7T/3 . 2m . -cr02 ..................... (3),

•f 2m^2 /A.

lf OT«2 = -2m ........................... W"

The quantity <GTO, thus defined, may be called the ' mean-radius '
of the whole system of circular vortices. Since m is constant for
each vortex, the constancy of the impulse requires that the mean-
radius shall be constant with respect to the time.

The formula for the kinetic energy (Art. 153 (4)) becomes, in
the present case,

T = 4*pjJ(w* — xv) 'sraydxd'Gr = ^-jrp^m (^u — xv) CT ...... (5).

159-161] CIRCULAR VORTICES. 255

Let us introduce a symbol x0, defined by

It is plain that the position of the circle (#0, tn-0) will depend only
on the strengths and the configuration of the vortices, and not on
the position of the origin on the axis of symmetry. This circle
may be called the 'circular axis' of the whole system of vortex
rings ; we have seen that it remains constant in radius. To find
its motion parallel to Ox, we have from (6) and (4),

(7),

since u and v are the rates of increase of x and w for any
particular vortex. By means of (5) we can put this in the form

] T

(x-x.)vn> ............ (8),

which will be of use to us later. The added term vanishes, since
= 0 on account of the constancy of the mean radius.

161. On account of the symmetry about Ox, there exists, in
the cases at present under consideration, a stream-function ty,
defined as in Art. 93, viz. we have

1 dilr
~^r

TV dx

, dv du 1 /cfiilr d2^ 1 d\lr\

whence 2o> = -= — = — — r. + —.11 n (2).

dx dm ty \ dx* d^ -BT dwj

It appears from Art. 148 (4) that at a great distance from the
vortices u, v are of the order R~s, and therefore ty will be of the
order R~l.

The formula for the kinetic energy may therefore be written
T = 7rpff(u* + ir) or dxdvr

f/y ^^ ^'*/r^ 7 ^

= TTO In v —• j- — u ~- dxaiz
Jj\ dx df»rj

by a partial integration, the terms at the limits vanishing.

256 VORTEX MOTION. [CHAP. VII

To determine ^jr in terms of the (arbitrary) distribution of
angular velocity (&>), we may make use of the formulae of Art. 145,
which give

F=0,

i t [(*<**

H= ^rjlj-^

ft)' COS ^' / 7 / 7 / 7 f

TS d^ dx d'ur

•(4),

where r = {(as — #')2 + •cr2 + ®"/2 — 2w«j' cos (^ — &'))*.

Since 27r^ denotes (Art. 93) the flux, in the direction of
^--negative, through the circle (a?, «r), we have

+4 II W — If

J J V dy o

where the integration extends over the area of this circle. By
Stokes' Theorem, this gives

the integral being taken round the circumference, or, in terms of
our present coordinates,

provided

f i ^ ^ i = r - _jBos^d0_ , ,

7 V, 'sr'} Jo {(a? - #')2 + ^2 + ^/2 ~ 2^OT/ cos ^1-
where ^ has been written for $ — &'.

It is plain that the function here denned is symmetrical with
respect to the two sets of variables a?, -BJ and x', *&'. It can be
expressed in terms of elliptic integrals, as follows. If we put

_

- ' ..............

* The vector whose components are F, G, H is now perpendicular to the
meridian plane xw. If we denote it by <S\ we have F= 0, G = - S sin ^, H = S cos ^,
so that (7) is equivalent to

\f= -TffS.

161-162] STREAM-FUNCTION. 257

we find

cosfl k 2 cos2 6-l

_
{(x - x'Y + ^2 + OT/S - 2*7*3-' cos 6}* ~ 2 («r«r')i ' (1 - k2 cos2

where, from comparison of coefficients,
Hence

where J^ (&), ^ (k) are the complete elliptic integrals of the first
and second kinds, with respect to the modulus k, defined by (9).

162. The stream-function for points at a distance from an
isolated circular vortex-filament, of strength m', whose coordinates
are of, -BT', is therefore given by

7T

The forms of the stream-lines corresponding to equidistant
values of ty, at points whose distances from the filament are great
in comparison with the dimensions of the cross-section, are shewn
on the next page*.

At points of the infinitely small section the modulus k of the
elliptic integrals in the value of ^ is nearly equal to unity. In
this case we have~t*

4

approximately, where k' denotes the complementary modulus
(1 — k-)$, so that in our case

* For another elliptic-integral form of (1), and for the most convenient method
of tracing the curves $ = const., see Maxwell, Electricity and Magnetism, Arts. 701,
702.

t See Cayley, Elliptic Functions, Cambridge, 1876, Arts. 72, 77; and Maxwell,
Electricity and Magnetism, Arts. 704, 705.

L. 17

258

VORTEX MOTION. [CHAP. VII

X'-

•x

162] STREAM LINES OF A VORTEX-RING. 259

nearly, if g denote the distance between two infinitely near points
(x, CT), (V, CT') in the same meridian plane. Hence at points
within the substance of the vortex the value of ty is of the order
m'vr log OT/€, where e is a small linear magnitude comparable with
the dimensions of the section. The velocity at the same point,
depending (Art. 93) on the differential coefficients of i/r, will
be of the order m'/e.

We can now estimate the magnitude of the velocity dx0/dt of
translation of the vortex-ring. By Art. 161 (3) T is of the order
prn'^'ur log -cr/e, and u is, as we have seen, of the order m'/e ; whilst
x — XQ is of course of the order e. Hence the second term on the
right-hand side of the formula (8) of Art. 160 is, in this case, small
compared with the first, and the velocity of translation of the
ring is of the order m'/^ . log cr/e, and approximately constant.

An isolated vortex-ring moves then, without sensible change
of size, parallel to its rectilinear axis with nearly constant
velocity. This velocity is small compared with that of the fluid
in the immediate neighbourhood of the circular axis, but may be
large compared with m'/w0, the velocity of the fluid at the centre
of the ring, with which it agrees in direction.

For the case of a circular section more definite results can be obtained
as follows. If we neglect the variations of OT' and o>' over the section,
the formulae (7) and (10) of Art. 161 give

or, if we introduce polar coordinates (s, %) in the plane of the section,

T-1)'** ..................... <«•

where a is the radius of the section. Now

0 JO

and this definite integral is known to be equal to 277 logs', or 27rlogs,
according as s'^s. Hence, for points within the section,

" s'ds'

(ii).

17—2

= - 2o>' wof* flog —0-2^ s'ds'-Mw, [a(\oS*2*-

J Q\ s / J S\ S

260 VORTEX MOTION. [CHAP. VII

The only variable part of this is the term -^o/t<r0s2; this shews that to our
order of approximation the stream -lines within the section are concentric
circles, the velocity at a distance s from the centre being a>'s. Substituting
in Art. 161 (3) we find

The last term in Art. 160 (8) is equivalent to

fi 8rao 7) ,-~\

{log V-*} ............... (m)-

in our present notation, m' denoting the strength of the whole vortex, this is
equal to 3m'2CT0/47r. Hence the formula for the velocity of translation of the
vortex becomes

"7/" m/ -1 "— " .(iv)*

163. If we have any number of circular vortex-rings, coaxial
or not, the motion of any one of these may be conceived as made
up of two parts, one due to the ring itself, the other due to the
influence of the remaining rings. The preceding considerations
shew that the second part is insignificant compared with the first,
except when two or more rings approach within a very small
distance of one another. Hence each ring will move, without
sensible change of shape or size, with nearly uniform velocity in
the direction of its rectilinear axis, until it passes within a short
distance of a second ring.

A general notion of the result of the encounter of two rings
may, in particular cases, be gathered from the result of Art. 147
(3). Thus, let us suppose that we have two circular vortices
having the same rectilinear axis. If the sense of the rotation be the
same for both, the two rings will advance, on the whole, in the same
direction. One effect of their mutual influence will be to increase
the radius of the one in front, and to contract the radius of
the one in the rear. If the radius of the one in front become
larger than that of the one in the rear, the motion of the former
ring will be retarded, whilst that of the latter is accelerated.
Hence if the conditions as to relative size and strength of the
two rings be favourable, it may happen that the second ring
will overtake and pass through the first. The parts played by
the two rings will then be reversed ; the one which is now in

* This result was first obtained by Sir W. Thomson, Phil. Mag., June, 1867.

162-163] MUTUAL INFLUENCE OF VORTEX-RINGS. 261

the rear will in turn overtake and pass through the other, and
so on, the rings alternately passing one through the other*.

If the rotations in the two rings be opposite, and such that
the rings approach one another, the mutual influence will be to
enlarge the radius of each ring. If the two rings be moreover
equal in size and strength, the velocity of approach will con
tinually diminish. In this case the motion at all points of the
plane which is parallel to the two rings, and half-way between
them, is tangential to this plane. We may therefore, if we
please, regard this plane as a fixed boundary to the fluid on
either side of it, and so obtain the case of a single vortex-ring
moving directly towards a fixed rigid wall.

The foregoing remarks are taken from von Helmholtz' paper.
He adds, in conclusion, that the mutual influence of vortex-rings
may easily be studied experimentally in the case of the (roughly)
semicircular rings produced by drawing rapidly the point of a
spoon for a short space through the surface of a liquid, the spots
where the vortex-filaments meet the surface being marked by
dimples. (Cf. Art. 28.) The method of experimental illustration
by means of smoke-rings f" is too well-known to need description
here. A beautiful variation of the experiment consists in forming
the rings in water, the substance of the vortices being coloured f.

For further theoretical researches on the motion of vortex-
rings, including the question of stability, and the determination of
the small oscillations, we must refer to the papers cited below §.

The motion of a vortex-ring in a fluid limited (whether
internally or externally) by a fixed spherical surface, in the case

* The corresponding case in two dimensions appears to have been worked out
very completely by Grobli; see Winkelmann, Handbuch der Physik, t. i., p. 447.
The same question has been discussed quite recently by Love, " On the Motion of
Paired Vortices with a Common Axis," Proc. Lond. Math. Soc.,t. xxv., p. 185 (1894).

t Reusch, "Ueber Ringbildung der Fliissigkeiten," Pogg. Ann., t. ex. (1860);
see also Tait, Recent Advances in Physical Science, London, 1876, c. xii.

J Reynolds, "On the Resistance encountered by Vortex Rings &c.", Brit. Ass.
Rep., 1876, Nature, t. xiv., p. 477.

§ J. J. Thomson, I. c. ante p. 239, and Phil. Trans., 1882.

W. M. Hicks, "On the Steady Motion and the Small Vibrations of a Hollow
Vortex," Phil. Trans. 1884.

Dyson, 1. c. ante p. 166.

The theory of ' Vortex- Atoms ' which gave the impulse to some of these investi
gations was suggested by Sir W. Thomson, Phil Mag., July, 1867.

262 VORTEX MOTION. [CHAP. VII

where the rectilinear axis of the ring passes through the centre of
the sphere, has been investigated by Lewis*, by the method of
1 images.'

The following simplified proof is due to Larmorf. The vortex-ring is
equivalent (Art. 148) to a spherical sheet of double-sources of uniform
density, concentric with the fixed sphere. The ' image ' of this sheet will,
by Art. 95, be another uniform concentric double-sheet, which is, again,
equivalent to a vortex-ring coaxial with the first. It easily follows from the
Art. last cited that the strengths (m\ m"} and the radii (or', or") of the vortex-
ring and its image are connected by the relation

(i).

The argument obviously applies to the case of a reentrant vortex of any
form, provided it lie on a sphere concentric with the boundary.

On the Conditions for Steady Motion.

164. In steady motion, i.e. when

du_ dv_ dw_

dt~ ' dt~ dt"

the equations (2) of Art. 6 may be written

du dv dw a ,
dx dx doc

Hence, if as in Art. 143 we put

du dv dw a , ,, _ , _ c?O _ 1 dp
dx dx doc dx p dx ' '

we have

It follows that

dx d')

dx dy ^ dz

* "On the Images of Vortices in a Spherical Vessel," Quart. Journ. Math.,
t. xvi., p. 338 (1879).

t " Electro-magnetic and other Images in Spheres and Planes," Quart. Journ.
Math.,t. xxiii., p. 94 (1889).

163-164] CONDITIONS FOR STEADY MOTION. 263

so that each of the surfaces ^' = const, contains both stream-lines
and vortex-lines. If further 8n denote an element of the normal
at any point of such a surface, we have

n^ ........................ (2),

where q is the current-velocity, co the rotation, and /3 the angle
between the stream-line and the vortex-line at that point.

Hence the conditions that a given state of motion of a fluid
may be a possible state of steady motion are as follows. It must
be possible to draw in the fluid an infinite system of surfaces
each of which is covered by a network of stream-lines and vortex-
lines, and the product qw sin /3 Sn must be constant over each
such surface, Sn denoting the length of the normal drawn to a
consecutive surface of the system.

These conditions may also be deduced from the considerations
that the stream-lines are, in steady motion, the actual paths of
the particles, that the product of the angular velocity into the
cross-section is the same at all points of a vortex, and that this
product is, for the same vortex, constant with regard to the
time *.

The theorem that the function ^', defined by (1), is constant
over each surface of the above kind is an extension of that of
Art. 22, where it was shewn that % is constant along a stream
line.

The above conditions are satisfied identically in all cases of
irrotational motion, provided of course the boundary-conditions be
such as are consistent with the steady motion.

In the motion of a liquid in two dimensions (xy) the product
cftn is constant along a stream-line ; the conditions in question
then reduce to this, that the angular velocity f must be constant
along each stream-line, or, by Art. 59,

»t.

where f(^r) is an arbitrary function o

* See a paper " On the Conditions for Steady Motion of a Fluid," Proc. Lond.
Math. Soc., t. ix., p. 91 (1878).

t Cf. Lagrange, Nouv. M6m. de VAcad. de Berlin, 1781, Oeuvres, t. iv., p. 720 ;
and Stokes, 1. c. p. 264.

264 VORTEX MOTION. [CHAP. VII

This condition is satisfied in all cases of motion in concentric circles
about the origin. Another obvious solution of (3) is

in which case the stream-lines are similar and coaxial conies. The angular
velocity at any point is ^ (A 4- C], and is therefore uniform.

Again, if we put / (\J/-) = - Fx//-, where k is a constant, and transform to
polar coordinates r, 6, we get

dr2 r dr
which is satisfied by

cos)
sin)

where Ja is a 'Bessel's Function.' This gives various solutions consistent
with a fixed circular boundary of radius a, the admissible values of k being
determined by

J3(ka) = 0 (iv).

The character of these solutions will be understood from the properties of
Bessel's Functions, of which some indication will be given in Chapter vni.

In the case of motion symmetrical about an axis (a?), we have
q . %7T'&r$n constant along a stream-line, VT denoting as in Art. 93
the distance of any point from the axis of symmetry. The con
dition for steady motion then is that the ratio &>/«r must be
constant along any stream-line. Hence, if i|r be the stream-
function, we must have, by Art. 161 (2),

denotes an arbitrary function of i/r.

An interesting example of (4) is furnished by the case of Hill's ' Spherical
Vortex f.' If we assume

-^ .................................... (v),

where r2=#2 + rar2, for all points within the sphere r = a, the formula (2)
of Art. 161 makes

so that the condition of steady motion is satisfied. Again it is evident, on
reference to Arts. 95, 96 that the irrotational flow of a stream with the

* This result is due to Stokes, "On the Steady Motion of Incompressible
Fluids," Camb. Trans., t. vii. (1842), Math, and Phys. Papers, t. i., p. 15.
t " On a Spherical Vortex," Phil. Trans., 1894, A.

164] SPHERICAL VORTEX. 265

general velocity -u parallel to the axis, past a fixed spherical surface
r=a, is given by

The two values of ^ agree when r = a-, this makes the normal velocity
continuous. In order that the tangential velocity may be continuous, the
values of d^/dr must also agree. Remembering that ar=rsin0, this gives
4 *«/~2 and therefore

The sum of the strengths of the vortex-filaments composing the spherical
vortex is 5u«.

The figure shews the stream-lines, both inside and outside the vortex;
they are drawn, as usual, for equidistant values of \//-.

If we impress on everything a velocity u parallel to #, we get a spherical
vortex advancing with constant velocity u through a liquid which is at rest at
infinity.

By the formulae of Arts. 160, 161, we readily find that the square of the
'mean-radius' of the vortex is fa2, the 'impulse' is 27rpa3u, and the energy is

CHAPTER VIII.

TIDAL WAVES.

165. ONE of the most interesting and successful applications
of hydrodynamical theory is to the small oscillations, under gravity,
of a liquid having a free surface. In certain cases, which are
somewhat special as regards the theory, but very important from
a practical point of view, these oscillations may combine to form
progressive waves travelling with (to a first approximation) no
change of form over the surface.

The term ' tidal,' as applied to waves, has been used in various
senses, but it seems most natural to confine it to gravitational
oscillations possessing the characteristic feature of the oceanic
tides produced by the action of the sun and moon. We have
therefore ventured to place it at the head of this Chapter, as
descriptive of waves in which the motion of the fluid is mainly
horizontal, and therefore (as will appear) sensibly the same for all
particles in a vertical line. This latter circumstance greatly
simplifies the theory.

It will be convenient to recapitulate, in the first place, some
points in the general theory of small oscillations which will
receive constant exemplification in the investigations which
follow*.

Let qlt q.2,...qn be n generalized coordinates serving to specify
the configuration of a dynamical system, and let them be so chosen
as to vanish in the configuration of equilibrium. The kinetic

* For a fuller account of the general theory see Thomson and Tait, Natural
Philosophy, kits. 337, ..., Lord Rayleigh, Theory of Sound, c. iv.,Routh, Elementary
Rigid Dynamics (5th ed.), London, 1891, c. ix.

165] SMALL OSCILLATIONS. 267

energy T will, as explained in Art. 133, be a homogeneous
quadratic function of the generalized velocities (ft, g^,..., say

2r=an212 + a22g22+... + 2a12?1?2 + ............ (1).

The coefficients in this expression are in general functions of the
coordinates <ft, q2,..., but in the application to small motions, we
may suppose them to be constant, and to have the values corre
sponding to (ft = 0, q2 = 0,.... Again, if (as we shall suppose) the
system is ' conservative,' the potential energy V of a small displace
ment is a homogeneous quadratic function of the component
displacements q1} (?2, ... , with (on the same understanding) constant
coefficients, say

... + 2c12g1g2 + ............ (2).

By a real* linear transformation of the coordinates (ft, ^2,... it
is possible to reduce T and V simultaneously to sums of squares ;
the new variables thus introduced are called the 'normal
coordinates' of the system. In terms of these we have

(3),
(4).

The coefficients a1}a2,... are called the 'principal coefficients of
inertia'; they are necessarily positive. The coefficients cl5c2,...
may be called the ' principal coefficients of stability ' ; they are all
positive when the undisturbed configuration is stable.

When given extraneous forces act on the system, the work
done by these during an arbitrary infinitesimal displacement
A(ft, Ag2, ... may be expressed in the form

(5).

The coefficients Qlt Q2> ••• are then called the 'normal components
of external force.'

In terms of the normal coordinates, the equations of motion
are given by Lagrange's equations (Art. 133 (17)), thus

_____

dtdqs dqs~ dqs s'

* The algebraic proof of this involves the assumption that one at least of the
functions T, V is essentially positive.

268 TIDAL WAVES. [CHAP. VIII

In the present application to infinitely small motions, these
take the form

9 f asqs + csqs = Qs . (6).

It is easily seen from this that the dynamical characteristics of the
normal coordinates are (1°) that an impulse of any normal type
produces an initial motion of that type only, and (2°) that a steady
extraneous force of any type maintains a displacement of that
type only.

To obtain the free motions of the system we put Qs = 0 in (6).
Solving we find

qs = A8cos(ffgt + €8) (7),

where crs = (cs/as)% (8)*,

and A8, €s are arbitrary constants. Hence a mode of free motion
is possible in which any normal coordinate qs varies alone, and
the motion of any particle of the system, since it depends
linearly on qs, will be simple-harmonic, of period 27r/crs, and
every particle will pass simultaneously through its equilibrium
position. The several modes of this character are called the
1 normal modes ' of vibration of the system ; their number is equal
to that of the degrees of freedom, and any free motion whatever
of the system may be obtained from them by superposition, with
a proper choice of the 'amplitudes ' (A8) and ' epochs ' (es).

In certain cases, viz. when two or more of the free periods
(27T/cr) of the system are equal, the normal coordinates are to a
certain extent indeterminate, i.e. they can be chosen in an infinite
number of ways. An instance of this is the spherical pendulum.
Other examples will present themselves later; see Arts. 187, 191.

If two (or more) normal modes have the same period, then
by compounding them, with arbitrary amplitudes and epochs, we
obtain a small oscillation in which the motion of each particle is
the resultant of simple-harmonic vibrations in different directions,
and is therefore, in general, elliptic-harmonic, with the same
period. This is exemplified in the conical pendulum ; an im
portant instance in our own subject is that of progressive waves in
deep water (Chap. ix.).

* The ratio o-/27r measures the ' frequency' of the oscillation. It is convenient,
however, to have a name for the quantity a- itself ; the term ' speed ' has been used in
this sense by Lord Kelvin and Prof. G. H. Darwin in their researches on the Tides.

165] THEORY OF NORMAL MODES. 269

If any of the coefficients of stability (cs) be negative, the
value of crs is pure imaginary. The circular function in (7) is then
replaced by real exponentials, and an arbitrary displacement
will in general increase until the assumptions on which the
approximate equation (6) is based become untenable. The un
disturbed configuration is then unstable. Hence the necessary
and sufficient condition of stability is that the potential energy V
should be a minimum in the configuration of equilibrium.

To find the effect of extraneous forces, it is sufficient to
consider the case where Qs varies as a simple-harmonic function of
the time, say

Qs=C8coa(at+€) ..................... (9),

where the value of a is now prescribed. Not only is this the
most interesting case in itself, but we know from Fourier's
Theorem that, whatever the law of variation of Qs with the time, it
can be expressed by a series of terms such as (9). A particular
integral of (9) is then

This represents the ' forced oscillation ' due to the periodic force
Qs. In it the motion of every particle is simple-harmonic, of the
prescribed period 27T/0-, and the extreme displacements coincide in
time with the maxima and minima of the force.

A constant force equal to the instantaneous value of the
actual force (9) would maintain a displacement

*

(11),

the same, of course, as if the inertia-coefficient as were null.
Hence (10) may be written

*?tr^Stf* ..................... (12X

where crs has the value (8). This very useful formula enables us
to write down the effect of a periodic force when we know that of
a steady force of the same type. It is to be noticed that qs and Qs
have the same or opposite phases according as cr J as, that is,
according as the period of the disturbing force is greater or less
than the free period. A simple example of this is furnished by a
simple pendulum acted on by a periodic horizontal force. Other

270 TIDAL WAVES. [CHAP. VIII

important illustrations will present themselves in the theory of the
tides*.

When a is very great in comparison with as, the formula (10)
becomes

C
qs = -- ~cos(<rt + 6) ............... (13);

the displacement is now always in the opposite phase to the force,
and depends only on the inertia of the system.

If the period of the impressed force be nearly equal to that of
the normal mode of order s, the amplitude of the forced oscillation,
as given by (12), is very great compared with qg. In the case of
exact equality, the solution (10) fails, and must be replaced by

qs = Bt sin (at + e) ..................... (14),

where, as is verified immediately on substitution, B = Cs/Zaas.
This gives an oscillation of continually increasing amplitude, and
can therefore only be accepted as a representation of the initial
stages of the disturbance.

Another very important property of the normal modes may be noticed,
although the use which we shall have occasion to make of it will be slight.
If by the introduction of constraints the system be compelled to oscillate
in any other manner, then if the character of this motion be known, the
configuration at any instant can be specified by one variable, which we will
denote by 6. In terms of this we shall have

fc-*t4

where the quantities B8 are certain constants. This makes

.)0 .............................. (i),

)P ............................ (ii).

Hence if 6 a cos (o-tf + e), the constancy of the energy (T+ F) requires

Hence o-2 is intermediate in value between the greatest and least of the
quantities cBlaa ; in other words, the frequency of the constrained oscillation
is intermediate between the greatest and least frequencies corresponding to
the normal modes of the system. In particular, when a system is modified
by the introduction of any constraint, the frequency of the slowest natural
oscillation is increased.

* Cf . T. Young, " A Theory of Tides," Nicholson's Journal, t. xxxv. (1813) ;
Miscellaneous Works, London, 1854, t. ii., p. 262.

165-166] APPROXIMATE TYPES. 271

Moreover, if the constrained type differ but slightly from a normal type
(a), o-2 will differ from c8/aa by a small quantity of the second order. This gives a
valuable method of estimating approximately the frequency in cases where
the. normal types cannot be accurately determined*.

The modifications which are introduced into the theory of
small oscillations by the consideration of viscous forces will be
noticed in Chapter xi.

Long Waves in Canals.

166. Proceeding now to the special problem of this Chapter,
let us begin with the case of waves travelling along a straight
canal, with horizontal bed, and parallel vertical sides. Let the
axis of x be parallel to the length of the canal, that of y
vertical and upwards, and let us suppose that the motion takes
place in these two dimensions x, y. Let the ordinate of the free
surface, corresponding to the abscissa x, at time t, be denoted by
77 -|- £/0j where y0 is the ordinate in the undisturbed state.

As already indicated, we shall assume in all the investigations
of this Chapter that the vertical acceleration of the fluid particles
may be neglected, or (more precisely) that the pressure at any
point (x, y) is sensibly equal to the statical pressure due to the
depth below the free surface, viz.

p-po = gp(yo + 'n-y) ..................... (i),

where p0 is the (uniform) external pressure.

This is independent of y, so that the horizontal acceleration is the
same for all particles in a plane perpendicular to x. It follows
that all particles which once lie in such a plane always do so ; in
other words, the horizontal velocity u is a function of as and
t only.

The equation of horizontal motion, viz.
du du 1 dp

_ _J_ y _ - — __ •*•

dt dx p dx '
is further simplified in the -ease of infinitely small motions by the

* These theorems are due to Lord Eayleigh, " Some General Theorems relating
to Vibrations," Proc. Lond. Math. Soc., t. iv., p. 357 (1873); Theory of Sound, c. iv.

272 TIDAL WAVES. [CHAP. VIII

omission of the term udu/dx, which is of the second order, so
that

If we put

then f measures the integral displacement of liquid past the
point x, up to the time t ; in the case of small motions it will, to
the first order of small quantities, be equal to the displacement
of the particle which originally occupied that position, or again
to that of the particle which actually occupies it at time t. The
equation (3) may now be written

The equation of continuity may be found by calculating the
volume of fluid which has, up to time t, entered the space bounded
by the planes x and x + $x\ thus, if h be the depth and b the
breadth of the canal,

The same result comes from the ordinary form of the equation of
continuity, viz.

[vdu , du

1 hus v = — I -j- ay = — V -j- .

*da;* * dx

if the origin be (for the moment) taken in the bottom of the canal. This
formula is of interest as shewing that the vertical velocity of any particle is
simply proportional to its height above the bottom. At the free surface we
have y = h+r), v = dr)/dt, whence (neglecting a product of small quantities)

dr,__ d^

dt~ dxdt

Prom this (5) follows by integration with respect to t.

166-167] WAVES IN UNIFORM CANAL. 273

Eliminating 77 between (4) and (5), we obtain

The elimination of £ gives an equation of the same form, viz.

d^-ah^ (7)

d?-g/ldx*'

The above investigation can readily be extended to the case of a uniform
canal of any form of section*. If the sectional area of the undisturbed fluid
be $, and the breadth at the free surface &, the equation of continuity is

(iv),

whence rj= ~^5E ....................................... (v)>

as before, provided h^=S/b, i.e. h now denotes the mean depth of the canal.
The dynamical equation (4) is of course unaltered,

167. The equations (6) and (7) are of a well-known type
which occurs in several physical problems, e.g. the transverse
vibrations of strings, and the motion of sound-waves in one
dimension.

To integrate them, let us write, for shortness,

<f=gh .............................. (8),

and x—ct = xl} x + ct = x2.

In terms of xl and #2 as independent variables, the equation (6)
takes the form

o.

The complete solution is therefore

% = F(a;-ct)+f(a; + ct) .................. (9),

where F,fa,rQ arbitrary functions.

The corresponding values of the particle- velocity and of the
surface- elevation are given by

|/c = - F1 (x - ct) +/'(* + ct), } ( }

rj/h=-F'(x-ct)-f(x + ct) j ......

* Kelland, Trans. R. S. Edin., t. xiv. (1839).
L. 18

274

TIDAL WAVES.

[CHAP, viii

The interpretation of these results is simple. Take first the
motion represented by the first term in (9), alone. Since F(x — ct)
is unaltered when t and x are increased by r and cr, respectively,
it is plain that the disturbance which existed at the point x
at time t has been transferred at time t + r to the point x + cr.
Hence the disturbance advances unchanged with a constant
velocity c in space. In other words we have a ' progressive wave '
travelling with constant velocity c in the direction of ^-positive.
In the same way the second term of (9) represents a progressive
wave travelling with velocity c in the direction of ^-negative.
And it appears, since (9) is the complete solution of (6), that any
motion whatever of the fluid, which is subject to the conditions
laid down in the preceding Art., may be regarded as made up of
waves of these two kinds.

The velocity (c) of propagation is, by (8), that ' due to ' half
the depth of the undisturbed fluid*.

The following table, giving in round numbers the velocity of
wave-propagation for various depths, will be of interest, later, in
connection with the theory of the tides.

h

c

c

2ira/c

(feet)

(feet per sec.)

(sea-miles per hour)

(hours)

312J

100

60

360

1250

200

120

180

5000

400

240

90

11250f

600

360

60

20000

800

480

45

The last column gives the time a wave would take to travel
over a distance equal to the earth's circumference (2?ra). In order
that a * long ' wave should traverse this distance in 24 hours, the
depth would have to be about 14 miles. It must be borne in
mind that these numerical results are only applicable to waves
satisfying the conditions above postulated. The meaning of these
conditions will be examined more particularly in Art. 169.

* Lagrange, Nouv. m6m. de VAcad. de Berlin, 1781, Oeuvres, t. i. p. 747.
t This is probably comparable in order of magnitude with the mean depth of
the ocean,

167-168] WAVE- VELOCITY. 275

168. To trace the effect of an arbitrary initial disturbance,
let us suppose that when £ = 0 we have

l/c = *(*), WA = *(*) .................. (11).

The functions F',f which occur in (10) are then given by

(*)+*(*)),) n«

(*)-*(*)} r

Hence if we draw the curves y = rjl, y = r)2, where

/IDA

the form of the wave-profile at any subsequent instant t is found
by displacing these curves parallel to x, through spaces ± ct,
respectively, and adding (algebraically) the ordinates. If, for
example, the original disturbance be confined to a length I of the
axis of x, then after a time Z/2c it will have broken up into two
progressive waves of length I, travelling in opposite directions.

In the particular case where in the initial state f = 0, and
therefore <£ (x) = 0, we have ^ = 773 ; the elevation in each of the
derived waves is then exactly half what it was, at corresponding
points, in the original disturbance.

It appears from (11) and (12) that if the initial disturbance be
such that f = ± fi/h . c, the motion will consist of a wave system
travelling in one direction only, since one or other of the functions
F' and/7 is then zero. It is easy to trace the motion of a surface-
particle as a progressive wave of either kind passes it. Suppose,
for example, that

£ = F(x-ct) ........................ (14),

and therefore f = crj/h .............................. (15).

The particle is at rest until it is reached by the wave ; it
then moves forward with a velocity proportional at each instant
to the elevation above the mean level, the velocity being in fact
less than the wave-velocity c, in the ratio of the surface-elevation
to the depth of the water. The total displacement at any time
is given by

T Irjcdt.

18—2

276 TIDAL WAVES. [CHAP. VIII

This integral measures the volume, per unit breadth of the canal,
of the portion of the wave which has up to the instant in question
passed the particle. Finally, when the wave has passed away, the
particle is left at rest in advance of its original position at a
distance equal to the total volume of the elevated water, divided
by the sectional area of the canal.

169. We can now examine under what circumstances the
solution expressed by (9) will be consistent with the assumptions
made provisionally in Art. 166.

The restriction to infinitely small motions, made in equation
(3), consisted in neglecting udujdx in comparison with du/dt. In
a progressive wave we have du/dt = ± cdujdx ; so that u must be
small compared with c, and therefore, by (15), 77 small compared
with h.

Again, the exact equation of vertical motion, viz.

Dv dp
?Di = -te-W'

gives, on integration with respect to y,

+i 2)v

This may be replaced by the approximate equation (1), pro
vided /3 (h + 77) be small compared with grj, where {3 denotes
the maximum vertical acceleration. Now in a progressive wave,
if X denote the distance between two consecutive nodes (i.e. points
at which the wave-profile meets the undisturbed level), the time
which the corresponding portion of the wave takes to pass a
particle is X/c, and therefore the vertical velocity will be of the
order TJC/X*, and the vertical acceleration of the order ?;c2/X2, where
77 is the maximum elevation (or depression). Hence the neglect
of the vertical acceleration is justified, provided A2/X2 is a small
quantity.

Waves whose slope is gradual, and whose length X is large
compared with the depth h of the fluid, are called ' long waves.'

* Hence, comparing with (15), we see that the ratio of the maximum vertical to
the maximum horizontal velocity is of the order h/\.

168-170] AIRY'S METHOD. 277

The requisite conditions will of course be satisfied in the

general case represented by equation (9), provided they are

satisfied for each of the two progressive waves into which the
disturbance can be analysed.

170. There is another, although on the whole a less con
venient, method of investigating the motion of ' long ' waves, in
which the Lagrangian plan is adopted, of making the coordinates
refer to the individual particles of the fluid. For simplicity, we
will consider only the case of a canal of rectangular section*. The
fundamental assumption that the vertical acceleration may be
neglected implies as before that the horizontal motion of all
particles in a plane perpendicular to the length of the canal will
be the same. We therefore denote by# + f the abscissa at time
t of the plane of particles whose undisturbed abscissa is x. If ij
denote the elevation of the free surface, in this plane, the equation
of motion of unit breadth of a stratum whose thickness (in the
undisturbed state) is $%, will be

where the factor (dpfdx) . $% represents the pressure-difference for
any two opposite particles x and x + &e on the two faces of the
stratum, while the factor h + 77 represents the area of the stratum.
Since the pressure about any particle depends only on its depth
below the free surface we may write

dp dn

•*• - fin _ L

dx~9pdxy
so that our dynamical equation is

The equation of continuity is obtained by equating the volumes
of a stratum, consisting of the same particles, in the disturbed and
undisturbed conditions respectively, viz. we have

* Airy, Encyc. Metrop., " Tides and Waves," Art. 192 (1845) ; see also Stokes,
"On Waves," Camb. and Dub. Math. Journ.,t. iv. (1849), Math, and Phys. Papers,
t. ii., p. 222. The case of a canal with sloping sides has been treated by McCowan,
" On the Theory of Long Waves...," Phil. Mag., March, 1892.

278 TIDAL WAVES. [CHAP. VIII

Between equations (1) and (2) we may eliminate either 77 or £ ;
the result in terms of f is the simpler, being

(3)*-

d*

This is the general equation of ' long ' waves in a uniform canal
with vertical sides.

So far the only assumption is that the vertical acceleration of
the particles may be neglected. If we now assume, in addition,
that r}\li is a small quantity, the equations (2) and (3) reduce to

#f . d'f
and -

The elevation 77 now satisfies the equation

This is in conformity with our previous result ; for the small-
ness of d^/dx means that the relative displacement of any two
particles is never more than a minute fraction of the distance
between them, so that it is (to a first approximation) now
immaterial whether the variable as be supposed to refer to a
plane fixed in space, or to one moving with the fluid.

171. The potential energy of a wave, or system of waves,
due to the elevation or depression of the fluid above or below the
mean level is, per unit breadth, gpjfydxdy, where the integra
tion with respect to y is to be taken between the limits 0 and ?/,
and that with respect to x over the whole length of the waves.
Effecting the former integration, we get

(1).
The kinetic energy is

(2).

* Airy, L c.

170-172] ENERGY. 279

In a system of waves travelling in one direction only we have

so that the expressions (1) and (2) are equal ; or the total energy
is half potential, and half kinetic.

This result may be obtained in a more general manner, as
follows*. Any progressive wave may be conceived as having been
originated by the splitting up, into two waves travelling in opposite
directions, of an initial disturbance in which the particle-velocity
was everywhere zero, and the energy therefore wholly potential.
It appears from Art. 168 that the two derived waves are symme
trical in every respect, so that each must contain half the original
store of energy. Since, however, the elevation at corresponding
points is for the derived waves exactly half that of the original
disturbance, the potential energy of each will by (1) be one-fourth
of the original store. The remaining (kinetic) part of the energy
of each derived wave must therefore also be one-fourth of the
original quantity.

172. If in any case of waves travelling in one direction only,
without change of form, we impress on the whole mass a velocity
equal and opposite to that of propagation, the motion becomes
steady, whilst the forces acting on any particle remain the same as
before. With the help of this artifice, the laws of wave-propa
gation can be investigated with great easef. Thus, in the present
case we shall have, by Art. 23 (4), at the free surface,

(1),

where q is the velocity. If the slope of the wave-profile be
everywhere gradual, and the depth h small compared with the
length of a wave, the horizontal velocity may be taken to be
uniform throughout the depth, and approximately equal to q.
Hence the equation of continuity is

q(h + 7)) = ch,

* Lord Rayleigh, "On Waves," PhiL Ma0.t April, 1876,
t Lord Kayleigh, I. c>

280 TIDAL WAVES. [CHAP. VIII

c being the velocity, in the steady motion, at places where the
depth of the stream is uniform and equal to h. Substituting
for q in (1), we have

Hence if r)/h be small, the condition for a free surface, viz.
p = const., is satisfied approximately, provided

which agrees with our former result.

173. It appears from the linearity of our equations that
any number of independent solutions may be superposed. For
example, having given a wave of any form travelling in one
direction, if we superpose its image in the plane x = 0, travelling
in the opposite direction, it is obvious that in the resulting
motion the horizontal velocity will vanish at the origin, and the
circumstances are therefore the same as if there were a fixed barrier
at this point. We can thus understand the reflexion of a wave at
a barrier ; the elevations and depressions are reflected unchanged,
whilst the horizontal velocity is reversed. The same results
follow from the formula

f = F(ct-x)-F(ct + x) .................. (1),

which is evidently the most general value of f subject to the
condition that f = 0 for x = 0.

We can further investigate without much difficulty the partial reflexion
of a wave at a point where there is an abrupt change in the section of the
canal. Taking the origin at the point in question, we may write, for the
negative side,

and for the positive side

»-*('.-£}.•

\ C2/

172-174] REFLECTION. 281

where the function F represents the original wave, and /, $ the reflected and
transmitted portions respectively. The constancy of mass requires that at
the point # = 0 we should have b^u^b^u^ where blt b.2 are the breadths at
the surface, and klt h2 are the mean depths. We must also have at the same
point »71=»/2) on account of the continuity of pressure*. These conditions give

*1

F(t)+f(t):

We thence find that the ratio of the elevations in corresponding parts of the
reflected and incident waves is

F &

The similar ratio for the transmitted wave is

The reader may easily verify that the energy contained in the reflected and
transmitted waves is equal to that of the original incident wave.

174. Our investigations, so far, relate to cases of free waves.
When, in addition to gravity, small disturbing forces X, Y act on
the fluid, the equation of motion is obtained as follows.

We assume that within distances comparable with the depth
h these forces vary only by a small fraction of their total value.
On this understanding we have, in place of Art. 166 (1),

Pj=f = (9- Y)(y. + v-y) ................ (1),

and therefore

1 d di

The last term may be neglected for the reason just stated, and if

* It will be understood that the problem admits only of an approximate treat
ment, on account of the non-uniform character of the motion in the immediate
neighbourhood of the point of discontinuity. The degree of approximation implied
in the above assumptions will become more evident if we suppose the suffixes to
refer to two sections /Sj and S2, one on each side of the origin 0, at distances from
0 which, though very small compared with the wave-length, are yet moderate
multiples of the transverse dimensions of the canal. The motion of the fluid will
be sensibly uniform over each of these sections, and parallel to the length. The
conditions in the text then express that there is no sensible change of level between
S1 and S2.

282 TIDAL WAVES. [CHAP. VIII

we further neglect the product of the small quantities F and
drj/dx, the equation reduces to

1 dp _ drj
~

_
~pdx~9dx

as before. The equation of horizontal motion then takes the
form

—

where X may be regarded as a function of x and t only. The
equation of continuity has the same form as in Art. 166, viz.

Hence, on elimination of ?;,

175. The oscillations of water in a canal of uniform section,
closed at both ends, may, as in the corresponding problem of
Acoustics, be obtained by superposition of progressive waves
travelling in opposite directions. It is more instructive, however,
with a view to subsequent more difficult investigations, to treat
the problem as an example of the general theory sketched in
Art. 165.

We have to determine (• so as to satisfy

together with the terminal conditions that f = 0 for # = 0 and
x — I, say. To find the free oscillations we put X = 0, and assume
that

f X COS (art + e),

where o- is to be found. On substitution we obtain

whence, omitting the time-factor,

i . crx n ax
A sin -- h B cos — ,
c c

174-175] DISTURBING FORCES. 283

The terminal conditions give B = 0, and

a-ljc = STT (3),

where s is integral. Hence the normal mode of order s is

given by

A . STTX fsirct \ /ylv

f = 4,sm-j-cosf -j- + esj (4),

where the amplitude As and epoch e8 are arbitrary.

In the slowest oscillation (s = 1), the water sways to and fro,
heaping itself up alternately at the two ends, and there is a node
at the middle (x = J I). The period (2Z/c) is equal to the time a
progressive wave would take to traverse twice the length of the
canal.

The periods of the higher modes are respectively £, J, J, • ••
of this, but it must be remembered, in this and in other similar
problems, that our theory ceases to be applicable when the length
l/s of a semi-undulation becomes comparable with the depth h.

On comparison with the general theory of Art. 165, it appears that the
normal coordinates of the present system are quantities qlt q2, ... such that
when the system is displaced according to any one of them, say qa) we have

. STTX

£•3,001 -j-;

and we infer that the most general displacement of which the system is
capable (subject to the conditions presupposed) is given by

where qlt qz, ... are arbitrary. This is in accordance with Fourier's Theorem.

When expressed in terms of the normal velocities and the normal co
ordinates, the expressions for T and V must reduce to sums of squares.
This is easily verified, in the present case, from the formula (i). Thus if S
denote the sectional area of the canal, we find

(ii),

and 2F=#p^ [*

* J o

where aa = i

284 TIDAL WAVES. [CHAP. VIII

It is to be noted that the coefficients of stability (c8) increase with the
depth.

Conversely, if we assume from Fourier's theorem that (i) is a sufficiently
general expression for the value of £ at any instant, the calculation just
indicated shews that the coefficients qa are the normal coordinates ; and the
frequencies can then be found from the general formula (8) of Art. 165 ; viz.
we have

ov, = (c./a.)* =
in agreement with (3).

176. As aD example of forced waves we take the case of a
uniform longitudinal force

X=fcos(o-t + 6) ........................ (5).

This will illustrate, to a certain extent, the generation of tides in
a land-locked sea of small dimensions. Assuming that f varies as
cos(cr£ + e), and omitting the time-factor, the equation (1) becomes

^+-% = _/
dx^ c2 * c2'

the solution of which is

f T. . orx T-. (TX
= -*-+DSUL — + #cos— ................. (6).

(72 C C

The terminal conditions give

Hence, unless sin crlfc = 0, we have D =f/&2 . tan cr//2c, so that
2/ ,.:„ aso „;_ <r(l — a)

<7*

and

hf o-(2a:-0 ' ......

i] = . sin - -^ - - . cos (crt + e)

If the period of the disturbing force be large compared with
that of the slowest free mode, <rl/2c will be small, and the formula
for the elevation becomes

(9),
approximately, exactly as if the water were devoid of inertia. The

175-176] WAVES IN A FINITE CANAL. 285

horizontal displacement of the water is always in the same
phase with the force, so long as the period is greater than
that of the slowest free mode, or <r//2c <-^TT. If the period be
diminished until it is less than the above value, the phase is
reversed.

When the period is exactly equal to that of a free mode of
odd order (s=l, 3, 5,...), the above expressions for f and rj
become infinite, and the solution fails. As pointed out in
Art. 165, the interpretation of this is that, in the absence of
dissipative forces (such as viscosity), the amplitude of the motion
becomes so great that the foregoing approximations are no longer
justified.

If, on the other hand, the period coincide with that of a free
mode of even order (s = 2, 4, 6,...), we have sin<rZ/c = 0,
cos0-Z/c = l, and the terminal conditions are satisfied indepen
dently of the value of D. The forced motion may then be
represented by

£=--^sin2^cos(o-£ + e) ............ (10)*.

(T £C

The above example is simpler than many of its class in that it is possible
to solve it without resolving the impressed force into its normal components.
If we wish to effect this resolution, we must calculate the work done during
an arbitrary displacement

.

Since X denotes the force on unit mass, we have

=pS I

J o

whence

. , , ~ 1 — cos Srr 07 ..

provided Ca= — — . pSlf

This vanishes, as we should expect, for all even values of s. The solution of
our problem then follows from the general formulae of Art. 165. The
identification (by Fourier's Theorem) of the result thus obtained with that
contained in the formulae (8) is left to the reader.

* In the language of the general theory, the impressed force has here no
component of the particular type with which it synchronizes, so that a vibration
of this type is not excited at all. In the same way a periodic pressure applied at
any point of a stretched string will not excite any fundamental mode which has a
node there, even though it synchronize with it.

TIDAL WAVES. [CHAP. VIII

Another very simple case of forced oscillations, of some interest
in connection with tidal theory, is that of a canal closed at one end
and communicating at the other with an open sea in which a
periodic oscillation

7] = a cos (vt + e) (11)

is maintained. If the origin be taken at the closed end, the
solution is obviously

cos —

s*

97 = a , . cos (at + e) (12),

cos —

c

I denoting the length. If o-l/c be small the tide has sensibly the
same amplitude at all points of the canal. For particular values
of I, (determined by cos o-l/c = 0), the solution fails through the
amplitude becoming infinite.

Canal Theory of the Tides.

177. The theory of forced oscillations in canals, or on open
sheets of water, owes most of its interest to its bearing on the
phenomena of the tides. The 'canal theory,' in particular, has
been treated very fully by Airy*. We will consider one or two of
the more interesting problems.

The calculation of the disturbing effect of a distant body on
the waters of the ocean is placed for convenience in an Appendix
at the end of this Chapter. It appears that the disturbing effect
of the moon, for example, at a point P of the earth's surface, may
be represented by a potential O whose approximate value is

(i-ewa) (i),

j^r

where M denotes the mass of the moon, D its distance from
the earth's centre, a the earth's radius, 7 the ' constant of gravi
tation,' and Sr the moon's zenith distance at the place P. This
gives a horizontal acceleration d£l/ad^, or

/sin 2^ (2),

* Encycl. Metrop., "Tides and Waves," Section vi. (1845). Several of the
leading features of the theory had been made out, by very simple methods, by
Young in 1813 and 1823 ; Miscellaneous Works, t. ii. pp. 262, 291,

176-178] CANAL THEORY OF THE TIDES. 287

towards the point of the earth's surface which is vertically beneath

the moon, provided

(3).

If E be the earth's mass, we may write g = yE/a2, whence
f 3 M

Putting M/E = £, a/D = ^, this gives f\g = 8'57 x 10~8. When
the sun is the disturbing body, the corresponding ratio is

It is convenient, for some purposes, to introduce a linear
magnitude H, defined by

If we put a = 21 x 106 feet, this gives, for the lunar tide, H = 1-80 ft.,
and for the solar tide H = '79 ft. It is shewn in the Appendix
that H measures the maximum range of the tide, from high water
to low water, on the ' equilibrium theory.'

178. Take now the case of a uniform canal coincident with
the earth's equator, and let us suppose for simplicity that the
moon describes a circular orbit in the same plane. Let f be the
displacement, relative to the earth's surface, of a particle of water
whose mean position is at a distance a?, measured eastwards, from
some fixed meridian. If n be the angular velocity of the earth's
rotation, the actual displacement of the particle at time t will
be f + nt, so that the tangential acceleration will be d^g/dt2. If we
suppose the 'centrifugal force' to be as usual allowed for in the
value of g, the processes of Arts. 166, 174 will apply without
further alteration.

If n' denote the angular velocity of the moon relative to the
fixed meridian*, we may write

*b = n't + as/a + e,
so that the equation of motion is

The free oscillations are determined by the consideration that f
is necessarily a periodic function of x, its value recurring whenever

That is, n' = n-nlt if Wj be the angular velocity of the moon in her orbit.

288 TIDAL WAVES. [CHAP. VIII

x increases by 2?ra. It may therefore be expressed, by Fourier's
Theorem, in the form

(2).

Substituting in (1), with the last term omitted, it is found that Ps
and Qs must satisfy the equation

The motion, in any normal mode, is therefore simple-harmonic, of
period 2?ra/5C.

For the forced waves, or tides, we find

c^H ( x \

whence y = »— - r^cos2 [n'fH --- he ............... (5).

2 c2 — n 2a2 V a J

The tide is therefore semi-diurnal (the lunar day being of course
understood), and is ' direct' or 'inverted,' i.e. there is high or low

water beneath the moon, according as c ria, in other words

according as the velocity, relative to the earth's surface, of a point
which moves so as to be always vertically beneath the moon, is
less or greater than that of a free wave. In the actual case of the
earth we have

(?/n'*a* = (gln'*a) . (h/a) = 311 h/a,

so that unless the depth of the canal were to greatly exceed such
depths as actually occur in the ocean, the tides would be inverted.

This result, which is sometimes felt as a paradox, comes under
a general principle referred to in Art. 165. It is a consequence
of the comparative slowness of the free oscillations in an equatorial
canal of moderate depth. It appears from the rough numerical
table on p. 274 that with a depth of 11250 feet a free wave would
take about 30 hours to describe the earth's semi-circumference,
whereas the period of the tidal disturbing force is only a little
over 12 hours.

The formula (5) is, in fact, a particular case of Art. 165 (12), for
it may be written

178-179] TIDE IN EQUATORIAL CANAL. 289

where rj is the elevation given by the ' equilibrium-theory/ viz.

2 \ a J

and a = 2rit <TO = 2c/a.

For such moderate depths as 10000 feet and under, n'2a2 is large
compared with gh; the amplitude of the horizontal motion, as
given by (4), is then //4/i/2, or g/4tn'2a . H, nearly, being approxi
mately independent of the depth. In the case of the lunar
tide this is equal to about 140 feet. The maximum elevation is
obtained from this by multiplying by 2 h/a ; this gives, for a depth
of 10000 feet, a height of only 133 of a foot.

For greater depths the tides would be higher, but still inverted,
until we reach the critical depth n'2a2/g, which is about 13 miles.
For depths beyond this limit, the tides become direct, and
approximate more and more to the value given by the equi
librium theory*.

179. The case of a circular canal parallel to the equator can
be worked out in a similar manner. If the moon's orbit be still
supposed to lie in the plane of the equator, we find by
spherical trigonometry

/ x \

cos S- = sin 8 cos [rit + — r—. + e (1),

V a sm 6 )

where 8 is the co-latitude, and x denotes the distance of any point
P of the canal from the zero meridian. This leads to

X = - j- • = -/sin 8 sin 2 (n't + X a + e) . . .(2),
dx \ asin.0 J

and thence to

cos 2

(n't + — ^7,+ e) ...... (3).

V a sin 6 )

2 c2 - n'2a2 sin2 8

Hence if ria > c the tide will be direct or inverted according as
8 ^ sin"1 c/n'a. If the depth be so great that c > ria, the tides
will be direct for all values of 8.

If the moon be not in the plane of the equator, but have a co -declination
A, the formula (1) is replaced by

cos ^ = cos 6 cos A + sin 6 sin A cos a (i),

* Cf. Young, I. c. ante p. 270.
L. 19

290 TIDAL WAVES. [CHAP. VIII

where a is the hour-angle of the moon from the meridian of P. For
simplicity, we will neglect the moon's motion in her orbit in comparison with
the earth's angular velocity of rotation (n) ; thus we put

„ ,

asm 6

and treat A as constant. The resulting expression for the component X of
the disturbing force is found to be

x- -si-

We thence obtain

. 0 .sin 20 sin 2 A cos

sin2 ^ sin2 A cos 2 (nt+

5-9 - 9 9 • o ,, . ..

z c2 - n2a? sin2 6 \ a sin 6

The first term gives a ' diurnal ' tide of period ^irjn ; this vanishes and
changes sign when the moon crosses the equator, i.e. twice a month. The
second term represents a semidiurnal tide of period TT/U, whose amplitude is
now less than before in the ratio of sin2 A to 1.

180. In the case of a canal coincident with a meridian we
should have to take account of the fact that the undisturbed
figure of the free surface is one of relative equilibrium under
gravity and centrifugal force, and is therefore not exactly circular.
We shall have occasion later on to treat the question of displace
ments relative to a rotating globe somewhat carefully ; for the
present we will assume by anticipation that in a narrow canal the
disturbances are sensibly the same as if the earth were at rest,
and the disturbing body were to revolve round it with the proper
relative motion.

If the moon be supposed to move in the plane of the equator,
the hour-angle from the meridian of the canal may be denoted by
n't + e, and if as be the distance of any point P on the canal from
the equator, we find

CT

cos ^ = cos - . cos (n't -f e) .................. (1).

Hence

X = — j- = - /sin 2 - . cos2 (n't -f e)
dx a

= --J/sm2-.{l+cos2(X£-f e)} ............ (2).

a

179-181] CANAL COINCIDENT WITH MERIDIAN. 291

Substituting in the equation (5) of Art. 174, and solving, we find

?; = jffcos2- + j f ,, cos2-.cos2(X£ + 6) .. .. (3).
4 a 4 c2-n 2a2 a

The first term represents a change of mean level to the extent

(4).

The fluctuations above and below the disturbed mean level
are given by the second term in (3). This represents a semi
diurnal tide ; and we notice that if, as in the actual case of the
earth, c be less than n'a, there will be high water in latitudes
above 45°, and low water in latitudes below 45°, when the moon
is in the meridian of the canal, and vice versa when the moon
is 90° from that meridian. These circumstances would be all
reversed if c were greater than na.

When the moon is not on the equator, but has a given declination,
the mean level, as indicated by the term corresponding to (4), has a coefficient
depending on the declination, and the consequent variations in it indicate a
fortnightly (or, in the case of the sun, a semi-annual) tide. There is also
introduced a diurnal tide whose sign depends on the declination. The reader
will have no difficulty in examining these points, by means of the general
value of ii given in the Appendix.

Wave-Motion in a Canal of Variable Section.

181. When the section (S, say) of the canal is not uniform,
but varies gradually from point to point, the equation of con
tinuity is, as in Art. 166 (iv),

where b denotes the breadth at the surface. If h denote the
mean depth over the width b, we have S = bh, and therefore

where h, b are now functions of x.

The dynamical equation has the same form as before, viz.

#£_ a^i

dP~ ~gdx ........................... (

19—2

292 TIDAL WAVES. [CHAP. VIII

Between (2) and (3) we may eliminate either rj or f ; the
equation in 77 is

d dr\

'l

The laws of propagation of waves in a rectangular canal of
gradually varying section were investigated by Green*. His
results, freed from the restriction to a special form of section,
may be obtained as follows.

If we introduce a variable 6 defined by

dxld6 = (gKp .................................... (i),

in place of #, the equation (4) transforms into

I/

where the accents denote differentiations with respect to 0. If b and h were
constants, the equation would be satisfied by rj — F(B-t), as in Art. 167; in
the present case we assume, for trial,

1 = B.F(0-t) .................................... (iii),

where 0 is a function of B only. Substituting in (ii), we find

The terms of this which involve F will cancel provided

or e = Cb~*k- ....................................... (v),

C being a constant. Hence, provided the remaining terms in (iv) may be
neglected, the equation (i) will be satisfied by (iii) and (v).

The above approximation is justified, provided we can neglect 0"/0' and
0'/0 in comparison with F'/F. As regards Q'/Q, it appears from (v) and
(iii) that this is equivalent to neglecting b~l . dbjdx and hrl . dhjdx in com
parison with r)~l.drj/dx. If, now, A denote a wave-length, in the general
sense of Art. 169, drj/dx is of the order rj/X, so that the assumption in
question is that \dbjdx and \dhjdx are small compared with b and h, re
spectively. In other words, it is assumed that the transverse dimensions of
the canal vary only by small fractions of themselves within the limits of a
wave-length. It is easily seen, in like manner, that the neglect of 0"/0' in
comparison with F'/F implies a similar limitation to the rates of change of
db\dx and dhjdx.

* "On the Motion of Waves in a Variable Canal of small depth and width."
Camb. Trans., i. vi. (1837), Math. Papers, p. 225; see also Airy, " Tides and Waves,"
Art. 260.

181] CANAL OF VARIABLE SECTION. 293

Since the equation (4) is unaltered when we reverse the sign of t, the
complete solution, subject to the above restrictions, is

1 = b-*h-*{F(0-t)+f(6 + t)} (vi),

when F and / are arbitrary functions.

The first term in this represents a wave travelling in the direction of
^-positive; the velocity of propagation is determined by the consideration
that any particular phase is recovered when dd and dt have equal values, and
is therefore equal to (#A)*, by (i), exactly as in the case of a uniform section.
In like manner the second term in (vi) represents a wave travelling in the
direction of ^'-negative. In each case the elevation of any particular part of
the wave alters, as it proceeds, according to the law b~^ h~^.

The reflection of a progressive wave at a point where the
section of a canal suddenly changes has been considered in Art.
173. The formulae there given shew, as we should expect, that
the smaller the change in the dimensions of the section, the
smaller will be the amplitude of the reflected wave. The case
where the transition from one section to the other is continuous,
instead of abrupt, comes under a general investigation of Lord
Rayleigh's*. It appears that if the space within which the
transition is completed be a moderate multiple of a wave-length
there is practically no reflection ; whilst in the opposite extreme
the results agree with those of Art. 173.

If we assume, on the basis of these results, that when the
change of section within a wave-length may be neglected a pro
gressive wave suffers no disintegration by reflection, the law of
amplitude easily follows from the principle of energy -f-. It
appears from Art. 17 1 that the energy of the wave varies as the
length, the breadth, and the square of the height, and it is easily
seen that the length of the wave, in different parts of the canal,
varies as the correspondiug velocity of propagation J, and therefore
as the square root of the mean depth. Hence, in the above notation,
?72&M is constant, or

t] oc b~^h~*,
which is Green's law above found.

* " On Reflection of Vibrations at the Confines of two Media between which the
Transition is gradual," Proc. Land. Math. Soc., t. xi. p. 51 (1880) ; Theory of Sound,
2nd ed., London, 1894, Art. 1486.

t Lord Rayleigh, 1. c. ante p. 279.

t For if P, Q be any two points of a wave, and P', Q' the corresponding points
when it has reached another part of the canal, the time from P to P' is the same
as from Q to Q', and therefore the time from P to Q is equal to that from P' to Q'.
Hence the distances PQ, P*Q' are proportional to the corresponding wave- velocities.

294 TIDAL WAVES. [CHAP. VIII

182. In the case of simple harmonic motion, assuming that
?; x cos (at + e), the equation (4) of the preceding Art. becomes

(1).

Some particular cases of considerable interest can be solved
with ease.

1°. For example, let us take the case of a canal whose breadth varies as
the distance from the end #=0, the depth being uniform ; and let us suppose
that at its mouth (x = a) the canal communicates with an open sea in which
a tidal oscillation

77 = acos(<r£ + e) ................................. (i),

is maintained. Putting h = const., 6oc#, in (1), we find

provided k2 = o-2/gh .................................... (iii).

The solution of (ii) which is finite when #=0 is

or, in the notation of Bessel's Functions,

t=AJQ(kx) .................................... (v).

Hence the solution of our problem is evidently

The curve y—JQ(x} is figured on p. 306; it indicates how the amplitude of
the forced oscillation increases, whilst the wave length is practically constant,
as we proceed up the canal from the mouth.

2°. Let us suppose that the variation is in the depth only, and that this
increases uniformly from the end #=0 of the canal, to the mouth, the remain
ing circumstances being as before. If, in (1), we put h=h(p/at K = <r2a/gh0,
we obtain

This solution of this which is finite for #=0 is

KX K%2

or i=*AJQ(2Ka) .................................... (ix),

whence finally, restoring the time-factor and determining the constant,

182]

WAVE-MOTION IN CONTRACTING CHANNELS.

295

The annexed diagram of the curve y = «70(V#), where, for clearness, the
scale adopted for y is 200 times that of #, shews how the amplitude continually
increases, and the wave-length diminishes, as we travel up the canal.

These examples may serve to illustrate the exaggeration of oceanic tides
which takes place in shallow seas arid in estuaries.

We add one or two simple problems of free oscillations.

3°. Let us take the case of a canal of uniform breadth, of length 2a, whose
bed slopes uniformly from either end to the middle. If we take the origin at
one end, the motion in the first half of the canal will be determined, as
above, by

(xi),

where /c = o-%/^A0, as before, h0 denoting the depth at the middle.

It is evident that the normal modes will fall into two classes. In the first
of these 77 will have opposite values at corresponding points of the two halves
of the canal, and will therefore vanish at the centre (x = a}. The values of a-
are then determined by

J0(2ja*) = 0 ................................. (xii),

viz. K being any root of this, we have

.(xiii).

In the second class, the value of 77 is symmetrical with respect to the
centre, so that drj/dx=Q at the middle. This gives

(xiv).

Some account of Bessel's Functions will be given presently, in connection
with another problem. It appears that the slowest oscillation is of the

296 TIDAL WAVES. [CHAP. VIII

asymmetrical class, and so corresponds to the smallest root of (xii), which
is 2x^a* = '765577, whence

_ 1.308

4°. Again, let us suppose that the depth of the canal varies according to
the law

where x now denotes the distance from the middle. Substituting in (1), with
b = const., we find

d (/, x2\ dn] o-2

, .N
-7----^ -r -^ ........................ (xvi).

ds ^'

If we put a'2 = n(n + l) ................................ (xvii),

this is of the same form as the general equation of zonal harmonics,
Art. 85 (1).

In the present problem n is determined by the condition that £ must be
finite for x/a= +1. This requires (Art. 86) that n should be integral; the
normal modes are therefore of the types

(xviii),

where Pn is a Legendre's Function, the value of a- being determined
by (xvii).

In the slowest oscillation (n=l\ the profile of the free surface is a
straight line. For a canal of uniform depth A0, and of the same length (2a),
the corresponding value of <r would be 7rc/2a, where c = (gh$. Hence in the
present case the frequency is less, in the ratio 2^/2/vr, or "9003.

The forced oscillations due to a uniform disturbing force

JT=/cos (<rt + e) ................................. (xix),

can be obtained by the rule of Art. 165 (12). The equilibrium form of the
free surface is evidently

(xx),

and, since the given force is of the normal type ?i=l, we have

where o-02 = 2grA0/a8.

182-183] SPECIAL PROBLEMS. 297

Waves of Finite Amplitude.

183. When the elevation rj is not small compared with the
mean depth h, waves, even in an uniform canal of rectangular
section, are no longer propagated without change of type. This
question was first investigated by Airy*, by methods of successive
approximation. He found that in a progressive wave different
parts will travel with different velocities, the wave-velocity corre
sponding to an elevation 77 being given approximately by

where c is the velocity corresponding to an infinitely small
amplitude.

A more complete view of the matter can be obtained by the
method employed by Riemann in treating the analogous problem
in Acoustics, to which reference will be made in Chapter x.

The sole assumption on which we are now proceeding is that
the vertical acceleration may be neglected. It follows, as ex
plained in Art. 166, that the horizontal velocity may be taken to
be uniform over any section of the canal. The dynamical equation
is

du du drj , .

*+tt^=-^ ....................... <*>•

as before, and the equation of continuity, in the case of a rect
angular section, is easily seen to be

where h is the depth. This may be written

drj drj du , .

—' '-+u—L=:-(h + <ri)— ..................... (3).

dt dx " dx

Let us now write

2P =/(,) + «, 2Q=/(,)-u ............... (4),

where the function f(rj) is as yet at our disposal. If we multiply
(3) ty/ 0?)» and add to (1), we get
dP dP

" Tides and Waves," Art. 198.

298 TIDAL WAVES. [CHAP. VIII

If we now determine /(?;) so that

(h + r,)f'W = g ........................ (5),

this may be written

dP dP

In the same way we find

The condition (5) is satisfied by

(8),

where c = (gh)*. The arbitrary constant has been chosen so as to
make P and Q vanish in the parts of the canal which are free from
disturbance, but this is not essential.

Substituting in (6) and (7) we find

-u|<# I

dQ
dx

It appears, therefore, that dP = 0, i. e. P is constant, for a
geometrical point moving with the velocity

whilst Q is constant for a point moving with the velocity

••••-•

Hence any given value of P travels forwards, and any given value
of Q travels backwards, with the velocities given by (10) and (11)
respectively. The values of P and Q are determined by those
of 1) and u, and conversely.

As an example, let us suppose that the initial disturbance
is confined to the space between x = a and x = b, so that P and Q
are initially zero for x < a and ec > b. The region within which P

183-184] WAVES OF FINITE AMPLITUDE. 299

differs from zero therefore advances, whilst that within which Q
differs from zero recedes, so that after a time these regions
separate, and leave between them a space within which P = 0,
•Q = 0, and the fluid is therefore at rest. The original disturbance
has now been resolved into two progressive waves travelling in
opposite directions.

In the advancing wave we have

(12),

so that the elevation and the particle- velocity are connected by a
definite relation (cf. Art. 168). The wave- velocity is given by (10)
and (12), viz. it is

To the first order of 7j/h, this is in agreement with Airy's result.

Similar conclusions can be drawn in regard to the receding
wave*.

Since the wave-velocity increases with the elevation, it appears
that in a progressive wave- system the slopes will become con
tinually steeper in front, and more gradual behind, until at length
a state of things is reached in which we are no longer justified in
neglecting the vertical acceleration. As to what happens after
this point we have at present no guide from theory ; observation
shews, however, that the crests tend ultimately to curl over and
' break.'

184. In the application of the equations (1) and (3) to tidal
phenomena, it is most convenient to follow the method of successive
approximation. As an example, we will take the case of a canal
communicating at one end (% = 0) with an open sea, where
the elevation is given by

rj = a cos crt
For a first approximation we have

du dn dn 1 du

-

* The above results can also be deduced from the equation (3) of Art. 170, to
which Biemann's method can be readily adapted.

300 TIDAL WAVES. [CHAP. VIII

the solution of which, consistent with (14), is

rj = a cos or (*--), M = — coso-U--) ................... (ii).

\ GJ C \ C/

For a second approximation we substitute these values of ?/ and u in (1) and'

(3), and obtain

du drt

dn 7 du go-a2 . ... / x\

:r = ~ h -j- ~ *-*- sm 2o" ( * ~ ~ )
dt dx c2 \ CJ

(iii).

Integrating these by the usual methods, we find, as the solution consistent
with (14),

= a cos 0-

/ x\
U--J —

. a
--3- x sin 2o-

A #\ \
It — ) ,

x sin 2o-

H)

.(iv).

The figure shews, with, of course, exaggerated amplitude, the profile of
the waves in a particular case, as determined by the first of these equations.
It is to be rioted that if we fix our attention on a particular point of the canal,
the rise and fall of the water do not take place symmetrically, the fall
occupying a longer time than the rise.

When analysed, as in (iv), into a series of simple -harmonic functions of
the time, the expression for the elevation of the water at any particular place
(#) consists of two terms, of which the second represents an ' over-tide,' or
' tide of the second order,' being proportional to a2 ; its frequency is double
that of the primary disturbance (14). If we were to continue the approxi
mation we should obtain tides of higher orders, whose frequencies are
3, 4, ... times that of the primary.

If, in place of (14), the disturbance at the mouth of the canal were given

by

£ = a cos <rt + a' cos (a-'t + e),

it is easily seen that in the second approximation we should obtain tides of
periods 2w/(<r + <r') and 2ir/(cr— <r'); these are called 'compound tides.' They
are closely analogous to the 'combination-tones' in Acoustics which were
first investigated by von Helmholtz*.

* "Ueber Combinationstone," Berl. Monatsber., May 22, 1856, Ges. Abh., t. i.,
p. 256, and " Theorie der Luftschwingungen in Kohren mit offenen Enden,"
Crelle, t. Ivii., p. 14 (1859), Ges. Abh., t. i., p. 318.

184-185] TIDES OF SECOND ORDER. 301

The occurrence of the factor x outside trigonometrical terms in (iv) shews
that there is a limit beyond which the approximation breaks down. The
condition for the success of the approximation is evidently that go-axfc*
should be small. Putting c2=gh, \ = <2irc/<r, this fraction becomes equal to
2;r (a/A) . (x/\). Hence however small the ratio of the original elevation (a)
to the depth, the fraction ceases to be small when as is a sufficient multiple of
the wave-length (X).

It is to be noticed that the limit here indicated is already being over
stepped in the right-hand portions of the figure above given ; and that
the peculiar features which are beginning to shew themselves on the rear
slope are an indication rather of the imperfections of the analysis than
of any actual property of the waves. If we were to trace the curve further,
we should find a secondary maximum and minimum of elevation developing
themselves on the rear slope. In this way Airy attempted to explain the
phenomenon of a double high-water which is observed -in some rivers ; but,
for the reason given, the argument cannot be sustained*.

The same difficulty does not necessarily present itself in the case of a canal
closed by a fixed barrier at a distance from the mouth, or, again, in the case
of the forced waves due to a periodic horizontal force in a canal closed at both
ends (Art. 176). Enough has, however, been given to shew the general
character of the results to be expected in such cases. For further details
we must refer to Airy's treatise f.

Propagation in Two Dimensions.

185. Let us suppose, in the first instance, that we have a
plane sheet of water of uniform depth h. If the vertical accelera
tion be neglected, the horizontal motion will as before be the same
for all particles in the same vertical line. The axes of x, y being
horizontal, let u, v be the component horizontal velocities at the
point (as, y), and let ? be the corresponding elevation of the free
surface above the undisturbed level. The equation of continuity
may be obtained by calculating the flux of matter into the
columnar space which stands on the elementary rectangle
viz. we have, neglecting terms of the second order,

a£<«%)8a + ^t

, dt , fdu dv\

whence ji=-h hj- + j- (1).

at \dx dy)

* Cf. McCowan, I, c. ante p. 277.

t " Tides and Waves," Arts. 198, ...and 308. See also G. H. Darwin, "Tides,
Encyc. Britann. (9th ed.) t. xxiii., pp. 362, 363 (1888).

302 TIDAL WAVES. [CHAP. VIII

The dynamical equations are, in the absence of disturbing

forces,

du _ dp dv _ dp

p~dt~~dx' pdt~~dyy

where we may write

p-fb-jp (* + ?-*),

if zQ denote the ordinate of the free surface in the undisturbed
state, and so obtain

du _ d£ dv _ d£ . .

dt-~gdx' dt~~9dy"'

If we eliminate u and v, we find

where c2 = gh as before.

In the application to simple-harmonic motion, the equations
are shortened if we assume a 'complex' time-factor ei(<rt+e), and
reject, in the end, the imaginary parts of our expressions. This
is of course legitimate, so long as we have to deal solely with
linear equations. We have then, from (2),

_ ig d% _ ig d£ ...

<j dx ' (7 dy

whilst (3) becomes

where A;2 = a2/c2 ........................... (6).

The condition to be satisfied at a vertical bounding wall is
obtained at once from (4), viz. it is

£=« .............................. ^

if &n denote an element of the normal to the boundary.

When the fluid is subject to small disturbing forces whose
variation within the limits of the depth may be neglected, the
equations (2) are replaced by

du _ d£ <ttl dv_ d£_<m
dt~~9dx~ dx' dt~ 9 dy dy ......... '

where H is the potential of these forces.

185-186] WAVES ON AN OPEN SHEET OF WATER. 303

If we put f=-n/$r ........................... (9),

so that f denotes the equilibrium-elevation corresponding to the
potential H, these may be written

du d /c, £X dv d /4, Sv

In the case of simple-harmonic motion, these take the forms

whence, substituting in the equation of continuity (1), we obtain
(V1^ + ^)? = V1f ...................... (12),

if Vx2 = d?/da? + cP/dy*, .................... (13),

and & = a*/gh, as before. The condition to be satisfied at a
vertical boundary is now

186. The equation (3) of Art. 185 is identical in form with
that which presents itself in the theory of the vibrations of a
uniformly stretched membrane. A still closer analogy, when
regard is had to the boundary conditions, is furnished by the theory
of cylindrical waves of sound*. Indeed many of the results
obtained in this latter theory can be at once transferred to our
present subject.

Thus, to find the free oscillations of a sheet of water bounded
by vertical walls, we require a solution of

<V,' + *•)£=<> ........................ (1),

subject to the boundary condition

d£/dn = 0 ........................... (2).

Just as in Art. 175 it will be found that such a solution is possible
only for certain values of k, and thus the periods (2ir/kc) of the
several normal modes are determined.

* Lord Eayleigh, Theory of Sound, Art. 339.

304 TIDAL WAVES. [CHAP. VIII

Thus, in the case of a rectangular boundary, if we take the
origin at one corner, and the axes of x} y along two of the sides,
the boundary conditions are that d^jdx = 0 for x = 0 and x = a,
and dQdy = 0 for y = 0 and y = 6, where a, b are the lengths of the
edges parallel to x, y respectively. The general value of f subject
to these conditions is given by the double Fourier's series

^v A mirx ^

?= 22.4m, „ COS—- COS-y^ .................. (3),

where the summations include all integral values of m, n from 0
to oo . Substituting in (1) we find

& = 'ir*(m*/a* + n*/li') ..................... (4).

If a > b, the component oscillation of longest period is got by
making m = l, n = 0, whence ka = 7r. The motion is then every
where parallel to the longer side of the rectangle. Cf. Art. 175.

187. In the case of a circular sheet of water, it is convenient
to take the origin at the centre, and to transform to polar
coordinates, writing

x = r cos 6, y = r sin 6.

The equation (1) of the preceding Art. becomes

.................. .

dr* r dr r* dd*
This might of course have been established independently.

As regards its dependance on 6, the value of f may, by
Fourier's Theorem, be supposed expanded in a series of cosines and
sines of multiples of 6 ; we thus obtain a series of terms of the
form

siri

(2).

It is found on substitution in (1) that each of these terms must
satisfy the equation independently, and that

This is the differential equation of Bessel's Functions*. Its

* Forsyth, Differential Equations, Art. 100.

186-187] CIRCULAR BASIN. 305

complete primitive consists, of course, of the sum of two definite
functions of rt each multiplied by an arbitrary constant, but in the
present problem we are restricted to a solution which shall be finite
at the origin. This is easily obtained in the form of an ascending
series ; thus, in the ordinary notation of Bessel's Functions, we have

where, on the usual convention as to the numerical factor,

7" M - 2* 1 1

s(Z ~2S77!(

Hence the various normal modes are given by

(5),
sin

where s may have any of the values 0, 1, 2, 3,..., and As is an
arbitrary constant. The admissible values of k are determined by
the condition that d£/dr = 0 for r = a, or

J8'(ka) = 0 ........................... (6).

The corresponding ' speeds' (<r) of the oscillations are then given

The analytical theory of Bessel's Functions is treated more or less fully in
various works to which reference is made below*. It appears that for large
values of 0 we may put

, -2-

where

I2 -4s2 _ (!2-4s2)(32-4s2)(52-4s2)
1.80 " 1.2.3(80)3

A

...(ii).

The series P, Q are of the kind known as ' semi-convergent,' i.e. although for
large values of 0 the successive terms may diminish for a time, they ultimately
increase again, but if we stop at a small term we get an approximately
correct result.

* Lommel, Studien ueber die BesseVschen Functionen, Leipzig, 1868 ; Heine,
Kugelfunctionen, Arts. 42,..., 57,...; Todhunter, Functions of Laplace, Lame, and
Bessel; Byerly, On Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal
Harmonics, Boston, U. S.A., 1893 ; see also Forsyth, Differential Equations, c. v.
An ample account of the matter, from the physical point of view, will be found in
Lord Rayleigh's Theory of Sound, cc. ix., xviii., with many interesting applications.

Numerical tables of the functions have been calculated by Bessel, and Hansen,
and (more recently) by Meissel (Berl. Alh., 1888). Hansen's tables are reproduced
by Lommel, and (partially) by Lord Rayleigh and Byerly.

L. 20

306

TIDAL WAVES.

[CHAP, vin

187] BESSEL'S FUNCTIONS. 307

It appears from (i) that Ja(z) belongs to the class of 'fluctuating
functions,' viz. as z increases the value of the function oscillates on both
sides of zero with a continually diminishing amplitude. The period of
the oscillations is ultimately 2jr.

The general march of the functions is illustrated to some extent by the
curves y = J0 (#), y = J\ (®\ which are figured on the opposite page. For the
sake of clearness, the scale of the ordinates has been taken five times as great
as that of the abscissae.

In the case s = 0, the motion is symmetrical about the origin,
so that the waves have annular ridges and furrows. The lowest
roots of

are given by

Jfca/7r = r2197, 2-2330, 3'2383, (8)*,

these values tending ultimately to the form &a/7r = ra + J, where
m is integral. In the rath mode of the symmetrical class there
are ra nodal circles whose radii are given by f =0 or

The roots of this are

kr/7r = '7655, 17571, 27546, (10)f.

For example, in the first symmetrical mode there is one nodal
circle r = '628a. The form of the section of the free surface by
a plane through the axis of z, in any of these modes, will be
understood from the drawing of the curve y — Jo (#).

When s > 0 there are s equidistant nodal diameters, in addition
to the nodal circles

J,(kr) = 0 (11).

It is to be noticed that, owing to the equality of the frequencies of
the two modes represented by (5), the normal modes are now to a
certain extent indeterminate ; viz. in place of cos sO or sin s6 we
might substitute coss(<9-as), where OLS is arbitrary. The nodal
diameters are then given by

Q — n — TT f19\

a* 2^ l**A

* Stokes, " On the Numerical Calculation of a class of Definite Integrals and
Infinite Series," Camb. Trans, t. ix. (1850), Math, and Phys. Papers, t. ii. p. 355.

It is to be noticed that ka/Tr is equal to r0/r, where T is the actual period, and TO
is the time a progressive wave would take to travel with the velocity (grfe)J over a
space equal to the diameter 2a.

t Stokes, I. c.

20—2

308

TIDAL WAVES.

[CHAP, viii

where m = 0, 1, 2,..., 5—1. The indeterminateness disappears,
and the frequencies become unequal, if the boundary deviate,
however slightly, from the circular form.

In the case of the circular boundary, we obtain by super
position of two fundamental modes of the same period, in different
phases, a solution

S=C,Js(kr).co8(<rt+86 + e) (13).

This represents a system of waves travelling unchanged round
the origin with an angular velocity a/s in the positive or
negative direction of 6. The motion of the individual particles
is easily seen from Art. 185 (4) to be elliptic-harmonic, one
principal axis of each elliptic orbit being along the radius
vector. All this is in accordance with the general theory referred
to in Art. 165.

The most interesting modes of the unsym metrical class are
those corresponding to 5 = 1, e.g.

f = AJ1(kr)cos0.cos(<rt+e) (14),

187]

where k is determined by

The roots of this are

NODAL LINES.

309

J1'(ka)=0 (15).

&a/7r=-586, 1-697, 2-717, (16)*.

We have now one nodal diameter (6 = JTT), whose position is,
however, indeterminate, since the origin of 6 is arbitrary. In the
corresponding modes for an elliptic boundary, the nodal diameter
would be fixed, viz. it would coincide with either the major or
the minor axis, and the frequencies would be unequal.

The accompanying diagrams shew the contour-lines of the free
surface in the first two modes of the present species. These lines
meet the boundary at right angles, in conformity with the general
boundary condition (Art. 186 (2)). The simple-harmonic vibrations

/ / 1

I I *

of the individual particles take place in straight lines perpen-
licular to the contour-lines, by Art. 185 (4). The form of the

* See Lord Kayleigh's treatise, Art. 339,

310 TIDAL WAVES. [CHAP. VIII

sections of the free surface by planes through the axis of z is given
by the curve y — Jl (x) on p. 306.

The first of the two modes here figured has the longest period
of all the normal types. In it, the water sways from side to side,
much as in the slowest mode of a canal closed at both ends
(Art. 175). In the second mode there is a nodal circle, whose
radius is given by the lowest root of Jj (kr) = 0 ; this makes

A comparison of the preceding investigation with the general theory of
small oscillations referred to in Art. 165 leads to several important properties
of Bessel's Functions.

In the first place, since the total mass of water is unaltered, we must have

/;

where £ has any one of the forms given by (5). For s>0 this is satisfied in
virtue of the trigonometrical factor cos sQ or sin s6 ; in the symmetrical case
it gives

tn,

0 (iv).

r
/«

Again, since the most general free motion of the system can be obtained
by superposition of the normal modes, each with an arbitrary amplitude and
epoch, it follows that any value whatever of £, which is subject to the
condition (iii), can be expanded in a series of the form

r) ..................... (v),

where the summations embrace all integral values of s (including 0) and, for
each value of s, all the roots k of (6). If the coefficients As, Bs be regarded
as functions of £, the equation (v) rnay be regarded as giving the value of the
surface-elevation at any instant. The quantities A8, Bs are then the normal
coordinates of the present system (Art. 165) ; and in terms of them the
formulae for the kinetic and potential energies must reduce to sums of
squares. Taking, for example, the potential energy

* The oscillations of a liquid in a circular basin of any uniform depth were
discussed by Poissoii, " Sur les petites oscillations de 1'eau contenue dans un
cylindre," Ann. de Gergonne, t. xix. p. 225 (1828-9); the theory of Bessel's
Functions had not at that date been worked out, and the results were consequently
not interpreted. The full solution of the problem, with numerical details, was
given independently by Lord Eayleigh, Phil. Mag., April, 1876.

The investigation in the text is limited, of course, to the case of a depth small
in comparison with the radius a. Poisson's and Lord Kayleigh's solution for the
case of finite depth will be noticed in the proper place in Chap. ix.

187-188] PROPERTIES OF BESSElAs FUNCTIONS. 311

this requires that II wlwzrd6dr = Q (vii),

where w1, ivz are any two terms of the expansion (v). If wlt w2 involve
cosines or sines of different multiples of 0, this is verified at once by integra
tions with respect to 6 ; but if we take

w1 oc Ja (^r) cos s0, w2 oc Js (k2r) cos s#,
where &15 &2 are any two distinct roots of (6), we get

, ---• =0 (viii)-

The general results of which (iv) and (viii) are particular cases, are

/JQ(kr}rdr= -jJ0'(ka) (ix),
o *

and

Jo'8(l^ 8 tf rc -kz_kz 2« 8 2a s 1« 1« 8 1« 2a

(x).

In the case of kl = k2 the latter expression becomes indeterminate ; the
evaluation in the usual manner gives

o ' 8*»'

For the analytical proofs of these formulae we must refer to the treatises cited
on p. 305.

The small oscillations of an annular sheet of water bounded by
concentric circles are easily treated, theoretically, with the help of
Bessel's Functions ' of the second kind.' The only case of any
special interest, however, is when the two radii are nearly equal ;
we then have practically a re-entrant canal, and the solution
follows more simply from Art. 178.

The analysis can also be applied to the case of a circular
sector of any angle*, or to a sheet of water bounded by two
concentric circular arcs and two radii.

188. As an example of forced vibrations, let us suppose that
the disturbing forces are such that the equilibrium elevation
would be

f = G cos sO . cos (at + e) (16).

See Lord Rayleigh, Theory of Sound, Art. 339.

312 TIDAL WAVES. [CHAP. VIII

This makes V^^O, so that the equation (12) of Art. 185
reduces to the form (1), above, and the solution is

£=AJ8 (kr) cos sd . cos (at + e) (17),

where A is an arbitrary constant. The boundary-condition (Art.
185 (14)), gives

AkaJs'(ka)=sC,

whence £= 0 ° ' Y, \ cos sO . cos (crt + e) (18).

KdJ s (fCd)

The case s = 1 is interesting as corresponding to a uniform
horizontal force ; and the result may be compared with that of
Art. 176.

From the case s = 2 we could obtain a rough representation of
the semi-diurnal tide in a polar basin bounded by a small circle
of latitude, except that the rotation of the earth is not as yet taken
into account.

We notice that the expression for the amplitude of oscillation
becomes infinite when J8 (ka) = 0. This is in accordance with a
general principle, of which we have already had several examples ;
the period of the disturbing force being now equal to that of one
of the free modes investigated in the preceding Art.

189. When the sheet of water is of variable depth, the
investigation at the beginning of Art. 185 gives, as the equation

of continuity,

d% _ d(hu) d(hv) .-.

di~ ~~dx~ ~dy~' '

The dynamical equations (Art. 185 (2)) are of course unaltered.
Hence, eliminating f, we find, for the free oscillations,

d /, d£\ d

If the time-factor be ei(fft+e), we obtain

dx \ dx) dy \ dy) g

When h is a function of r, the distance from the origin, only,
this may be written

188-189] BASIN OF VARIABLE DEPTH. 313

As a simple example we may take the case of a circular basin which
shelves gradually from the centre to the edge, according to the law

Introducing polar coordinates, and assuming that £ varies as cos sQ or sin sO,
the equation (4) takes the form

This can be integrated by series. Thus, assuming

,m

L»J

where the trigonometrical factors are omitted, for shortness, the relation
between consecutive coefficients is found to be

= m (m - 2) - ^ -

Lm — 2>

or, if we write — Tr-«*n(n— 2)— ^ (iv),

where 7& is not as yet assumed to be integral,

The equation is therefore satisfied by a series of the form (iii), beginning with
the term Aa(r/a)8, the succeeding coefficients being determined by putting
m=5 + 2, s + 4,... in (v). We thus find

(m-t-

8\al I 2(&? + 2) az

or in the usual notation of hypergeometric series

where a = -^7i + is, /3

Since these make y — a — /3 = 0, the series is not convergent for r = a, unless it
terminate. This can only happen when n is integral, of the form
The corresponding values of o- are then given by (iv).

In the symmetrical modes (5=0) we have

where j may be any integer greater than unity. It may be shewn that this
expression vanishes for^ - 1 values of r between 0 and a, indicating the exist
ence of j - I nodal circles. The value of or is given by

(«)•

314 TIDAL WAVES. [CHAP. VIII

Thus the gravest symmetrical mode (./ = 2) has a nodal circle of radius *707a \
and its frequency is determined by o-2 = tyk^a?.

Of the unsymmetrical modes, the slowest, for any given value of *, is that
for which n = s + 2, in which case we have

rs
£= A8 — cos s6 cos (a-t + e),

the value of o- being given by

.................................... (x).

The slowest mode of all is that for which s — l, n = 3; the free surface is
then always plane. It is found on comparison with Art. 187 (16) that the
frequency is '768 of that of the corresponding mode in a circular basin of
uniform depth A0, and of the same radius.

As in Art. 188 we could at once write down the formula for the tidal
motion produced by a uniform horizontal periodic force ; or, more generally,
for the case when the disturbing potential is of the type

Q oc rs cos s6 cos (o-t + e).

190. We proceed to consider the case of a spherical sheet, or
ocean, of water, covering a solid globe. We will suppose for the
present that the globe does not rotate, and we will also in the first
instance neglect the mutual attraction of the particles of the water.
The mathematical conditions of the question are then exactly the
same as in the acoustical problem of the vibrations of spherical
layers of air * .

Let a be the radius of the globe, h the depth of the fluid ; we
assume that h is small compared with a, but not (as yet) that it is
uniform. The position of any point on the sheet being specified
by the angular coordinates 6, co, let u be the component velocity of
the fluid at this point along the meridian, in the direction of 6
increasing, and v the component along the parallel of latitude, in
the direction of &> increasing. Also let ? denote the elevation of
the free surface above the undisturbed level. The horizontal
motion being assumed, for the reasons explained in Art. 169, to be
the same at all points in a vertical line, the condition of con
tinuity is

-fa (uha sin OBco) 80 + -y- (vha$0) Sco = — a sin OBco . aS0 . -£ ,
du cLa) at

where the left-hand side measures the flux out of the columnar

* Discussed in Lord Kayleigh's Theory of Sound, c. xvm.

189-191] SPHERICAL SHEET OF WATER. 315

space standing on the element of area a sin 6$co . a$6, whilst the
right-hand member expresses the rate of diminution of the volume
of the contained fluid, owing to fall of the surface. Hence

d£_ 1 _ ( d (hu sin 0) d(hv)}
dt~ a sin 0 I d6 da> }"

If we neglect terms of the second order in u, v, the dynamical
equations are, on the same principles as in Arts. 166, 185,

du d d£l dv d c?H ,.

dt~ add~ad6' dt ~ asinOda) ~ a sin 0da>' ' '
where U denotes the potential of the extraneous forces.
If we put

(3),

these may be written

du_ gd - dv_ g d
dt~~ad@(^~Qj K-^iam?*^ Vri

Between (1) and (4) we can eliminate u, v, and so obtain an equa
tion in f only.

In the case of simple-harmonic motion, the time-factor being
ei(<rt+e)} the equations take the forms

,,_ i (d(husin0) d(hv)
^~ ~

191. We will now consider more particularly the case of
uniform depth. To find the free oscillations we put ?=0; the
equations (5) arid (6) of the preceding Art. then lead to

1 d . dfr 1 d*

nx

This is identical in form with the general equation of spherical
surface-harmonics (Art. 84 (2)). Hence, if we put

o--a2/gh = n (n + 1) (2),

a solution of (1) will be

?=#„ (3),

where Sn is the general surface-harmonic of order n.

316 TIDAL WAVES. [CHAP. VIII

It was pointed out in Art. 87 that Sn will not be finite over
the whole sphere unless n be integral. Hence, for an ocean
covering the whole globe, the form of the free surface at any
instant is, in any fundamental mode, that of a ' harmonic spheroid '

r=a + h + Sncos((rt + €) .................. (4),

and the speed of the oscillation is given by

<r=[n(n + I)}*.(gh)*la ..................... (5),

the value of n being integral.

The characters of the various normal modes are best gathered
from a study of the nodal lines (Sn = 0) of the free surface. Thus,
it is shewn in treatises on Spherical Harmonics* that the zonal
harmonic Pn (/it) vanishes for n real and distinct values of p lying
between ± 1, so that in this case we have n nodal circles of
latitude. When n is odd one of these coincides with the equator.
In the case of the tesseral harmonic

sm

the second factor vanishes for n — s values of /z, and the trigono
metrical factor for 2s equidistant values of CD. The nodal lines
therefore consist of n — s parallels of latitude and 2s meridians.
Similarly the sectorial harmonic

sm
has as nodal lines 2n meridians.

These are, however, merely special cases, for since there are
2n+l independent surface-harmonics of any integral order n,
and since the frequency, determined by (5), is the same for each
of these, there is a corresponding degree of indeterminateness in
the normal modes, and in the configuration of the nodal lines *fv

We can also, by superposition, build up various types of
progressive waves; e.g. taking a sectorial harmonic we get a
solution in which

f oc(l - ffin cos (na> - at + e) ............... (6);

this gives a series of meridional ridges and furrows travelling

* For references, see p. 117.

f Some interesting varieties are figured in the plates to Maxwell's Electricity and
Magnetism, t. i.

191] NORMAL MODES OF A SPHERICAL SHEET. 317

round the globe, the velocity of propagation, as measured at the
equator, being

It is easily verified, on examination, that the orbits of the

particles are now ellipses having their principal axes in the

directions of the meridians and parallels, respectively. At the
equator these ellipses reduce to straight lines.

In the case n = I, the harmonic is always of the zonal type.
The harmonic spheroid (4) is then, to our order of approximation,
a sphere excentric to the globe. It is important to remark,
however, that this case is, strictly speaking, not included in our
dynamical investigation, unless we imagine a constraint applied to
the globe to keep it at rest ; for the deformation in question of
the free surface would involve a displacement of the centre of mass
of the ocean, and a consequent reaction on the globe. A corrected
theory for the case where the globe is free could easily be investi
gated, but the matfcer is hardly important, first because in such
a case as that of the Earth the inertia of the solid globe is so
enormous compared with that of the ocean, and secondly because
disturbing forces which can give rise to a deformation of the type
in question do not as a rule present themselves in nature. It
appears, for example, that the first term in the expression for the
tide-generating potential of the sun or moon is a spherical har
monic of the second order (see Appendix).

When n = 2, the free surface at any instant is approximately
ellipsoidal. The corresponding period, as found from (5), is then
'816 of that belonging to the analogous mode in an equatorial
canal (Art. 178).

For large values of n the distance from one nodal line to
another is small compared with the radius of the globe, and the
oscillations then take place much as on a plane sheet of water.
For example, the velocity, at the equator, of the sectorial waves
represented by (6) tends with increasing n to the value (gh)*, in
agreement with Art. 167.

From a comparison of the foregoing investigation with the general theory
of Art. 165 we are led to infer, on physical grounds alone, the possibility of
the expansion of any arbitrary value of £ in a series of surface harmonics, thus

318 TIDAL WAVES. [CHAP. VIII

the coefficients being the normal coordinates of the system. Again, since the
products of these coefficients must disappear from the expressions for the
kinetic and potential energies, we are led to the 'conjugate' properties of
spherical harmonics quoted in Art. 88. The actual calculation of the potential
and kinetic energies will be given in the next Chapter, in connection with an
independent treatment of the same problem.

The effect of a simple-harmonic disturbing force can be
written down at once from the formula (12) of Art. 165. If
the surface-value of fl be expanded in the form

............................. (8),

where £ln is a surface-harmonic of integral order n, then the
various terms are normal components of force, in the generalized
sense of Art. 133 ; and the equilibrium value of f corresponding
to any one term Hn is

?» = -£«. ............................ (9).

Hence, for the forced oscillation due to this term, we have

where <r measures the ' speed ' of the disturbing force, and <rn that
of the corresponding free oscillation, as given by (5). There is no
difficulty, of course, in deducing (10) directly from the equations
of the preceding Art.

192. We have up to this point neglected the mutual attrac
tion of the parts of the liquid. In the case of an ocean covering
the globe, and with such relations of density as we meet with in the
actual earth and ocean, this is not insensible. To investigate its
effect in the case of the free oscillations, we have only to sub
stitute for nn, in the last formula, the gravitation-potential of the
displaced water. If the density of this be denoted by p, whilst p0
represents the mean density of the globe and liquid combined, we
have*

................... II),

and

* See, for example, Eouth, Analytical Statics, Cambridge 1892, t. ii. pp. 91-92.

191-192] EFFECT OF GRAVITATION OF WATER. 319

7 denoting the gravitation-constant, whence

Substituting in (10) we find

where an is now used to denote the actual speed of the oscillation,
and crn' the speed calculated on the former hypothesis of no
mutual attraction. Hence the corrected speed is given by

For an ellipsoidal oscillation (n — 2), and for p/p0 = '18 (as in the
case of the Earth), we find from (14) that the effect of the mutual
attraction is to lower the frequency in the ratio of '94 to 1.

The slowest oscillation would correspond to n = l, but, as
already indicated, it would be necessary, in this mode, to imagine
a constraint applied to the globe to keep it at rest. This being
premised, it appears from (15) that if p>/?0the value of oy5 is
negative. The circular function of t is then replaced by real
exponentials; this shews that the configuration in which the
surface of the sea is a sphere concentric with the globe is one
of unstable equilibrium. Since the introduction of a constraint
tends in the direction of stability, we infer that when p> p0 the
equilibrium is a fortiori unstable when the globe is free. In
the extreme case when the globe itself is supposed to have no
gravitative power at all, it is obvious that the water, if disturbed,
would tend ultimately, under the influence of dissipative forces, to
collect itself into a spherical mass, the nucleus being expelled.

It is obvious from Art. 165, or it may easily be verified inde
pendently, that the forced vibrations due to a given periodic
disturbing force, when the gravitation of the water is taken into
account, will be given by the formula (10), provided Hn now
denote the potential of the extraneous forces only, and crn have
the value given by (15).

* This result was given by Laplace, Mecanique Celeste, Livre ler, Art. 1 (1799).
The free and the forced oscillations of the type n = 2 had been previously investi
gated in his " Eecherches sur quelques points du systeme du monde," Mem. de
I'Acad. roy. des Sciences, 1775 [1778]; Oeuvres Completes, t. ix., pp. 109,....

320 TIDAL WAVES. [CHAP. VIII

193. The oscillations of a sea bounded by meridians, or
parallels of latitude, or both, can also be treated by the same
method*. The spherical harmonics involved are however, as a
rule, no longer of integral order, and it is accordingly difficult to
deduce numerical results.

In the case of a zonal sea bounded by two parallels of latitude, we assume

* (i),

where /z = cos#, and p(p), q(fi) are the two functions of //, containing
(1— /n2)*s as a factor, which are given by the formula (2) of Art. 87. It will
be noticed that p (/z) is an even, and q (/z) an odd function of /*.

If we distinguish the limiting parallels by suffixes, the boundary conditions
are that u=0 for /M = /XI and/u = fi2. For the free oscillations this gives, by
Art. 190 (6),

3 (ii),

0 (iii),

whence

which is the equation to determine the admissible values of n. The speeds
(o-) corresponding to the various roots are given as before by Art. 191 (5).

If the two boundaries are equidistant from the equator, we have fi2= — ^
The above solutions then break up into two groups ; viz. for one of these we
have

5 = 0, p'M = 0 ................................ (v),

and for the other

0 .............................. (vi).

In the former case £ has the same value at two points symmetrically
situated on opposite sides of the equation ; in the latter the values at these
points are numerically equal, but opposite in sign.

If we imagine one of the boundaries to be contracted to a point (say
/z2 = 1), we pass to the case of a circular basin. The values of p' (1) and q' (1)
are infinite, but their ratio can be evaluated by means of formulae given in
Art. 85. This gives, by (iii), the ratio A : B, and substituting in (ii) we get
the equation to determine n. A more interesting method of treating this
case consists, however, in obtaining, directly from the differential equation of
surface-harmonics, a solution which shall be finite at the pole /i = l. This
involves a change of variable, as to which there is some latitude of choice.
Perhaps the simplest plan is to write, for a moment,

2 = J(l-/x) = sin2^ .............................. (vii).

* Ct'. Lord Eayleigh, I. c. ante p. 314.

193] CASE OF A POLAR SEA. 321

Assuming SH=(l -/**)*' w°^J 5o> ........................... (viii),

the differential equation in wt which is given in Art. 87, becomes, in terms of
the new variable,

=0 ...... (ix).

The solution of this which is finite for 2 = 0 is given by the ascending series

) ................................................... (x).

Hence the expression adapted to our case is

2) 1 cosj

"•J«nJ*

1.2. (5 + 1) (5 + 2)

where the admissible values of n are to be determined from the condition
that d£/d0=Q for 0 = 0V

The actual calculation of the roots of the equation in n, for any arbitrary
value of 0ly would be difficult. The main interest of the investigation consists,
in fact, in the transition to the plane problem of Art. 187, and in the connection
which we can thus trace between Bessel's Functions and Spherical Harmonics.
If we put a = oo , a0 = r, we get the case of a plane sheet of water, referred to
polar coordinates r, «o. Making, in addition, n0 = kr, so that n is now infinite,
the formula (xi) gives

( fc2r2 £4r4 1 cos

2(25 + 2) 2.4(25 + 2)(2s + 4)
or {ccJt(Jkr)c.s\8a> (xii),

in the notation of Art. 187 (4). We thus obtain Bessel's Functions as limiting
forms of Spherical Harmonics of infinite order t.

* When n (as well as s) is integral, the series terminates, and the expression
differs only by a numerical factor from the tesseral harmonic denoted by

T* ^ sin I SW' in Art< 87' In the case s = 0 we obtain one of the expansions of the
zonal harmonic given by Murphy, Elementary Principles of the Theories of Electri
city..., Cambridge, 1833, p. 7. (The investigation is reproduced by Thomson and
Tait, Art. 782.)

t This connection appears to have been first explicitly noticed by Mehler,
"Ueber die Vertheilung der statischen Elektricitat in einem von zwei Kugelkalotten
begrenzten Korper," Crelle, t. Ixviii. (1868). It was investigated independently by
Lord Kayleigh, " On the Kelation between the Functions of Laplace and Bessel,"
Proc. Lond. Math. Soc., t. ix., p. 61 (1878) ; see also the same author's Theory of
Sound, Arts. 336, 338.

L, 21

322 TIDAL WAVES. [CHAP. VIII

If the sheet of water considered have as boundaries two
meridians (with or without parallels of latitude), say o> = 0 and
co = a, the condition that v = 0 at these restricts us to the factor
cos 56), and gives so. — mir, where m is integral. This determines
the admissible values of s, which are not in general integral *.

Tidal Oscillations of a Rotating Sheet of Water.

194. The theory of the tides on an open sheet of water is
seriously complicated by the fact of the earth's rotation. If,
indeed, we could assume that the periods of the free oscillations,
and of the disturbing forces, were small compared with a day, the
preceding investigations would apply as a first approximation,
but these conditions are far from being fulfilled in the actual
circumstances of the Earth.

The difficulties which arise when we attempt to take the
rotation into account have their origin in this, that a particle
having a motion in latitude tends to keep its angular momentum
about the earth's axis unchanged, and so to alter its motion in
longitude. This point is of course familiar in connection with
Hadley's theory of the trade- winds *f*. Its bearing on tidal theory
seems to have been first recognised by MaclaurinJ.

195. Owing to the enormous inertia of the solid body of the
earth compared with that of the ocean, the effect of tidal reactions
in producing periodic changes of the angular velocity is quite
insensible. This angular velocity will therefore for the present be
treated as constant §.

The theory of the small oscillations of a dynamical system
about a state of equilibrium relative to a solid body which rotates
with constant angular velocity about a fixed axis differs in some
important particulars from the theory of small oscillations about
a state of absolute equilibrium, of which some account was given

* The reader who wishes to carry the study of the problem further in this
direction is referred to Thomson and Tait, Natural Philosophy (2nd ed.), Appendix
B, " Spherical Harmonic Analysis."

t " Concerning the General Cause of the Trade Winds," Phil. Trans. 1735.

J De Causa Physicd Fluxus et Eefluxus Maris, Prop. vii. : " Motus aquas turbatur
ex inaequali velocitate qua corpora circa axem Teme motu diurno deferuntur" (1740).

§ The secular effect of tidal friction in this respect will be noticed later (Chap.

XI.).

193-195] MOTION RELATIVE TO A ROTATING SOLID. 323

in Art. 165. It is therefore worth while to devote a little space
to it before entering on the consideration of special problems.

Let us take a set of rectangular axes x, y, z> fixed relatively to
the solid, of which the axis of z coincides with the axis of rotation,
and let n be the angular velocity of the rotation. The equations
of motion of a particle m relative to these moving axes are known
to be

m(x — 2m/ — n^x) = X, ^

m(y + 2nx-n2y)=Y, I .................. (1),

mz =Z }

where X, F, Z are the impressed forces on the particle. Let us
now suppose that the relative coordinates (x, y, z) of any particle
can be expressed in terms of a certain number of independent quan
tities ql} ^2,.... If we multiply the above equations by dxjdqS)
dy/dqs, dz/dqs, and add, and denote by S a summation embracing
all the particles of the system, we obtain

/ dx ..dy ..dz\ ^ ^ f . dy . dx
\x^- + y-r- + z -3- } + 2n2<m(x-f- -y -,-
\ dqs ydqs dqsJ \ dqs y dqs

= mtf + f + + + -...

dqs \ dqs dqs dqj

There is a similar equation for each of the generalized coordinates
qs-

Now, exactly as in Hamilton's proof* of Lagrange's equations,
the first term in (2) may be replaced by

dt dqs dqs '
where ^ = iSm(^ + ^2 + ^2) .................. (3),

i.e. ® denotes the energy of the relative motion, supposed expressed
in terms of the generalized coordinates qS) and the generalized
velocities qs. Again, we may write

+ Y + z_ ......... (4)

dqs dqs dqsj dqs
where Fis the potential energy, and Qs is the generalized com-

* See ante p. 201 (footnote).

21—2

324 TIDAL WAVES. [CHAP. VIII

ponent of extraneous force corresponding to the coordinate qt.

Also, since

dx . dx .

X = -j— (ft + j— (72 + • . .,

efyi dq^
dy . dy .

*-S*+i*^-

we have

^ /. dy . dx\ ^ (d(x,y) . d(x,y) . }

Sm (x -/- -y-r- = 2mJ j/ \gi + j/ xga + '-r-

V % <y%/ (<* (&>?«) d(q2,qs)* j

We will write, for shortness,

Finally, we put

rf«in*Sm(«Mvy*) ..................... (6),

viz. T0 denotes the energy of the system when rotating with the
solid, without relative motion, in the configuration (q1} q.2) ...).

With these notations, the typical equation (2) takes the form

and it is to be particularly noticed that the coefficients [r, s~\ are
subject to the relations

[r,s] = -|>,r], [s,s] = 0 .................. (8).

The conditions for relative equilibrium, in the absence of ex
traneous forces, are found by putting ^ = 0, g2 = 0, ... in (7), or more
simply from (2). In either way we obtain

which shews that the equilibrium value of the expression V- T0is
' stationary.'

196. We will now suppose the coordinates qs to be chosen
so as to vanish in the undisturbed state. In the case of a small
disturbance, we may then write

. + 2al2q,q2 + ...... (1),

.+ 2c1,qlq2+ ...... (2),

* Of. Thomson and Tait, Natural Philosophy (2nd ed.), Part i. p. 319. It
should be remarked that these equations are a particular case of Art. 139 (14),
obtained, with the help of the relations (7) of Art. 141, by supposing the rotating
solid to be free, but to have an infinite moment of inertia.

195-197] GENERAL EQUATIONS. 325

where the coefficients may be treated as constants. The terms of
the first degree in V — TQ have been omitted, on account of the
'stationary' property.

In order to simplify the equations as much as possible, we will
further suppose that, by a linear transformation, each of these
expressions is reduced, as in Art. 165, to a sum of squares; viz.

22T = a1£12 + a2g22+ (3),

2(F-ro)= ciqi*+c2q*+ (4).

The quantities qlt q2) ... may be called the 'principal coordinates'
of the system, but we must be on our guard against assuming that
the same simplicity of properties attaches to them as in the case
of no rotation. The coefficients a1? a2j... and clf c2,... may be
called the ' principal coefficients ' of inertia and of stability, respec
tively. The latter coefficients are the same as if we were to
ignore the rotation, and to introduce fictitious ' centrifugal ' forces
(mnzx, mnzy, 0) acting on each particle in the direction outwards
from the axis.

If we further write, for convenience, /3rs in place of [r, s], then,
in terms of the new coordinates, the equation (7) of the preceding
Art. gives, in the case of infinitely small motions,

If we multiply these equations by qlt q2, ... in order, and add,
then taking account of the relation

Prs =—Psr (6),

wefmd jt(^+V-T,) = qiql + Q4, + (7).

This might have been obtained without approximation from the
exact equations (7) of Art. 195. It may also be deduced directly
from first principles.

197. To investigate the free motions of the system, we put
Qi = ®> Q2= 0, ... in (5), and assume, in accordance with the usual
method of treating linear equations,

ft = 41^, q* = AteV,..., (8).

TIDAL WAVES. [CHAP. VIII

Substituting, we find

Eliminating the ratios Al : A2 : J3 : ..., we get the equation

A, &A, •••

, &.X, ...

/332X,

= 0 (10),

or, as we shall occasionally write it, for shortness,

A(X) = 0 (11).

The determinant A (X) comes under the class called by Cay ley
'skew-determinants,' in virtue of the relation (6). If we re
verse the sign of X, the rows and columns are simply interchanged,
and the value of the determinant therefore unaltered. Hence,
when expanded, the equation (10) will involve only even powers of
X, and the roots will be in pairs of the form

* = ± (p + ia).

In order that the configuration of relative equilibrium should
be stable it is essential that the values of p should all be zero,
for otherwise terms of the forms e±pt cos at and e±ptsinat would
present themselves in the realized expression for any coordinate
qs. This would indicate the possibility of an oscillation of
continually increasing amplitude.

In the theory of absolute equilibrium, sketched in Art. 165,
the necessary and sufficient condition of stability is simply that
the potential energy must be a minimum in the configuration of
equilibrium. In the present case the conditions are more com
plicated*, but we may readily shew that if the expression for
V— TO be essentially positive, in other words if the coefficients
GI, C2, ... in (4) be all positive, the equilibrium will be stable.
This follows at once from the equation (7), which gives, in the
case of free motion,

® + (F- ro) = const (12),

* They have been investigated by Kouth, On the Stability of a Given State of
Motion ; see also his Advanced Rigid Dynamics (4th ed,), London, 1884.

197] CONDITION OF SECULAR STABILITY. 327

shewing that under the present supposition neither ?& nor V— T0
can increase beyond a certain limit depending on the initial
circumstances.

Hence stability is assured if V— T0 is a minimum in the
configuration of relative equilibrium. But this condition is not
essential, and there may even be stability with V— TQ a maximum,
as will be shewn presently in the particular case of two degrees of
freedom. It is to be remarked, however, that if the system be
subject to dissipative forces, however slight, affecting the relative
coordinates qlt q%, ..., the equilibrium will be permanently or
'secularly' stable only if V— T0 is a minimum. It is the
characteristic of such forces that the work done by them on
the system is always negative. Hence, by (7), the expression
f& + (V— T0) will, so long as there is any relative motion of the
system, continually diminish, in the algebraical sense. Hence if
the system be started from relative rest in a configuration such that
V— TO is negative, the above expression, and therefore a fortiori
the part V— T0, will assume continually increasing negative
values, which can only take place by the system deviating more
and more from its equilibrium-configuration.

This important distinction between ' ordinary ' or kinetic, and
' secular ' or practical stability was first pointed out by Thomson
and Tait*. It is to be observed that the above investigation pre
supposes a constant angular velocity (n) maintained, if necessary, by
a proper application of force to the rotating solid. When the solid
is free, the condition of secular stability takes a somewhat different
form, to be referred to later (Chap. XII.).

To examine the character of a free oscillation, in the case
of stability, we remark that if \ be any root of (10), the equations
(9) give '

where An, A,-,, Ar3, ... are the minors of any row in the determi
nant A, and G is arbitrary. It is to be noticed that these minors
will as a rule involve odd as well as even powers of X, and so

* Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincare, " Sur
1'equilibre d'une masse fluide animee d'un mouvement de rotation," Acta Mathe*
matica, t. vii. (1885),

328 TIDAL WAVES. [CHAP. VIII

assume unequal values for the two oppositely signed roots (± X,) of
any pair. If we put X = + ia, the general symbolical value of qs
corresponding to any such pair of roots may be written

qs = C&r, (itr) e*
If we put

we get a solution of our equations in real form, involving two
arbitrary constants K, e ; thus

ql = FI (a2) . K cos (fft + e)- afj_ (a2) . K sin (fft + e),

qz = F2 (o-2) . K cos (trt + e) - af, (cr2) . K sin (at + e),

q3 = F3 (o-2) . K cos (<rf + e) - 0/3 (o-2) . # sin (at + e), " '( '

The formulae (14) express what may be called a 'natural
mode ' of oscillation of the system. The number of such possible
modes is of course equal to the number of pairs of roots of (10),
i.e. to the number of degrees of freedom of the system.

If £, 77, f denote the component displacements of any particle
from its equilibrium position, we have

c. _dx dx
dql ^ dqz ^2

dq^1 ' /7/».*a ' -' (15>-

= i — <7i +
dqi

Substituting from (15), we obtain a result of the form

£ = P . K cos (at + e) + P' • K sin (at + e), 1
tl=Q.Kcoa(fft + e)+Q'.KBm(fft + e), \ (16),

where P, P', Q, Q', R, Rf are determinate functions of the mean
position of the particle, involving also the value of a, and there
fore different for the different normal modes, but independent of
the arbitrary constants K, e. These formulae represent an elliptic-

* We might have obtained the same result by assuming, in (5),

a — A fi(fft + e) n - A „*(<* + *) „ - A /»*'(<r*+e)

2l — Al6 ' #2 — A2e > <l3 — A3e J ...... '

where A1, A%, A3, ... are real, and rejecting, in the end, the imaginary parts.

197-198] FREE AND FORCED OSCILLATIONS. 329

harmonic motion of period 2-Tr/cr ; the directions

tlP=<n/Q = yR, and SIP'=<nlQf=SIR (if),

being those of two semi-conjugate diameters of the elliptic orbit,
of lengths (P2 + Q2 + Ri2 . K, and (P/2 + Q'2 + R'rf . Kt respectively.
The positions and forms and relative dimensions of the elliptic
orbits, as well as the relative phases of the particles in them,
are in each natural mode determinate, the absolute dimensions
and epochs being alone arbitrary.

198. The symbolical expressions for the forced oscillations
due to a periodic disturbing force can easily be written down. If
we assume that Q1; Q2> ••• all varj as &"*, where a- is prescribed,
the equations (5) give, omitting the time-factors,

.(18).

The most important point of contrast with the theory of the
' normal modes ' in the case of no rotation is that the displacement
of any one type is no longer affected solely by the disturbing force
of that type. As a consequence, the motions of the individual
particles are, as is easily seen from (15), now in general elliptic-
harmonic.

As in Art. 165, the displacement becomes very great when
A (ia) is very small, i. e. whenever the ' speed ' a- of the disturbing
force approximates to that of one of the natural modes of free
oscillation.

When the period of the disturbing forces is infinitely long, the
displacements tend to the ' equilibrium-values '

as is found by putting cr = 0 in (18), or more simply from the
fundamental equations (5). This conclusion must be modified,
however, when any one or more of the coefficients of stability
Cj, c2, ... is zero. If, for example, ^ = 0, the first row and
column of the determinant A (X) are both divisible by X, so

330 TIDAL WAVES. [CHAP. VIII

that the deter minantal equation (10) has a pair of zero roots.
In other words we have a possible free motion of infinitely
long period. The coefficients of Q2) Qs, >•• on the right-hand side
of (18) then become indeterminate for cr = 0, and the evaluated
results do not as a rule coincide with (19). This point is of some
importance, because in the hydrodynamical applications, as we
shall see, steady circulatory motions of the fluid, with a constant
deformation of the free surface, are possible when no extraneous
forces act; and as a consequence forced tidal oscillations of long
period do not necessarily approximate to the values given by the
equilibrium theory of the tides. Cf. Arts. 207, 210.

In order to elucidate the foregoing statements we may consider more in
detail the case of two degrees of freedom. The equations of motion are then
of the forms

The equation determining the periods of the free oscillations is

or «1a2X4 + (o^Cg + «2ci + ft2) X2

For ' ordinary ' stability it is sufficient that the roots of this quadratic in X2
should be real and negative. Since «15 «2 are essentially positive, it is easily
seen that this condition is in any case fulfilled if cx , c2 are both positive, and
that it will also be satisfied even when c15 c2 are both negative, provided /32 be
sufficiently great. It will be shewn later, however, that in the latter case the
equilibrium is rendered unstable by the introduction of dissipative forces.

To find the forced oscillations when Qlt Q2 vary as ei<Tt, we have, omitting
the time-factor,

whence

(Ct-o^Q
» fa -*%*,) fa -

(v)

Let us now suppose that c2=0, or, in other words, that the displacement
q2 does not affect the value of V— TQ. We will also suppose that Q2 = Q> *'•#.
that the extraneous forces do no work during a displacement of the type q2.
The above formulae then give

198-199] TWO DEGREES OF FREEDOM. 331

In the case of a disturbance of long period we have or = 0, approximately, and
therefore

The displacement ql is therefore less than its equilibrium-value, in the
ratio 1 : l+p2/^^; and it is accompanied by a motion of the type q2
although there is no extraneous force of the latter type (cf. Art. 210).
We pass, of course, to the case of absolute equilibrium, considered in
Art. 165, by putting /3 = 0.

199. Proceeding to the hydrodynamical examples, we begin
with the case of a plane horizontal sheet of water having in the
undisturbed state a motion of uniform rotation about a vertical
axis*. The results will apply without serious qualification to
the case of a polar or other basin, of not too great dimensions, on
a rotating globe.

Let the axis of rotation be taken as axis of z. The axes of x
and y being now supposed to rotate in their own plane with the
prescribed angular velocity n, let us denote by u, v, w the
velocities at time t, relative to these axes, of the particle which
then occupies the position (x, y, z).

The actual velocities of the same particle, parallel to the in
stantaneous positions of the axes, will be u — ny, v + nx, w.

After a time $t, the particle in question will occupy, relatively
to the axes, the position (x + u$t, y + v$t, z + wSt), and therefore
the values of its actual component velocities parallel to the new
positions of the axes will be

~m & + n (x

* Sir W. Thomson, "On Gravitational Oscillations of Rotating Water," Phil*
., Aug. 1880.

332 TIDAL WAVES. [CHAP. VIII

where

D d , d t d ( d „'

tfC^TT-l-* j-+tfj- + 10 3~ ............... (1),

Ite dt dx dy dz

as usual. These are in directions making with the original axis
of x angles whose cosines are 1, — nSt, 0, respectively, so that the
velocity parallel to this axis at time t + St is

u + -yji $t — n (y + vSt) — (v + nx) nSt.

Hence, and by similar reasoning, we obtain, for the component
accelerations in space, the expressions

- 2m, - n'x, Dt + 2nU-tfy, ......... (2)*.

In the present application, the relative motion is assumed
to be infinitely small, so that we may replace DjDt by d/dt.

200. Now let ZQ be the ordinate of the free surface when
there is relative equilibrium under gravity alone, so that

^0 = i-(«3 + ya)+ const ................... (3),

as in Art. 27. For simplicity we will suppose that the slope of
this surface is everywhere very small, in other words, if r be the
greatest distance of any part of the sheet from the axis of rotation,
n2r/g is assumed to be small.

If ZQ + f denote the ordinate of the free surface when disturbed,
then on the usual assumption that the vertical acceleration of the
water is small compared with g, the pressure at any point (x, y, z)
will be given by

+£-z) ..................... (4),

, I dp d£ I dp d£

whence — -f- = — n*x — g -j- , — j=~ n y — 9 j~ •
p dx y dx p dy 9 dy

The equations of horizontal motion are therefore
du 9 d£ d£l

dt y dx dx '

dv « d£ dQ

dt dy dy

where H denotes the potential of the disturbing forces.

* These are obviously equivalent to the expressions for the component accelera
tions which appear on the left-hand sides of Art. 195 (1).

199-200] ROTATING SHEET OF WATER. 333

If we write ? = -*% .............................. (6)>

these become

The equation of continuity has the same form as in Art. 189
viz.

£ 4(fe) 4(ftv)
cft~ cte <fy

where ^ denotes the depth, from the free surface to the bottom, in
the undisturbed condition. This depth will not, of course, be
uniform unless the bottom follows the curvature of the free
surface as given by (3).

If we eliminate £ — £ from the equations (7), by cross-differentiation,
we find

d

or, writing u — d^dt^ v = drj/dt,

and integrating with respect to t,

dv du fd£ d\

- — + 2n l-f+ — )=const (ii).

dx dy \dx dy) ^ '

This is merely the expression of von Helmholtz' theorem that the product of
the angular velocity

. (dv du\

rc + i -y-- ^-1

z \dx dy)

and the cross-section

of a vortex-filament, is constant.

In the case of a simple-harmonic disturbance, the time-factor
being e™*, the equations (7) and (8) become

-,

and fV?=--- ...... (10).

dx dy

334 TIDAL WAVES. [CHAP. VIII

From (9) we find

d
• + Zw^-J(C-?)» |

(11),

(T — 'm

and if we substitute from these in (10), we obtain an equation in
f only.

In the case of uniform depth the result takes the form

Vi2£ + ^^2f=Vi2? (12),

where Vj2 = d?lda? + d~/dy2, as before.

When £=0, the equations (7) and (8) can be satisfied by constant values of
u, v, £ provided certain conditions are satisfied. We must have

and therefore 7=° ................................. (iv)-

d(x,y}

The latter condition shews that the contour-lines of the free surface must be
everywhere parallel to the contour-lines of the bottom, but that the value of
£ is otherwise arbitrary. The flow of the fluid is everywhere parallel to the
contour-lines, and it is therefore further necessary for the possibility of such
steady motions that the depth should be uniform along the boundary (sup
posed to be a vertical wall). When the depth is everywhere the same, the
condition (iv) is satisfied identically, and the only limitation on the value of
£ is that it should be constant along the boundary.

201. A simple application of these equations is to the case of
free waves in an infinitely long uniform straight canal *.

If we assume f=ae<k^ct-^+myt v = 0 ..................... (1),

the axis of x being parallel to the length of the canal, the equa
tions (7) of the preceding Art., with the terms in f omitted, give

cu = g%, 2nu = — gm% ..................... (2),

whilst, from the equation of continuity (Art. 200 (8)),

c£=hu .............................. (3).

* Sir W. Thomson, I.e. ante p. 331,

200-202] STRAIGHT CANAL. 335

We thence derive

c2 = ght m — — 2n/c ..................... (4).

The former of these results shews that the wave-velocity is
unaffected by the rotation.

When expressed in real form, the value of f is

%=ae-™y!ccos{k(ct-x)+e} ............... (5).

The exponential factor indicates that the wave-height increases as
we pass from one side of the canal to the other, being least on the
side which is forward, in respect of the rotation. If we take
account of the directions of motion of a water-particle, at a crest
and at a trough, respectively, this result is easily seen to be in
accordance with the tendency pointed out in Art. 194*.

The problem of determining the free oscillations in a rotating
canal of finite length, or in a rotating rectangular sheet of water,
has not yet been solved.

202. We take next the case of a circular sheet of water
rotating about its centre f.

If we introduce polar coordinates r, 6, and employ the symbols
R, © to denote displacements along and perpendicular to the
radius vector, then since R = iaR, © = iV®, the equations (9) of
Art. 200 are equivalent to

whilst the equation of continuity (10) becomes

d(hRr) d(hQ) .

rdr rd6

TT r>

Hence R —

9

—

and substituting in (2) we get the differential equation in f.

* For applications to tidal phenomena see Sir W. Thomson, Nature, t. xix.
pp. 154, 571 (1879).

t The investigation which follows is a development of some indications given
by Lord Kelvin in the paper referred to.

336 TIDAL WAVES. [CHAP. VIII

In the case of uniform depth, we find

d? 1 d 1 d?
where Vf = ^2 + - ^ + - ^ .................. (5),

and K = (<72 - 4m*)/gh ....................... (6).

This might have been written down at once from Art. 200 (12).

The condition to be satisfied at the boundary (r = a, say)
is R = 0, or

203. In the case of the free oscillations we have f= 0. The way
in which the imaginary i enters into the above equations, taken
in conjunction with Fourier's theorem, suggests that 6 occurs
in the form of a factor eis&, where s is integral. On this supposi
tion, the differential equation (4) becomes

dr2 r dr
and the boundary-condition (7) gives

for r = a.

The equation (8) is of Bessel's form, except that K is not, in
the present problem, necessarily positive. The solution which is
finite for r = 0 may be written

r = 4/8(«,r) ........................... (10),

where

1"

According as K is positive or negative, this differs only by a
numerical factor from Js(fc^r) or Is(/c'*r), where K is written
for — K, and Is (z) denotes the function obtained by making all
the signs + on the right-hand side of Art. 187 (4)*.

* The functions I,(z) have been tabulated by Prof. A. Lodge, Brit. Ass. Eep.
1889.

202-203] FREE OSCILLATIONS. 337

In the case of symmetry about the axis (s = 0), we have, in

real form,

/c*r).cos(o-£ + e) .................. (12),

where K is determined by

JoV)=0 ........................... (13).

The corresponding values of a are then given by (6). The
free surface has, in the various modes, the same forms as in
Art. 187, but the frequencies are now greater, viz. we have

o-2 = <r02 + 4^2 ........................ (14),

where <r0 is the corresponding value of cr when there is no
rotation. It is easily seen, however, on reference to (3), that the
relative motions of the fluid particles are no longer purely radial ;
the particles describe, in fact, ellipses whose major axes are in the
direction of the radius vector.

When s > 0 we have

?=4/i(«,r).cos((r$ + a0+€) ............... (15),

where the admissible values of K, and thence of <r, are determined
by (9), which gives

«a5/.(*,«) + ^/.(*,a) = 0 ............ (16).

The formula (15) represents a wave rotating relatively to the
water with an angular velocity a/s, the rotation of the wave being
in the same direction with that of the water, or the opposite,
according as a/n is negative or positive.

Some indications as to the values of <r may be gathered from a graphical
construction. If we put Ka2=#, we have, from (6),

«r/2»= ±(1 +*/£)* .............................. (i),

where /3 = 4n*a?lgh .................................... (ii).

It is easily seen that the quotient of

«/•(*» «) b7 a^/«(K'a)

is a function of /ca2, or #, only. Denoting this function by <£ (a?), the equation
(16) may be written

f(*)±(V+*/^-0 .............................. (iii).

The curve y= -<£(#) .................................... (*v)

L. 22

338

TIDAL WAVES.

[CHAP, viii

can be readily traced by means of the tables of the functions Js (0), Ia (z\ and
its intersections with the parabola

3/2=l+*//3 (v),

will give, by their ordinates, the values of <r/2»i. The constant /3, on which
the positions of the roots depend, is equal to the square of the ratio
2na/(ffh)t which the period of a wave travelling round a circular canal of
depth h and perimeter 2na bears to the half-period (irjri) of the rotation of
the water.

The accompanying figures indicate the relative magnitudes of the lower
roots, in the cases s = I and s = 2, when /3 has the values 2, 6, 40, respectively*.

y

With the help of these figures we can trace, in a general way, the changes
in the character of the free modes as /3 increases from zero. The results may
be interpreted as due either to a continuous increase of n, or to a continuous
diminution of h. We will use the terms 'positive' and 'negative' to distin
guish waves which travel, relatively to the water, in the same direction as the
rotation and the opposite.

When |3 is infinitely small, the values of x are given by «//(#*) = 0; these
correspond to the vertical asymptotes of the curve (iv). The values of a-
then occur in pairs of equal and oppositely-signed quantities, indicating that
there is now no difference between the velocity of positive and negative waves.
The case is, in fact, that of Art. 187 (13).

* For clearness the scale of y has been taken to be 10 times that of x,

203]

GRAPHICAL DETERMINATION OF THE ROOTS.

339

As /3 increases, the two values of a- forming a pair become unequal in
magnitude, and the corresponding values of x separate, that being the greater
for which o-/2?i is positive. When /3=s (a + 1) the curve (iv) and the parabola
(v) touch at the point (0, - 1), the corresponding value of o- being -2n. As
ft increases beyond this critical value, one value of x becomes negative, and
the corresponding (negative) value of cr/2n becomes smaller and smaller.

Hence, as ft increases from zero, the relative angular velocity becomes
greater for a negative than for a positive wave of (approximately) the same
type ; moreover the value of o- for a negative wave is always greater than 2n.

(3-6

= 40

9-3

26-4

45-0

70-9

As the rotation increases, the two kinds of wave become more and more
distinct in character as well as in * speed.' With a sufficiently great value of
ft we may have one, but never more than one, positive wave for which
<r is numerically less than 2w. Finally, when ft is very great, the value of <r
corresponding to this wave becomes very small compared with n, whilst the
remaining values tend all to become more and more nearly equal to ±

If we use a zero suffix to distinguish the case of ^=0, we find

22—2

340 TIDAL WAVES. [CHAP. VIII

where #0 refers to the proper asymptote of the curve (iv). This gives the
' speed ' of any free mode in terms of that of the corresponding mode when
there is no rotation.

204. As a sufficient example of forced oscillations we may
assume

(17),

where the value of a- is now prescribed.

This makes V2f = 0, and the equation (4) then gives

f=4/-,(*,r)e« <*+"+•> .................. (18),

where A is to be determined by the boundary-condition (7),
whence

.c ......... (19).

d ,, , . 2sn

This becomes very great when the frequency of the disturbance
is nearly coincident with that of a free mode of corresponding
type.

From the point of view of tidal theory the most interesting cases are
those of s = l with o-=n, and s = 2 with o- = 2rc, respectively. These would
represent the diurnal and semidiurnal tides due to a distant disturbing body
whose proper motion may be neglected in comparison with the rotation n.

In the case of s = l we have a uniform horizontal disturbing force.
Putting, in addition, <r = n, we find without difficulty that the amplitude of
the tide-elevation at the edge (r = a) of the basin has to its 'equilibrium-value'
the ratio

where « = JV(3j8). With the help of Lodge's tables we find that this ratio has
the values

1-000, -638, -396,

for/3= 0, 12, 48, respectively.

When o- = 2?i, we have K = 0, /,(*, r) = r8, and thence, by (17), (18), (19),

i.e. the tidal elevation has exactly the equilibrium -value.

This remarkable result can be obtained in a more general manner; it
holds whenever the disturbing force is of the type

provided the depth h be a function of r only. If we revert to the equations

204-205] FORCED OSCILLATIONS. 341

(1), we notice that when (r — ^n they are satisfied by £=£, Q = iR To deter
mine R as a function of r, we substitute in the equation of continuity (2),
which gives

d(hR) 8-1 ID / x r \

_^-_Afl*-x(r) ........................... (iv).

The arbitrary constant which appears on integration of this equation is to be
determined by the boundary-condition.

In the present case we have x (r) = Crs/a*. Integrating, and making R — Q
for r = a, we find,

The relation e = iR shews that the amplitudes of R and 0 are equal, while
their phases differ always by 90° ; the relative orbits of the fluid particles are
in fact circles of radii

described each about its centre with angular velocity 2n in the negative
direction. We may easily deduce that the path of any particle in space is an
ellipse of semi-axes r + T described about the origin with harmonic motion in
the positive direction, the period being 2rr/n. This accounts for the peculiar
features of the case. For if ( have always the equilibrium-value, the hori
zontal forces due to the elevation exactly balance the disturbing force, and
there remain only the forces due to the undisturbed form of the free surface
(Art. 200 (3)). These give an acceleration gdz^dr, or n2r, to the centre,
where r is the radius-vector of the particle in its actual position. Hence all
the conditions of the problem are satisfied by elliptic-harmonic motion of
the individual particles, provided the positions, the dimensions, and the
' epochs ' of the orbits can be adjusted so as to satisfy the condition of con
tinuity, with the assumed value of £ The investigation just given resolves
this point.

205. We may also briefly notice the case of a circular basin
of variable depth, the law of depth being the same as in Art. 189,
viz.

Assuming that R, 0, £ all vary as e (<rt+se+^ ^ an(j that h is a function of
r only, we find, from Art. 202 (2), (3),

Introducing the value of h from(l), we have, for the free oscillations

342 TIDAL WAVES. [CHAP. VIII

This is identical with Art. 189 (ii), except that we now have

3.2 _ 4^2 4ns

in place of o-2/^A0- The solution can therefore be written down from the
results of that Art., viz. if we put

ghQ a-

we have £=AS K\ F (a, /3, y,

and the condition of convergence at the boundary r = a requires that

k=s + 2j (v),

where j is some positive integer. The values of a- are then given by (iii).

The forms of the free surface are therefore the same as in the case of no
rotation, but the motion of the water-particles is different. The relative orbits
are in fact now ellipses having their principal axes along and perpendicular to
the radius vector • this follows easily from Art. 202 (3).

In the symmetrical modes (5 = 0), the equation (iii) gives

where o-0 denotes the ' speed ' of the corresponding mode in the case of
no rotation, as found in Art. 189.

For any value of s other than zero, the most important modes are those for
which £ = s + 2. The equation (iii) is then divisible by <r + 2?i, but this is an
extraneous factor ; discarding it, we have the quadratic

(vii),
whence o-=n±(n2 + '2,sgk()la2}^ ........................ (viii).

This gives two waves rotating round the origin, the relative wave-velocity
being greater for the negative than for the positive wave, as in the case
of uniform depth (Art. 203). With the help of (vii) the formula reduce to

(r\8 f,

5)' *-*j

the factor et being of course understood in each case. Since G = iR,

the relative orbits are all circles. The case ,s = l is noteworthy; the free
surface is then always plane, and the circular orbits have all the same
radius.

When k > s-f 2, we have nodal circles. The equation (iii) is then a cubic
in a-/2n ; it is easily seen that its roots are all real, lying between — co and

205-206] BASIN OF VARIABLE DEPTH. 343

-1, —1 and 0, and + 1 and -{-co, respectively. As a numerical example, in
the case of s=l, k = 5, corresponding to the values

2, 6, 40

of 4:n2az/ghQ, we find

( + 2-889 +1-874 +1-180,

o-/2n = <- 0-125 -0-100 -0'037,

(-2-764 -1-774 -M43.

The first and last root of each triad give positive and negative waves of a
somewhat similar character to those already obtained in the case of uniform
depth. The smaller negative root gives a comparatively slow oscillation which,
when the angular velocity n is infinitely small, becomes a steady rotational
motion, without elevation or depression of the surface.

The most important type of forced oscillations is such that

(x).

We readily verify, on substitution in (ii), that

£ 0»~A 7~"2 0™_N ^,2 » \XV'

We notice that when <r = Zn the tide-height has exactly the equilibrium-
value, in agreement with Art. 204.

If o-j, o-2 denote the two roots of (vii), the last formula may be written

Tides on a Rotating Globe.

206. We proceed to give some account of Laplace's problem
of the tidal oscillations of an ocean of (comparatively) small depth
covering a rotating globe*. In order to bring out more clearly the
nature of the approximations which are made on various grounds,
we shall adopt a method of establishing the fundamental equations
somewhat different from that usually followed.

When in relative equilibrium, the free surface is of course a
level-surface with respect to gravity and centrifugal force ; we
shall assume it to be a surface of revolution about the polar axis,
but the ellipticity will not in the first instance be taken to be
small.

* " Kecherches sur quelques points du systeme du monde," Mem. de VAcad. roy.
des Sciences, 1775 [1778] and 1776 [1779]; Oeuvres Completes, t. ix. pp. 88, 187.
The investigation is reproduced, with various modifications, in the Mecanique
Celeste, Livre 4me, c. i. (1799).

344 TIDAL WAVES. [CHAP. VIII

We adopt this equilibrium-form of the free surface as a surface
of reference, and denote by 6 and o> the co-latitude (i.e. the angle
which the normal makes with the polar axis) and the longitude,
respectively, of any point upon it. We shall further denote by
z the altitude, measured outwards along a normal, of any point
above this surface.

The relative position of any particle of the fluid being specified
by the three orthogonal coordinates 0, o>, z, the kinetic energy of
unit mass is given by

where R is the radius of curvature of the meridian-section of the
surface of reference, and tzr is the distance of the particle from
the polar axis. It is to be noticed that .R is a function of 6 only,
whilst -or is a function of both 6 and z ; and it easily follows from
geometrical considerations that

d^l(R -f z) dO = cos 0, dvr/dz = sin 6 (2).

The component accelerations are obtained at once from (1) by
Lagrange's formula. Omitting terms of the second order, on
account of the restriction to infinitely small motions, we have

1 / d dT dT\ v * 1

...(3).

I (ddT dT\ (d™f>±

— -y -TT — 3— = OTft) + 2ll -ja V +

•GT \ctt da) d(oj \du

ddT dT .. , 2 LO ., dsr
-J--J. — -j- = z — (nt + 2n«) & y-
dt dz dz ^ J dz

Hence, if we write u, v, w for the component relative velocities of a
particle, viz.

u=(R + z}Q) v = '&a)) w = z ............ (4),

and make use of (2), the hydrodynamical equations may be put in
the forms

at

C^-+2nucos0 + 2nwsm0= - -
dt vr dco \p

dt dz \p

........................... (5),

206] LAPLACE'S PROBLEM. 345

where M* is the gravitation-potential due to the earth's attraction,
whilst n denotes the potential of the extraneous forces.

So far the only approximation consists in the omission of terms
of the second order in u, v, w. In the present application, the
depth of the sea being small compared with the dimensions of the
globe, we may replace R + z by R. We will further assume that
the effect of the relative vertical acceleration on the pressure may
be neglected, and that the vertical velocity is small compared with
the horizontal velocity. The last of the equations (5) then re
duces to

Let us integrate this between the limits z and f, where f
denotes the elevation of the disturbed surface above the surface
of reference. At the surface of reference (z = 0) we have

¥-£ra2OT2= const.,

by hypothesis, and therefore at the free surface (z = f)
¥ - -n2OT2 = const.

provided g= (^ - Jw2O .................. (7).

Here g denotes the value of apparent gravity at the surface of
reference; it is of course, in general, a function of 6. The
integration in question then gives

= const. +0?+Xl ............ (8),

the variation of H with z being neglected. Substituting from (8)
in the first two of equations (5), we obtain, with the approxima
tions above indicated,

where ?=-fl/$r ........................... (10).

These equations are independent of z> so that the horizontal
motion may be assumed to be sensibly the same for all particles
in the same vertical line.

346 TIDAL WAVES. [CHAP. VIII

As in Art. 190, this last result greatly simplifies the equation
of continuity. In the present case we find without difficulty

_ d(hv)\

dt~ w tide d<» } ......

207. It is important to notice that these equations involve no
assumptions beyond those expressly laid down ; in particular,
there is no restriction as to the ellipticity of the meridian, which
may be of any degree of oblateness.

In order, however, to simplify the question as far as possible,
without sacrificing any of its essential features, we will now take
advantage of the circumstance that in the actual case of the earth
the ellipticity is a small quantity, being in fact comparable with
the ratio (n*a/g) of centrifugal force to gravity at the equator, which
is known to be about -^. Subject to an error of this order of
magnitude, we may put R = a, is = a sin 6, g = const., where a is
the earth's mean radius. We thus obtain*

du _ , d , -T,.

with ,_ + ......... (2),

dt a sin 0 { du da) )

this last equation being identical with Art. 190 (1).

Two conclusions of some interest in connection with our previous work
follow at once from the form of the equations (1). In the first place, if u, V
denote the velocities along and perpendicular to any horizontal direction s, we
easily find, by transformation of coordinates

-2»v cos 6= --Q ....................... (i).

In the case of a narrow canal, the transverse velocity v is zero, and the
equation (i) takes the same form as in the case of no rotation ; this has
been assumed by anticipation in Art. 180. The only effect of the rotation
in such cases is to produce a slight slope of the wave-crests and furrows
in the direction across the canal, as investigated in Art. 201.

Again, by comparison of (1) with Art. 200 (7), we see that the oscillations
of a sheet of water of relatively small dimensions, in colatitude 6, will take
place according to the same laws as those of a plane sheet rotating about
a normal to its plane with angular velocity n cos 6.
* Laplace, I.e. ante p. 343.

206-208] GENERAL EQUATIONS. 347

As in Art. 200, free steady motions are possible, subject to certain con
ditions. Putting f=0, we find that the equations (1) and (2) are satisfied by
constant values of u, v, £, provided

q df a d£

a, __ _ _ <j_ _ _ i M — y _ _ ±. (11)

Zna sin 6 cos & da ' 2rai cos 6 d& "
d (h sec 0, t]

and rf(tfl.) -° .............................. <U1>-

The latter condition is satisfied by any assumption of the form

(iv),

and the equations (ii) then give the values of w, v. It appears from (ii) that
the velocity in these steady motions is everywhere parallel to the contour-lines
of the disturbed surface.

If h is constant, or a function of the latitude only, the only condition
imposed on £ is that it should be independent of <o ; in other words the eleva
tion must be symmetrical about the polar axis.

208. We will now suppose that the depth h is a function of 6
only, and that the barriers to the sea, if any, coincide with parallels
of latitude. Assuming, further, that H, u, v, fall vary as
where 6' is integral, we find

iav + 2nu cos 0 = -"— (f- f)

.., . u 1 (c£ (Aw sin $)

with i«rf=— A\~ ~TQ ' ~

Solving for u, v, we get

d

V = 7, ^^ 3

If we put, for shortness,
these may be written

v = —

4m(/2-cos26>)V

/cos 0d?
—- -T + » cosec 6

348 TIDAL WAVES. [CHAP. VIII

The formulae for the component displacements (f, rj, say), can
be written down from the relations u = f, v = r), or u = zcrf , -y = i(rrj.
It appears that in all cases of periodic disturbing forces the fluid
particles describe ellipses having their principal axes along the
meridians and the parallels of latitude, respectively.

Substituting from (7) in (4) we obtain the differential equation
inf:

1 d f Asinfl /d£

2 _ CQS2 Q VA +

30-+*? cosec* e

.................. (8).

In the case of the free oscillations we have f = 0. The manner
in which the boundary -conditions (if any), or the conditions of
finiteness, then determine the admissible values off, and thence of
cr, will be understood by analogy, in a general way, from Arts. 191,
193. For further details we must refer to the paper cited below*.
A practical solution of the problem, even in the case (s = 0) of
symmetry about the axis, with uniform depth, has not yet been
worked out.

The more important problem of the forced oscillations, though
difficult, can be solved for certain laws of depth, and for certain
special values of a which correspond more or less closely to the
main types of tidal disturbance. To this we now proceed.

209. It is shewn in the Appendix to this Chapter that the
tide-generating potential, when expanded in simple-harmonic
functions of the time, consists of terms of three distinct types.

The first type is such that the equilibrium tide-height would
be given by

The corresponding forced waves are called by Laplace the ' Oscilla
tions of the First Species'; they include the lunar fortnightly

* Sir W. Thomson, " On the General Integration of Laplace's Differential
Equation of the Tides," Phil. Mag., Nov. 1875.

t In strictness, 6 here denotes the geocentric latitude, but the difference between
this and the geographical latitude may be neglected in virtue of the assumptions in
troduced in Art. 207.

208-209] CASE OF SYMMETRY ABOUT AXIS. 349

and the solar semi-annual tides, and, generally, all the tides of
long period. Their characteristic is symmetry about the polar axis.

Putting s = 0 in the formulae of the preceding Art. we have

ia- d?

4m(/2-cos20)d<9'

er cos 6 d

~

Id (hu sin 0) , .

and ^o-f = -- s—^ — — 7^ - - .................. (3).

a sin 0 dd

The equations (2) shew that the axes of the elliptic orbit of
any particle are in the ratio of / : cos 6. Since / is small, the
ellipses are very elongated, the greatest length being from E. to
W., except in the neighbourhood of the equator. At the equator
itself the motion of the particles vanishes.

Eliminating u, v between (2) and (3), or putting ,9 = 0 in Art.
208 (8), we find

1 d

Te ' cos

We shall consider only the case of uniform depth (h = const.).
Writing /z for cos 6, the equation then becomes

where f$ = 4ima/h= 4>ri*a*/gh ..................... (6).

The complete primitive of this equation is necessarily of the form

r =<#>(/*) +^»+£/G") .................. w,

where c/> (/z), F(/JL) are even functions, and /(/*) is an odd function,
of IJL, and the constants At B are arbitrary. In the case of an
ocean completely covering the globe, it is not obvious at first sight
that there is any limitation to the values of A and B, although on
physical grounds we are assured that the solution of the problem is
uniquely determinate, except for certain special values of the ratio
/(= c7/2n), which imply a coincidence between the 'speed' of the
disturbing force and that of one of the free oscillations of sym
metrical type. The difficulty disappears if we consider first, for a
moment, the case of a zonal sea bounded by two parallels of

350 TIDAL WAVES. [CHAP. VIII

latitude. The constants A, B are then determined by the
conditions that u = 0 at each of these parallels. If the boundaries
in question are symmetrically situated on opposite sides of the
equator, the constant B will be zero, and the odd function f(fi)
may be disregarded ab initio. By supposing the boundaries to
contract to points at the poles we pass to the case of an unlimited
ocean. If we address ourselves in the first instance to this latter
form of the problem, the one arbitrary constant (A) which it is
necessary to introduce is determined by the condition that the
motion must be finite at the poles.

210. The integration of the equation (5) has been treated by
Lord Kelvin* and Prof. G. H. Darwin f.

We assume
1

(8).

This leads to

r = A - i/'JV + {(B, -f*Bs) ^+...

+ |(^-/f%-i)^ + ......... (9),

where A is arbitrary ; and makes

-B^)^ + ......... (10).

Substituting in (5), and equating coefficients of the several powers
of p, we find

^O ..................... (11),

Q ............... (12),

. O

and thenceforward

* Sir W. Thomson, " On the ' Oscillations of the First Species ' in Laplace's
Theory of the Tides," Phil. Mag., Oct. 1875.

t " On the Dynamical Theory of the Tides of Long Period," Proc. Roy. Soc.,
Nov. 5, 1886 ; Encyc. Britann., Art, " Tides,"

209-210] TIDES OF LONG PERIOD. 351

It is to be noticed that (12) may be included under the typical
form (13), provided we write B^ = - 2H'.

These equations determine Blt B3, ... B2j+l, ... in succession, in
terms of A, and the solution thus obtained would be appropriate,
as already explained, to the case of a zonal sea bounded by two
parallels in equal N. and S. latitudes. In the case of an ocean
covering the globe, it would, as we shall prove, give infinite
velocities at the poles, except for one definite value of A, to be
determined.

Let us write

B2j+lIB2j^ = Nj+l (14);

we shall shew, in the first place, that as j increases Nj must tend
either to the limit 0 or to the limit 1. The equation (13) may be
written

•"i ~~ 1 ~~

Hence, when j is large, either

approximately, or Nj+l is not small, in which case JV}+2 will be
nearly equal to 1, and the values of Nj+3, Nj+4) ... will tend more
and more nearly to 1, the approximate formula being

"" ' ~

Hence, with increasing j, Nj tends to one or other of the forms
(16) and (17).

In the former case (16), the series (8) will be convergent for
/A = ± 1, and the solution is valid over the whole globe.

In the other event (17), the product NlfNz — •$}+!> and
therefore the coefficient BZj+l, tends with increasing j to a finite
limit other than zero. The series (8) will then, after some finite
number of terms, become comparable with 1 4-/A2 + /u,4 + ..., or
(1 — /A2)"1, so that we may write

M

352 TIDAL WAVES. [CHAP. VIII

where L and M are functions of p which remain finite when
p=±l. Hence, from (2),

which makes u infinite at the poles.

It follows that the conditions of our problem can only be
satisfied if Nj tends to the limit zero ; and this consideration, as
we shall see, restricts us to a determinate value of the hitherto
arbitrary constant A.

The relation (15) may be put in the form

and by successive applications of this we find

(2j+2)(2y + 3) (2j + 4)(2j

, . 1 . ,_ _ +&c

2j(2j + lV (2j+2)(2j+3)+ (2/+4.)(%'+B) '

.................. (21),

on the present supposition that Nj+k tends with increasing k to the
limit 0, in the manner indicated by (16). In particular, this
formula determines the value of Nlf Now

£1 = J\riJ8_1 = _2^1#/,
and the equation (11) then gives

NtH' .................. (22);

in other words, this is the only value of A which is consistent with
a zero limit of Nj, and therefore with a finite motion at the poles.
Any other value of A, differing by however little, if adopted as a
starting-point for the successive calculation of B1} B3) ... will
inevitably lead at length to values of Nj which approximate to the
limit 1.

For this reason it is not possible, as a matter of practical
Arithmetic, to calculate B1} B3, ... in succession in the above

210] NUMERICAL SOLUTION. 353

manner ; for this would require us to start with exactly the right
value of A, and to observe absolute accuracy in the subsequent
stages of the work. The only practical method is to use the
formulas

BJH' = - 2^, B3 = N2B1} B5 = N3B3,

or BJH' = -2Nlf BJH' = - 2^2,

where the values of Nlt N2y N3, ... are to be computed from tlfjtf m
continued fraction (21). It is evident a posteriori that the solutio^f X %
thus obtained will satisfy all the conditions of the problem, anji r"
that the series (9) will converge with great rapidity. The mo^i U. *
convenient plan of conducting the calculation is to assumes +J
roughly approximate value, suggested by (16), for one of tkfe j£
ratios JV} of sufficiently high order, and thence to compute

N}^,N^,... N.z, Nt

in succession by means of the formula (20). The values of the
constants A, Bl} B3, ..., in (9), are then given by (22) and (23).
For the tidal elevation we find

- N,N, . . . Ni_t (1 -f*N,) yli- ...... (24).

In the case of the lunar fortnightly tide, / is the ratio of a
sidereal day to a lunar month, and is therefore equal to about ^,
or more precisely '0365. This makes f2 = '00133. It is evident
that a fairly accurate representation of this tide, and a fortiori of
the solar semi-annual tide, and of the remaining tides of long
period, will be obtained by putting /= 0; this materially shortens
the calculations.

The results will involve the value of /3, = 4n*a*/gr£. For
/3 = 40, which corresponds to a depth of 7260 feet, we find in
this way

f/JT^-1515- T0000yu,2+ l-5153yu,4 - l'2120/*8+ '6063/A8- -2076/I-10
-f -0516/*12 - -0097/414 + -001 V5 - -0002y*18 ...... (25) *,

whence, at the poles (/JL = ± 1),

* The coefficients in (25) and (26) differ only slightly from the numerical
values obtained by Prof. Darwin for the case /= -0365.

L. 23

354 TIDAL WAVES. [CHAP. VIII

and, at the equator (/* = 0),

Again, for £ = 10, or a depth of 29040 feet, we get
S/H = '2359 - l-OOOO/*2 + -589V - '1623/t6

+ -0258/t8 - -0026/*10 + -0002/Lt12 ....... (26).

This makes, at the poles,

?=-Jtf'x '470,
and, at the equator,

For /3 = 5, or a depth of 58080 feet, we find

?/#' - '2723 - rOOOO/i2 + -340V-

- -0509/A6 + '0043/A8 - -000 V° ......... (27).

This gives, at the poles,

f=-f#'x -651,
and, at the equator,

?= JJI'x-817.

Since the polar and equatorial values of the equilibrium tide are
— \H' and \H ', respectively, these results shew that for the depths
in question the long-period tides are, on the whole, direct, though
the nodal circles will, of course, be shifted more or less from the
positions assigned by the equilibrium theory. It appears, more
over, that, for depths comparable with the actual depth of the sea,
the tide has less than half the equilibrium value. It is easily
seen from the form of equation (5), that with increasing depth,
and consequent diminution of /3, the tide height will approximate
more and more closely to the equilibrium value. This tendency is
illustrated by the above numerical results.

It is to be remarked that the kinetic theory of the long-
period tides was passed over by Laplace, under the impression
that practically, owing to the operation of dissipative forces,
they would have the values given by the equilibrium theory.
He proved, indeed, that the tendency of frictional forces must
be in this direction, but it has been pointed out by Darwin*
that in the case of the fortnightly tide, at all events, it is
doubtful whether the effect would be nearly so great as Laplace
supposed. We shall return to this point later.

* I.e. ante p. 350.

210-212] FREE OSCILLATIONS. 355

211. It remains to notice how the free oscillations are deter
mined. In the case of symmetry with respect to the equator, we
have only to put H' = 0 in the foregoing analysis. The conditions
of convergency for yu, = + 1 determine Nz, N3, JV4, ... exactly as
before; whilst equation (12) gives N2 = 1 — /3/2/2 . 3, and there
fore, by (20),

__

— -° ...... <28><

which is equivalent to ^ = x . This equation determines the
admissible values of / 1'= <r/2w). The constants in (9) are then
given by

where A is arbitrary.

The corresponding theory for the asymmetrical oscillations
may be left to the reader. The right-hand side of (8) must now
be replaced by an even function of yu,.

212. In the next class of tidal motions (Laplace's ' Oscillations
of the Second Species ') which we shall consider, we have

f»jr'«m£ooe0.<xw(<r* + » + e) ............... (1),

where <r differs not very greatly from n. This includes the lunar
and solar diurnal tides.

In the case of a disturbing body whose proper motion could be
neglected, we should have o- = n, exactly, and therefore /= J. In
the case of the moon, the orbital motion is so rapid that the actual
period of the principal lunar diurnal tide is very appreciably
longer than a sidereal day*; but the supposition that/=J sim
plifies the formula so materially that we adopt it in the following

* It is to be remarked, however, that there is an important term in the harmo
nic development of fi for which 0 = 11 exactly, provided we neglect the changes in
the plane of the disturbing body's orbit. This period is the same for the sun as for
the moon, and the two partial tides thus produced combine into what is called the
1 luni-solar ' diurnal tide.

23—2

356 TIDAL WAVES. [CHAP. VIII

investigation*. We shall find that it enables us to calculate the
forced oscillations when the depth follows the law

h = (I-qcos*0)hQ ........................ (2),

where q is any given constant.

Taking an exponential factor ei(nt+ta+e) , and therefore putting
s = l, /= ^, in Art. 208 (7), and assuming

f' = Csin0cos<9 ........................ (3),

we find u= — i<rC/m, v = a-C/m.cos0 ............... (4).

Substituting in the equation of continuity (Art. 208 (4)), we get

ma
which is consistent with the law of depth (2), provided

C=- -— H"

mi • • 2qhJma $ /h_x

This gives f=- Y; — f ..................... (7).

1 —

One remarkable consequence of this formula is that in
the case of uniform depth (q = 0) there is no diurnal tide, so
far as the rise and fall of the surface is concerned. This result
was first established (in a different manner) by Laplace, who
attached great importance to it as shewing that his kinetic theory
is able to account for the relatively small values of the diurnal
tide which are given by observation, in striking contrast to what
would be demanded by the equilibrium-theory.

But, although with a uniform depth there is no rise and fall,
there are tidal currents. It appears from (4) that every particle
describes an ellipse whose major axis is in the direction of the
meridian, and of the same length in all latitudes. The ratio of the
minor to the major axis is cos 6, and so varies from 1 at the poles
to 0 at the equator, where the motion is wholly N. and S.

213. Finally, we have to consider Laplace's 'Oscillations of the
Third Species,' which are such that

f = Jff"/sina0.cos(<r$ + 2G> + €) ............... (1),

* Taken with very slight alteration from Airy ("Tides and Waves," Arts. 95...),
and Darwin (Encyc, Britann., t. xxiii., p. 359).

212-214] DIURNAL AND SEMI-DIURNAL TIDES. 357

where a is nearly equal to 2?i. This includes the most important of
all the tidal oscillations, viz. the lunar and solar semi-diurnal tides.

If the orbital motion of the disturbing body were infinitely
slow we should have a = 2ra, and therefore /= 1 ; for simplicity
we follow Laplace in making this approximation, although it is
a somewhat rough one in the case of the principal lunar tide*.

A solution similar to that of the preceding Art. can be obtained
for the special law of depth

li = hQ$itfd ........................... (2)f.

Adopting an exponential factor e* lmt+2m+e} , and putting therefore
f= it s = 2, we find that if we assume

?'=asin20 ........................... (3)

the equations (7) of Art. 208 give

„ .„ ~ ...

u=— Ocot 6, 0=-— C — . a ............ (4),

m 2m sm 6

whence, substituting in Art. 208 (4),

£= 2^0/ma. (7sm20 ..................... (5).

Putting f = £" + £ and substituting from (1) and (3), we find

0 ,_

and therefore b = ~ n - rn — ? ........................ (')•

1 - '2/io/ma *

For such depths as actually occur in the ocean we have 2/&0 < ma,
and the tide is therefore inverted.

It may be noticed that the formulae (4) make the velocity
infinite at the poles.

214. For any other law of depth a solution can only be
obtained in the form of an infinite series.

In the case of uniform depth we find, putting s = 2, /=!,
4raa//i = £ in Art. 208 (8),

(1 - /*ay |J + {/3 (1 - ^ - 2p* - 6} ?' - - /3 (1 - ^)2 ? p • -(8),

* There is, however, a ' luni-solar ' semi-diurnal tide whose speed is exactly 2/i
if we neglect the changes in the planes of the orbits. Cf. p. 355, footnote.
t Cf. Airy and Darwin, II. cc.

358 TIDAL WAVES. [CHAP. VIII

where //, is written for cos 6. In this form the equation is some
what intractable, since it contains terms of four different dimensions
in yLt. It simplifies a little, however, if we transform to

!/,=(!- /*2)*, =• sin 0,
as independent variable ; viz. we find

which is of three different dimensions in v.

To obtain a solution for the case of an ocean covering the
globe, we assume

Substituting in (9), and equating coefficients, we find

jB0 = 0, B2 = 0, 0.54 = 0 (11),

and thenceforward

2j (2j + 6) B2j+4 - 2j (2j + 3) B2j+2 + (3B2j = 0 (13).

These equations give B6, Bs, ... J92j, ... in succession, in terms of
B4, which is so far undetermined. It is obvious, however, from the
nature of the problem, that, except for certain special values of h
(and therefore of /?), which are such that there is a free oscil
lation of corresponding type (s = 2) having the speed 2/i, the
solution must be unique. We shall see, in fact, that unless B4
have a certain definite value the solution above indicated will
make the meridian component (u) of the velocity discontinuous
at the equator*.

The argument is in some respects similar to that of Art. 210.
If we denote by Nj the ratio B2j+2/B2j of consecutive coefficients,
we have, from (13),

2? + 3 8 1

zj + 0

from which it appears that, with increasing j, Nj must tend to
one or other of the limits 0 and 1. More precisely, unless the
limit of NJ be zero, the limiting form of Nj+l will be

•6), or 1-3/2J,

* In the case of a polar sea bounded by a small circle of latitude whose angular
radius is <^TT, the value of B4 is determined by the condition that u = Q, or d£'ldi> = 0,
at the boundary.

214] SEMI-DIURNAL TIDES: UNIFORM DEPTH. 359

approximately. This is the same as the limiting form of the ratio
of the coefficients of i$ and i/2-*"2 in the expansion of (1 — z>2)*. We
infer that, unless B4 have such a value as to make Nao= 0, the
terms of the series (10) will become ultimately comparable with
those of (1 — v")%, so that we may write

? = L + (l-v*)*M ..................... (15),

where L, M are functions of v which do not vanish for v = 1. Near
the equator (y = 1) this makes

Hence, by Art. 208 (7), u would change from a certain finite
value to an equal but opposite value as we cross the equator.

It is therefore essential, for our present purpose, to choose the
value of .Z?4 so that Nx = 0. This is effected by the same method
as in Art. 210. Writing (13) in the form

we see that Nj must be given by the converging continued fraction

_ /3__ 0 J3 _

¥_ 2j(2j + 6) (2; + 2)(2/+8) (2j+4)(2y + 10) n ,

j " " '

_

2J + 6 2J + 8 2J + 10 k

This holds from j = 2 upwards, but it appears from (12) that it
will give also the value of A^ (not hitherto defined), provided we
use this symbol for B^jH'". We have then

Bt = N,H'", Be=N,Bt, Ba = N3Be,....

Finally, writing f = f + f, we obtain

.lvf> + N1N,Ns* + ......... (19).

As in Art. 210, the practical method of conducting the calcula
tion is to assume an approximate value for JV}+1, where j is a
moderately large number, and then to deduce Nj, JV}_,,... Na, A^
in succession by means of the formula (17).

360 TIDAL WAVES. [CHAR VIII

The above investigation is taken substantially from the very remarkable
paper written by Lord Kelvin * in vindication of Laplace's treatment of the
problem, as given in the Mecanique Celeste. In the passage more especially
in question, Laplace determines the constant B± by means of the continued
fraction for N11 without, it must be allowed, giving any adequate justifica
tion of the step ; and the soundness of this procedure had been disputed by
Airy f, and after him by FerrelJ.

Laplace, unfortunately, was not in the habit of giving specific references,
so that few of his readers appear to have become acquainted with the original
presentment § of the kinetic theory, where the solution for the case in question
is put in a very convincing, though somewhat different, form. Aiming in the
first instance at an approximate solution by means of & finite series, thus :

(i),

Laplace remarks || that in order to satisfy the differential equation, the
coefficients would have to fulfil the conditions

as is seen at once by putting J32k + 4 = 0, B21c + 6 — Q,... in the general relation (13).

We have here k + 1 equations between k constants. The method followed
is to determine the constants by means of the first k relations ; we thus
obtain an exact solution, not of the proposed differential equation (9), but of the
equation as modified by the addition of a term @B2k + 2 1/2* + 6 to the right-hand
side. This is equivalent to an alteration of the disturbing force, and if we can
obtain a solution such that the required alteration is very small, we may
accept it as an approximate solution of the problem in its original form IF.

Now, taking the first k relations of the system (ii) in reverse order,
we obtain -52fc + 2 ^n terms of Z?^, thence J32Jfc in terms of jB2fc~i> an(^ so on> Until5
finally, B± is expressed in terms of H'" ; and it is obvious that if k be large
enough the value of -52& + 2, and the consequent adjustment of the disturbing

* Sir W. Thomson, " On an Alleged Error in Laplace's Theory of the Tides,"
Phil. Mag., Sept. 1875.

t " Tides and Waves," Art. 111.

+ "Tidal Kesearches," U.S. Coast Survey Rep., 1874, p. 154.

§ " Kecherches sur quelques points du systeme du monde," Mem. de VAcad.
roy. des Sciences, 1776 [1779] ; Oeuvres Completes, t. ix., pp. 187....

|| Oeuvres, t. ix., p. 218. The notation has been altered.

H It is remarkable that this argument is of a kind constantly employed by Airy
himself in his researches on waves.

214]

LAPLACE'S SOLUTION.

361

force which is required to make the solution exact, will be very small. This
will be illustrated presently, after Laplace, by a numerical example.

The process just given is plainly equivalent to the use of the continued
fraction (17) in the manner already explained, starting with j + ! = >(•, and
•#j~»/?/2fc (84+3). The continued fraction, as such, does not, however, make
its appearance in the memoir here referred to, but was introduced in the
Mecanique Celeste, probably as an after-thought, as a condensed expression of
the method of computation originally employed.

The following table gives the numerical values of the coeffi
cients of the several powers of v in the formula (19) for f/JEP", in
the cases j3 = 40, 20, 10, 5, 1, which correspond to depths of 7260,
14520, 29040, 58080, 290400, feet, respectively*. The last line
gives the value of QH'" for v — 1, i.e. the ratio of the amplitude
at the equator to its equilibrium-value. At the poles (y = 0),
the tide has in all cases the equilibrium-value zero.

, = 40

,=20

13=10

0=1

I/2

+ i-oooo

+ 1-0000

+ 1-0000

+ 1-0000

+ 1-0000

V*

+ 20-1862

-0-2491

+ 6-1960

+ 0-7504

+0-1062

J>6

+ 10-1164

-1-4056

+ 3-2474 I +0-1566

+ 0-0039

*

- 13-1047

-0-8594

+0-7238

+0-0157

+0-0001

vw

-15-4488

-0-2541

+0-0919

+ 0-0009

1/12

- 7-4581

- 0-0462

+0-0076

1/14

- 2-1975

-0-0058

+ 0-0004

1/16

- 0-4501

- 0-0006

j/18

- 0-0687

1/20

- 0-0082

1/22

- 0-0008

"24

- o-oooi

- 7-434

-1-821

+ 11-267

+ 1-924

+ 1-110

We may use the above results to estimate the closeness of the approxima
tion in each case. For example, when j3 = 40, Laplace finds B.^= - -000004 //'" ;
the addition to the disturbing force which is necessary to make the solution
exact would then be — -00002 #"'V°, and would therefore bear to the actual
force the ratio - -00002 v28.

It appears from (19) that near the poles, where v is small, the
tides are in all cases direct. For sufficiently great depths, ft will

* The first three cases were calculated by Laplace, I.e. ante p. 360 ; the last by
Lord Kelvin. The results have been roughly verified by the present writer.

362 TIDAL WAVES. [CHAP. VIII

be very small, and the formulae (17) and (19) then shew that the
tide has everywhere sensibly the equilibrium value, all the coeffi
cients being small except the first, which is unity. As h is
diminished, /3 increases, and the formula (17) shews that each
of the ratios Nj will continually increase, except when it changes
sign from -f to — by passing through the value oo . No singu
larity in the solution attends this passage of Nj through oo ,
except in the case of Nlf since, as is easily seen, the product
Nj^Nj remains finite, and the coefficients in (19) are therefore all
finite. But when JVj = oo , the expression for f becomes infinite,
shewing that the depth has then one of the critical values
already referred to.

The table above given indicates that for depths of 29040 feet,
and upwards, the tides are everywhere direct, but that there is
some critical depth between 29040 feet and 14520 feet, for which
the tide at the equator changes from direct to inverted. The
largeness of the second coefficient in the case 0 = 40 indicates that
the depth could not be reduced much below 7260 feet before
reaching a second critical value.

Whenever the equatorial tide is inverted, there must be one
or more pairs of nodal circles (f=0), symmetrically situated on
opposite sides of the equator. In the case of /3 = 40, the position
of the nodal circles is given by z; = '95, or 0 = 90° + 18°, approxi
mately *.

215. We close this chapter with a brief notice of the question
of the stability of the ocean, in the case of rotation.

It has been shewn in Art. 197 that the condition of secular
stability is that V— T0 should be a minimum in the equilibrium
configuration. If we neglect the mutual attraction of the elevated
water, the application to the present problem is very simple. The
excess of the quantity V— T0 over its undisturbed value is evidently

8 (1),

where M* denotes the potential of the earth's attraction, SS is an
element of the oceanic surface, and the rest of the notation is as

* For a fuller discussion of these points reference may be made to the original
investigation of Laplace, and to Lord Kelvin's papers.

214-215] STABILITY OF THE OCEAN. 363

before. Since "^ - £?i2w2 is constant over the undisturbed level
(z = 0), its value at a small altitude z may be taken to be cjz + const.,
where, as in Art. 206,

Since f/£dS = Q, on account of the constancy of volume, we find
from (1) that the increment of V— TQ is

iffgraa (3).

This is essentially positive, and the equilibrium is therefore
secularly stable*.

It is to be noticed that this proof does not involve any
restriction as to the depth of the fluid, or as to smallness of the
ellipticity, or even as to symmetry of the undisturbed surface with
respect to the axis of rotation.

If we wish to take into account the mutual attraction of the
water, the problem can only be solved without difficulty when the
undisturbed surface is nearly spherical, and we neglect the varia
tion of g. The question (as to secular stability) is then exactly the
same as in the case of no rotation. The calculation for this case
will find an appropriate place in the next chapter. The result, as
we might anticipate from Art. 192, is that the ocean is stable if,
and only if, its density be less than the mean density of the Earth*.

* Cf. Laplace, Mecaniquc Celeste, Livre 4mo, Arts. 13, 14.

APPENDIX.

ON TIDE-GENERATING FORCES.

a. IF, in the annexed figure, 0 and C be the centres of the earth and of
the disturbing body (say the moon), the potential of the moon's attraction at
a point P near the earth's surface will be - yM/CP, where M denotes the

moon's mass, and y the gravitation-constant. If we put OC=Z>, OP = r, and
denote the moon's (geocentric) zenith-distance at P, viz. the angle POC, by ^,
this potential is equal to

yM

We require, however, not the absolute accelerative effect on P, but the
acceleration relative to the earth. Now the moon produces in the whole
mass of the earth an acceleration yM/D2* parallel to OC, and the potential of
a uniform field of force of this intensity is evidently

yM

~ rj2-r cos ^-

Subtracting this from the former result we get, for the potential of the relative
attraction on P,

This function G is identical with the 'disturbing-function' of planetary
theory.

* The effect of this is to produce a monthly inequality in the motion of the
earth's centre about the sun. The amplitude of the inequality in radius vector is
about 3000 miles ; that of the inequality in longitude is about 1". Laplace,
Mecaniqne Celeste, Livre 6m% Art. 30, and Livre 13me, Art. 10.

EQUILIBRIUM THEORY. 365

Expanding in powers of rfD, which is in our case a small quantity, and
retaining only the most important term, we find

Considered as a function of the position of P, this is a zonal harmonic of the
second degree, with OC as axis.

The reader will easily verify that, to the order of approximation adopted,
Q is equal to the joint potential of two masses, each equal to -|J/, placed, one
at Ct and the other at a point C' in CO produced such that OC' = OC*.

b. In the ' equilibrium- theory ' of the tides it is assumed that the free surface
takes at each instant the equilibrium-form which might be maintained if the
disturbing body were to retain unchanged its actual position relative to the
rotating earth. In other words, the free surface is assumed to be a level-
surface under the combined action of gravity, of centrifugal force, and of the
disturbing force. The equation to this level-surface is

^-2OT2 + G = const

where n is the angular velocity of the rotation, or denotes the distance of any
point from the earth's axis, and ¥ is the potential of the earth's attraction.
If we use square brackets [ ] to distinguish the values of the enclosed quanti
ties at the undisturbed level, and denote by £ the elevation of the water
above this level due to the disturbing potential Q, the above equation is equi
valent to

approximately, where djdz is used to indicate a space-differentiation along the
normal outwards. The first term is of course constant, and we therefore
have

(v),

where, as in Art. 206, g = [~^- (* - ^2w2) 1 ........................... (vi).

Evidently, g denotes the value of * apparent gravity ' ; it will of course vary
more or less with the position of P on the earth's surface.

It is usual, however, in the theory of the tides, to ignore the slight
variations in the value of g, and the effect of the ellipticity of the undisturbed
level on the surface- value of O. Putting, then, r = a, g = yE/a*, where E
denotes the earth's mass, and a the mean radius of the surface, we have,
from (ii) and (v),

(vii),

where #=

as in Art. 177. Hence the equilibrium -form of the free surface is a harmonic
* Thomson and Tait, Natural Philosophy, Art. 804.

366 . ON TIDE-GENERATING FORCES.

spheroid of the second order, of the zonal type, having its axis passing through
the disturbing body.

C. Owing to the diurnal rotation, and also to the orbital motion of the
disturbing body, the position of the tidal spheroid relative to the earth is
continually changing, so that the level of the water at any particular place
will continually rise and fall. To analyse the character of these changes, let
6 be the co-latitude, and o> the longitude, measured eastward from some fixed
meridian, of any place P, and let A be the north-polar-distance, and a the
hour-angle west of the same meridian, of the disturbing body. We have,

then,

cos S- = cos A cos 6 + sin A sin 6 cos (a + o>) .................. (ix),

and thence, by (vii),

+ ^Hsin 2 A sin 26 cos (a + o>)

+ pTsin2Asin20cos2(a + a))-|-C' ..................... (x).

Each of these terms may be regarded as representing a partial tide, and
the results superposed.

Thus, the first term is a zonal harmonic of the second order, and gives a
tidal spheroid symmetrical with respect to the earth's axis, having as nodal
lines the parallels for which cos2 0 = £, or 0 = 90° ±35° 16'. The amount of the
tidal elevation in any particular latitude varies as cos2 A — J. In the case
of the moon the chief fluctuation in this quantity has a period of about a
fortnight ; we have here the origin of the ' lunar fortnightly ' or ' declina-
tional3 tide. When the sun is the disturbing body, we have a 'solar semi
annual' tide. It is to be noticed that the mean value of cos2A-^ with
respect to the time is not zero, so that the inclination of the orbit of the
disturbing body to the equator involves as a consequence a permanent change
of mean level. Of. Art. 180.

The second term in (x) is a spherical harmonic of the type obtained by
putting ?i = 2, s = l in Art. 87 (6). The corresponding tidal spheroid has as
nodal lines the meridian which is distant 90° from that of the disturbing
body, and the equator. The disturbance of level is greatest in the meridian
of the disturbing body, at distances of 45° N. and S. of the equator. The
oscillation at any one place goes through its period with the hour-angle a,
i.e. in a lunar or solar day. The amplitude is, however, not constant, but
varies slowly with A, changing sign when the disturbing body crosses the
equator. This term accounts for the lunar and solar ' diurnal ' tides.

The third term is a sectorial harmonic (n = 2, a = 2), and gives a tidal
spheroid having as nodal lines the meridians which are distant 45° E. and W.
from that of the disturbing body. The oscillation at any place goes through
its period with 2a, i. e. in half a (lunar or solar) day, and the amplitude varies
as sin2 A, being greatest when the disturbing body is on the equator. We
have here the origin of the lunar and solar ' semi-diurnal' tides.

ANALYSIS OF DISTURBING FORCES. 367

The ' constant ' C is to be determined by the consideration that, on account
of the invariability of volume, we must have

J/fdS = 0 .................................... (xi),

where the integration extends over the surface of the ocean. If the ocean
cover the whole earth we have (7=0, by the general property of spherical
surface-harmonics quoted in Art. 88. It appears from (vii) that the greatest
elevation above the undisturbed level is then at the points S = 0, ^=180°, i.e.
at the points where the disturbing body is in the zenith or nadir, and the
amount of this elevation is \H. The greatest depression is at places where
^ = 90°, i. e. the disturbing body is on the horizon, and is \H. The greatest
possible range is therefore equal to H.

In the case of a limited ocean, C does not vanish, but has at each instant a
definite value depending on the position of the disturbing body relative to the
earth. This value may be easily written down from equations (x) and (xi) ;
it is a sum of spherical harmonic functions of A, a, of the second order, with
constant coefficients in the form of surface-integrals whose values depend on
the distribution of land and water over the globe. The changes in the value
of (7, due to relative motion of the disturbing body, give a general rise and
fall of the free surface, with (in the case of the moon) fortnightly, diurnal, and
semi-diurnal periods. This * correction to the equilibrium-theory,' as usually
presented, was first fully investigated by Thomson and Tait*. The necessity
for a correction of the kind, in the case of a limited sea, had however been
recognized by D. Bernoullif.

d. We have up to this point neglected the mutual attraction of the par
ticles of the water. To take this into account, we must add to the disturbing
potential Q the gravitation-potential of the elevated water. In the case of an
ocean covering the earth, the correction can be easily applied, as in Art. 192.
Putting W = 2 in the formulae of that Art,, the addition to the value of Q. is
— f p/po'#£ > and we thence find without difficulty

It appears that all the tides are increased, in the ratio (1— fp/po)"1. If we
assume p/p0 = *18, this ratio is 1*12.

e. So much for the equilibrium-theory. For the purposes of the kinetic
theory of Arts. 206 — 214, it is necessary to suppose the value (x) of £ to be
expanded in a series of simple-harmonic functions of the time. The actual

* Natural Philosophy, Art. 808; see also Prof. G. H. Darwin, "On the Cor
rection to the Equilibrium Theory of the Tides for the Continents," Proc. Roy. Soc.,
April 1, 1886. It appears as the result of a numerical calculation by Prof. H. H.
Turner, appended to this paper, that with the actual distribution of land and water
the correction is of little importance.

t Traite sur le Flux et Reflux de la Mer, c. xi. (1740). This essay, as well as the
one by Maclaurin cited on p. 322, and another on the same subject by Euler, is
reprinted in Le Seur and Jacquier's edition of Newton's Principia.

ON TIDE-GENEKATING FORCES.

expansion, taking account of the variations of A and a, and of the distance
D of the disturbing body, (which enters into the value of _£T), is a somewhat
complicated problem of Physical Astronomy, into which we do not enter here*.

Disregarding the constant (7, which disappears in the dynamical equations
(1) of Art. 207, the constancy of volume being now secured by the equation of
continuity (2), it is easily seen that the terms in question will be of three
distinct types.

First, we have the tides of long period, for which

C=#'(cos20-i).cos(o-* + e) (xiii).

The most important tides of this class are the ' lunar fortnightly ' for which,
in degrees per mean solar hour, o- = l°'098, and the 'solar semi-annual' for
which o- = 0° -082.

Secondly, we have the diurnal tides, for which

^^''smflcostf.cos^ + w + e) (xiv),

where a- differs but little from the angular velocity n of the earth's rotation.
These include the 'lunar diurnal' [o- = 13°'943], the 'solar diurnal3 [0- = 14°'959],
and the 'luni-solar diurnal3 [tr = w = 150t041].

Lastly, we have the semi-diurnal tides, for which

£=#"'sm20.cos(o-£-f-2a> + f) (xv)f,

where <r differs but little from Zn. These include the ' lunar semi-diurnal '
[0- = 28° -984], the 'solar semi-diurnal' [o- = 30°], and the 'luni-solar semi
diurnal >[o-=2n = 30° -082].

For a complete enumeration of the more important partial tides, and for
the values of the coefficients H', H", H"' in the several cases, we must refer
to the papers by Lord Kelvin and Prof. G. H. Darwin, already cited. In the
Harmonic Analysis of Tidal Observations, which is the special object of these
investigations, the only result of dynamical theory which is made use of is the
general principle that the tidal elevation at any place must be equal to the
sum of a series of simple-harmonic functions of the time, whose periods are
the same as those of the several terms in the development of the disturbing
potential, and are therefore known a priori. The; amplitudes and phases of
the various partial tides, for any particular port, are then determined by
comparison with tidal observations extending over a sufficiently long period J.

* Reference may be made to Laplace, Mecanique Celeste, Livre 13me, Art. 2 ; to
the investigations of Lord Kelvin and Prof. G. H.JDarwin in the Brit. Ass. Reports
for 1868, 1872, 1876, 1883, 1885 ; and to the Art. on " Tides," by the latter author,
in the Encyc. Britann. (9th ed.).

f It is evident that over a small area, near the poles, which may be treated as
sensibly plane, the formulae (xiv) and (xv) make

f ocrcos(o-f + w + e), and £ccr2cos((r£ + 2w + e),

respectively, where r, w are plane polar coordinates. These forms have been used
by anticipation in Arts. 188, 204.

+ It is of interest to note, in connection with Art. 184, that the tide-gauges,
being situated in relatively shallow water, are sensibly affected by certain tides of
the second order, which therefore have to be taken account of in the general
scheme of Harmonic Analysis.

HARMONIC ANALYSIS. 369

We thus obtain a practically complete expression which can be used for
the systematic prediction of the tides at the port in question.

f. One point of special interest in the Harmonic Analysis is the deter
mination of the long-period tides. It has been already stated that owing to
the influence of dissipative forces these must tend to approximate more or less
closely to their equilibrium values. Unfortunately, the only long-period tide,
whose coefficient can be inferred with any certainty from the observations,
is the lunar fortnightly, and it is at least doubtful whether the dissipative
forces are sufficient to produce in this case any great effect in the direction
indicated. Hence the observed fact that the fortnightly tide has less than
its equilibrium value does not entitle us to make any inference as to elastic
yielding of the solid body of the earth to the tidal distorting forces exerted by
the moon*.

* Prof. G. H. Darwin, I.e. ante p. 350.

CHAPTER IX.

SURFACE WAVES.

216. WE have now to investigate, as far as possible, the laws
of wave-motion in liquids when the restriction that the vertical
acceleration may be neglected is no longer imposed. The most
important case not covered by the preceding theory is that of
waves on relatively deep water, where, as will be seen, the agita
tion rapidly diminishes in amplitude as we pass downwards from
the surface ; but it will be understood that there is a continuous
transition to the state of things investigated in the preceding
chapter, where the horizontal motion of the fluid was sensibly the
same from top to bottom.

We begin with the oscillations of a horizontal sheet of water,
and we will confine ourselves in the first instance to cases where
the motion is in two dimensions, of which one (#) is horizontal,
and the other (y) vertical. The elevations and depressions of the
free surface will then present the appearance of a series of parallel
straight ridges and furrows, perpendicular to the plane xy.

The motion, being assumed to have been generated originally
from rest by the action of ordinary forces, will be necessarily
irrotational, and the velocity-potential <f> will satisfy the equation

*-o ........................ (i),

*

with the condition -f- = 0 . . (2)

dn

at a fixed boundary.

216] GENERAL CONDITIONS. 371

To find the condition which must be satisfied at the free
surface (p = const.), let the origin 0 be taken at the undisturbed
level, and let Oy be drawn vertically upwards. The motion being
assumed to be infinitely small, we find, putting ^L — gy in the
formula (4) of Art. 21, and neglecting the square of the velocity (q),

Hence if tj denote the elevation of the surface at time t above the
point (x, 0), we shall have, since the pressure there is uniform,

9 L

provided the function F(t\ and the additive constant, be supposed
merged in the value of d*f>/cU. Subject to an error of the order
already neglected, this may be written

Since the normal to the free surface makes an infinitely small
angle (dy/dx) with the vertical, the condition that the normal
component of the fluid velocity at the free surface must be equal
to the normal velocity of the surface itself gives, with sufficient
approximation,

drj Vd\

This is in fact what the general surface condition (Art. 10 (3))
becomes, if we put F(x, y, 2,t)=y-rj, and neglect small quanti
ties of the second order.

Eliminating ?? between (5) and (6), we obtain the condition

to be satisfied when y = 0.

In the case of simple-harmonic motion, the time-factor being
condition becomes

24—2

372 SURFACE WAVES. [CHAP. IX

217. Let us apply this to the free oscillations of a sheet of
water, or a straight canal, of uniform depth h, and let us suppose
for the present that there are no limits to the fluid in the direction
of x, the fixed boundaries, if any, being vertical planes parallel to xy.

Since the conditions are uniform in respect to x, the simplest
supposition we can make is that </> is a simple-harmonic function
of x ; the most general case consistent with the above assumptions
can be derived from this by superposition, in virtue of Fourier's
Theorem.

We assume then

(/> = P cos kx . e'>e+e) ........................ (1),

where P is a function of y only. The equation (1) of Art. 216
gives

whence P = Aeky + Be~ky ........................ (3).

The condition of no vertical motion at the bottom is dcf>/dy = 0
for y = — h, whence

=, say.
This leads to

(/> = C cosh k(y + h) cos kx . ei(<Tt+e) ............... (4).

The value of a is then determined by Art. 216 (8), which gives

o-2 = gk tanh kh ........................ (5).

Substituting from (4) in Art. 216 (5), we find

ia-C
77= —cothM COB fa?. **<•**•) ............... (6),

y

or, writing a = - aC/g . cosh kh,

and retaining only the real part of the expression,

77 = a cos kx . sin (crt + e) .................. (7).

This represents a system of ' standing waves,' of wave-length
X = 2-7T/&, and vertical amplitude a. The relation between the
period (2?r/<r) and the wave-length is given by (5). Some numeri
cal examples of this dependence will be given in Art. 218.

217] STANDING WAVES. 373

In terms of a we have

ga cosh k (y + h) , , , , /CA

6 = —Z -- ^r — cos kx . cos (at + e) ......... (8),

cr cosh kh

and it is easily seen from Art. 62 that the corresponding value of
the stream-function is

qa sinh k (y -f h) . 7 , . >. /m

Jr=v -- ±2— — > sm kx . cos (at + e) ......... (9).

or cosh kh

If x, y be the coordinates of a particle relative to its mean
position (x, y), we have

(ft~ cfo' d* dy"

if we neglect the differences between the component velocities at
the points (x, y) and (x + x, y + y), as being small quantities of
the second order.

Substituting from (8), and integrating with respect to t, we find

cosh k (y + h) . . . , x

x. = — a - . , . . — - sm KX . sm (at 4- e),
sinh kh

y = a -- . U; , — - cos kx . sin (cr^ + e)
sinh kh

where a slight reduction has been effected by means of (5). The
motion of each particle is rectilinear, and simple-harmonic, the
direction of motion varying from vertical, beneath the crests and
hollows (kx—m'jr)^ to horizontal, beneath the nodes (kx = (m + J) TT).
As we pass downwards from the surface to the bottom the ampli
tude of the vertical motion diminishes from a cos kx to 0, whilst
that of the horizontal motion diminishes in the ratio cosh kh : 1.

When the wave-length is very small compared with the
depth, kh is large, and therefore tanh kh = l. The formulae (11)
then reduce to

x = - ae*y sin &# . sin (<r£ + e),

( .

y = ae1® cos kx . sin (crt + e) }

with a^ = gk ........................... (13)*

The motion now diminishes rapidly from the surface down-

* This case may of course be more easily investigated independently.

374

SURFACE WAVES.

[CHAP, ix

wards ; thus at a depth of a wave-length the diminution of
amplitude is in the ratio e~'2n or 1/535. The forms of the lines
of (oscillatory) motion (-vjr = const.), for this case, are shewn
in the annexed figure.

In the above investigation the fluid is supposed to extend to infinity in
the direction of x, and there is consequently no restriction to the value of k.
The formulae also, give, however, the longitudinal oscillations in a canal of finite
length, provided k have the proper values. If the fluid be bounded by the
vertical planes x — 0, x = I (say ), the condition d(f)/dx = 0 is satisfied at both ends
provided sin^ = 0, or kl = mn, where m = l, 2, 3, .... The wave-lengths of the
normal modes are therefore given by the formula \ = 2l/m, Cf. Art. 175.

218. The investigation of the preceding Art. relates to the
case of ' standing ' waves ; it naturally claims the first place, as a
straightforward application of the usual method of treating the
free oscillations of a system about a state of equilibrium.

In the case, however, of a sheet of water, or a canal, of uniform
depth, extending to infinity in both directions, we can, by super
position of two systems of standing waves of the same wave-length,
obtain a system of progressive waves which advance unchanged
with constant velocity. For this, it is necessary that the crests
and troughs of one component system should coincide (horizon
tally) with the nodes of the other, that the amplitudes of the two
systems should be equal, and that their phases should differ by a
quarter-period.

Thus if we put v — Vi ± v-2 (1),

where % = a sin kx cos at, 7j2 = a cos kx sin at (2),

we get 97 = a sin (kx ±a-t} (3),

which represents (Art. 167) an infinite train of waves travelling

217-218] PROGRESSIVE WAVES. 375

in the negative or positive direction of x, respectively, with
the velocity c given by

y ..................... (4),

k \k J

where the value of a has been substituted from Art. 217 (5). In
terms of the wave-length (X) we have

When the wave-length is anything less than double the depth,
we have tanh kh=l, sensibly, and therefore

»'=(£)' ..................... <«>'•

On the other hand when X is moderately large compared with h
we have tanh kh = kh, nearly, so that the velocity is independent
of the wave-length, being given by

(7),

as in Art. 167. This formula is here obtained on the assumption
that the wave-profile is a curve of sines, but Fourier's theorem
shews that this restriction is now unnecessary.

It appears, on tracing the curve y = (tanh x)jx} or from the
numerical table to be given presently, that for a given depth h
the wave-velocity increases constantly with the wave-length,
from zero to the asymptotic value (7).

Let us now fix our attention, for definiteness, on a train of
simple-harmonic waves travelling in the positive direction, i.e. we
take the lower sign in (1) and (3). It appears, on comparison
with Art. 217 (7), that the value of % is deduced by putting
e = ^TT, and subtracting £TT from the value of &#f, and that of rj.2
by putting e = 0, simply. This proves a statement made above as
to the relation between the component systems of standing waves,

* Green, "Note on the Motion of Waves in Canals," Camb. Trans., t. vii. (1839);
Mathematical Papers, p. 279.

t This is of course merely equivalent to a change of the origin from which x is
measured,

376 SURFACE WAVES. [CHAP. IX

and also enables us to write down at once the proper modifications
of the remaining formulae of the preceding Art.

Thus, we find, for the component displacements of a particle,

cosh k (y + h) n ^
x = xx - x.j = a - — . ,,, — - cos UUf - crt),

Smhkh . ..(8).

sinh k (y + Ti) . /7 J
y = yi-ya=a . -' am (to -««

This shews that the motion of each particle is elliptic-harmonic,
the period (2?r/cr, = X/c,) being of course that in which the dis
turbance travels over a wave-length. The semi-axes, horizontal
and vertical, of the elliptic orbits are

cosh k (y + h) siuhlc(y + h)

(L - : — \ — T~l -- dilU. (Jb - : — ; — f^ -- ,

sinh kh smh kh

respectively. These both diminish from the surface to the bottom
(y = — h), where the latter vanishes. The distance between the
foci is the same for all the ellipses, being equal to a cosech kh. It
easily appears, on comparison of (8) with (3), that a surface-particle
is moving in the direction of wave-propagation when it is at a crest,
and in the opposite direction when it is in a trough*.

When the depth exceeds half a wave-length, e~kh is very small,
and the formulae (8) reduce to

x = aeky cos (kx - a-t), \ ,~.

"

so that each particle describes a circle, with constant angular
velocity cr, = (^X/27r)^t. The radii of these circles are given by
the formula a^y, and therefore diminish rapidly downwards.

In the following table, the second column gives the values of sech kh
corresponding to various values of the ratio A/A. This quantity measures the
ratio of the horizontal motion at the bottom to that at the surface. The third
column gives the ratio of the vertical to the horizontal diameter of the elliptic
orbit of a surface particle. The fourth and fifth columns give the ratios
of the wave-velocity to that of waves of the same length on water of infinite
depth, and to that of ' long ' waves on water of the actual depth, respectively.

* The results of Arts. 217, 218, for the case of finite depth, were given, substan
tially, by Airy, "Tides and Waves," Arts. 160... (1845).
t Green, /. c. ante p. 875.

218]

ORBITS OF PARTICLES.

377

fc/X

sech kh

tanh kh

cl(gk-y

cl(gh)l

o-oo

1-000

o-ooo

o-ooo

1-000

•01

•998

•063

•250

•999

•02

•992

•125

•354

•997

•03

•983

•186

•432

•994

•04

•969

•246

•496

•990

•05

•953

•304

•552

•984

•06

•933

•360

•600

•977

•07

•911

•413

•643

•970

•08

•886

•464

•681

•961

•09

•859

•512

•715

•951

•10

•831

•557

•746

•941

•20

•527

•850

•922

•823

•30

•297

•955

•977

•712

•40

•161

•987

•993

•627

•50

•086

•996

•998

•563

•60

•046

•999

•999

•515

•70

•025

1-000

1-000

•477

•80

•013

1-000

1-000

•446

•90

•007

1-000

1-000

•421

1-00

•004

1-000

1-000

•399

00

•000

1-000

1-000

•000

The annexed tables of absolute values of periods and wave-velocities are
abridged from Airy's treatise*. The value of g adopted by him is 32'16 f.s.s.

Depth of
water,
in feet

1
0-442

10

]

1-873

Length of
100

^eriod of ws
17-645

wave, in fee
1000

ive, in secoi
176-33

4
10,000

ids
1763-3

1

10

0-442

1-398

5-923

55-80

557-62

100

0-442

1-398

4-420

18-73

176-45

1000

0-442

1-398

4-420

13-98

59-23

10,000

0-442

1-398

4-420

13-98

44-20

Depth of

Length of wave, in feet

water,

in feet

1

10

100

1000

10,000

00

Wave- velocity, in feet per second

1

2-262

5-339

5-667

5-671

5-671

5-671

10

2-262

7-154

16-88

17-92

17-93

17-93

100

2-262

7-154

22-62

53-39

56-67

56-71

1000

2-262

7-154

22-62

71-54

168-8

179-3

10,000

2-262

7-154

22-62

71-54

226-2

567-1

" Tides and Waves," Arts. 169, 170.

378 SURFACE WAVES. [CHAP. IX

The possibility of progressive waves advancing with unchanged form is of
course limited, theoretically, to the case of uniform depth ; but the foregoing
numerical results shew that practically a variation in the depth will have no
appreciable influence, provided the depth everywhere exceeds (say) half the
wave-length.

We remark, finally, that the theory of progressive waves may
be obtained, without the intermediary of standing waves, by
assuming at once, in place of Art. 217 (1),

The conditions to be satisfied by P are exactly the same as before,
and we easily find, in real form,

rj = asm(kx — at} (11),

qa cosh k(y + h)

<p = — Vi — - cos (KM — at] (12),

a cosh kh

with the same determination of a as before. From (12) all the
preceding results as to the motion of the individual particles can
be inferred without difficulty.

219. The energy of a system of standing waves of the simple-
harmonic type is easily found. If we imagine two vertical planes
to be drawn at unit distance apart, parallel to xy, the potential
energy per wave-length of the fluid between these planes is, as in
Art. 171,

Substituting the value of 77 from Art. 217 (7), we obtain

%gpa?\ . sin2 (at + e) (1).

The kinetic energy is, by the formula (1) of Art. 61,

Substituting from Art. 217 (8), and remembering the relation
between <r and k, we obtain

2X . cos2 (at + e) ..................... (2).

The total energy, being the sum of (1) and (2), is constant,

218-220] ENERGY. 379

and equal to lgptf\. We may express this by saying that the
total energy per unit area of the water-surface is j<7/oa2.

A similar calculation may be made for the case of progressive
waves, or we may apply the more general method explained in
Art. 171. In either way we find that the energy at any instant
is half potential and half kinetic, and that the total amount, per
unit area, is ^gpa2. In other words, the energy of a progressive
wave-system of amplitude a is equal to the work which would be
required to raise a stratum of the fluid, of thickness a, through a
height -J-a.

220. So long as we confine ourselves to a first approximation
all our equations are linear; so that it' (f>l, <£2, ... be the velocity-
potentials of distinct systems of waves of the simple-harmonic
type above considered, then

will be the velocity-potential of a possible form of wave-motion,
with a free surface. Since, when <£ is determined, the equation of
the free surface is given by Art. 216 (5), the elevation above the
mean level at any point of the surface, in the motion given by (1),
will be equal to the algebraic sum of the elevations due to the
separate systems of waves </>x, </>2, ••• Hence each of the latter
systems is propagated exactly as if the others were absent, and
produces its own elevation or depression at each point of the
surface.

We can in this way, by adding together terms of the form given in
Art. 218 (12), with properly chosen values of a, build up an analytical ex
pression for the free motion of the water in an infinitely long canal, due to any
arbitrary initial conditions. Thus, let us suppose that, when £ = 0, the equa
tion of the free surface is

?=/(#), ....................................... (i),

and that the normal velocity at the surface is then F '(#), or, to our order of
approximation,

The value of 0 is found to be

si"/

380 SURFACE WAVES. [CHAP. IX

and the equation of the free surface is

] .............. (iv).

_ -x am fat

These formulae, in which c is a function of k given by Art. 218 (4), may be
readily verified by means of Fourier's expression for an arbitrary function as
a definite integral, viz.

/(#) = - ^ dk If* dX/(A)cos£(X-o?)t ............. (v).

7T J 0 (J -oo J

When the initial conditions are arbitrary, the subsequent motion is made
up of systems of waves, of all possible lengths, travelling in either direction,
each with the velocity proper to its own wave-length. Hence, in general, the
form of the free surface is continually altering, the only exception being when
the wave-length of every component system which is present in sensible
amplitude is large compared with the depth of the fluid. In this case the
velocity of propagation (glifi is independent of the wave-length, so that, if we
have waves travelling in one direction only, the wave-profile remains un
changed in form as it advances, as in Art. 167.

In the case of infinite depth, the formulae (iii), (iv) take the simpler forms

" cos k (X " * sn

............ (vi),

The problem of tracing out the consequences of a limited initial disturbance,
in this case, received great attention at the hands of the earlier investigators
in the subject, to the neglect of the more important and fundamental pro
perties of simple- harmonic trains. Unfortunately, the results, even on the
simplest suppositions which we may make as to the nature and extent of the
original disturbance, are complicated and difficult of interpretation. We shall
therefore content ourselves with the subjoined references, which will enable
the reader to make himself acquainted with what has been achieved in this
branch of the subject*.

* Poisson, " M6moire sur la Theorie des Ondes," Mem. de VAcad. roy. des
Sciences, 1816.

Cauchy, 1. c. ante p. 18.

Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of
any Depth, or in a Dispersive Medium," Proc. Roy. Soc., Feb. 3, 18 7.

W. Burnside, " On Deep- Water Waves resulting from a Limited Original
Disturbance," Proc. Lond. Math. Soc., t. xx., p. 22 (1888).

220-221] EFFECT OF ARBITRARY INITIAL CONDITIONS. 381

221. A remarkable result of the dependence of the velocity of
propagation on the wave-length is furnished by the theory of
group-velocity. It has often been noticed that when an isolated
group of waves, of sensibly the same length, is advancing over
relatively deep water, the velocity of the group as a whole is less
than that of the waves composing it. If attention be fixed on a
particular wave, it is seen to advance through the group, gradually
dying out as it approaches the front, whilst its former position in
the group is occupied in succession by other waves which have
come forward from the rear.

The simplest analytical representation of such a group is
obtained by the superposition of two systems of waves of the same
amplitude, and of nearly but not quite the same wave-length.
The corresponding equation of the free surface will be of the form

77 = a sin k (x — ct) + a sin k' (x — c't)

/I, TJ T[>n Tff n' \ /]f> I Iff

-~ / IV ^"" Iv /VC/ fif L/ , \ * / IV |^ A/

= 2a cos

If k, k' be very nearly equal, the cosine in this expression varies
very slowly with x ; so that the wave -profile at any instant has
the form of a curve of sines in which the amplitude alternates
gradually between the values 0 and 2a. The surface therefore
presents the appearance of a series of groups of waves, separated
at equal intervals by bands of nearly smooth water. The motion
of each group is then sensibly independent of the presence of the
others. Since the interval between the centres of two successive
groups is 27r/(& — k'), and the time occupied by the system in
shifting through this space is %7r/(kc — k'c'), the group- velocity is
(kc — k'c')/(k — k'), or d (kc)/dk, ultimately. In terms of the wave
length X (= 27T/&), the group- velocity is

»

This result holds for any case of waves travelling through a
uniform medium. In the present application we have

(3),

382 SURFACE WAVES. [CHAP. IX

and therefore, for the group-velocity,

dk sinh2M

The ratio which this bears to the wave-velocity c increases as Ich
diminishes, being -| when the depth is very great, and unity when
it is very small, compared with the wave-length.

The above explanation seems to have been first suggested by
Stokes*. The question was attacked from another point of view
by Prof. Osborne Reynolds f, by a calculation of the energy propa
gated across a vertical plane of particles. In the case of infinite
depth, the velocity-potential corresponding to a simple-harmonic
train of waves

r) = a sin k (x — ct) ........................ (5),

is (f> = aceky cos k (x — ct) ..................... (6),

as may be verified by the consideration that for y = 0 we must
have drj/dt = — dfy/dy. The variable part of the pressure is pd(f>/dt,
if we neglect terms of the second order, so that the rate at which
work is being done on the fluid to the right of the plane x is

- f p £ dy = pa"k*c3 sin2 k (x - ct) f

J _oo ClX J —

(7),

since c2 = g/k. The mean value of this expression is %gpa?c. It
appears on reference to Art. 219 that this is exactly one-half of
the energy of the waves which cross the plane in question per
unit time. Hence in the case of an isolated group the supply of
energy is sufficient only if the group advance with half the
velocity of the individual waves.

It is readily proved in the same manner that in the case

* Smith's Prize Examination, 1876. See also Lord Bayleigh, Theory of Sound,
Art. 191.

t " On the Bate of Progression of Groups of Waves, and the Bate at which
Energy is Transmitted by Waves," Nature, t. xvi., p. 343 (1877). Professor
Beynolds has also constructed a model which exhibits in a very striking manner
the distinction between wave-velocity and group-velocity in the case of the
transverse oscillations of a row of equal pendulums whose bobs are connected
by a string,

221] GROUP-VELOCITY. 383

of a finite depth h the average energy transmitted per unit
time is

which is, by (4), the same as

Hence the rate of transmission of energy is equal to the group-
velocity, d(kc)/dk, found independently by the former line of
argument.

These results have a bearing on such questions as the ' wave-
resistance' of ships. It appears from Art. 227, below, in the
two-dimensional form of the problem, that a local disturbance
of pressure advancing with velocity c [< (gh)*] over still water of
depth h is followed by a simple-harmonic train of waves of the
length (27T/&) appropriate to the velocity c, and determined there
fore by (3); whilst the water in front of the disturbance is sensibly
at rest. If we imagine a fixed vertical plane to be drawn in the rear
of the disturbance, the space in front of this plane gains, per unit
time, the additional wave-energy \gpa?c, where a is the amplitude
of the waves generated. The energy transmitted across the plane
is given by (8). The difference represents the work done by the
disturbing force. Hence if R denote the horizontal resistance
experienced by the disturbing body, we have

As c increases from zero to (gh)*, kh diminishes from oc to 0, and
therefore R diminishes from %gpa* to Of.

When c > (gh)*, the water is unaffected beyond a certain small
distance on either side, and the wave-resistance R is then zeroj.

* Lord Kayleigh, " On Progressive Waves," Proc. Lond. Math. Soc., t. ix., p. 21
(1877); Theory of Sound, t. i., Appendix.

t It must be remarked, however, that the amplitude a due to a disturbance of
given character will also vary with c.

+ Cf. Sir W. Thomson " On Ship Waves," Proc. Inst. Hech. Eny., Aug. 3, 1887;
Popular Lectures and Addresses, London, 1889-94, t. iii., p. 450. A formula equi
valent to (10) was given in a paper by the same author, " On Stationary Waves in
Flowing Water," PlriL Mag., Nov. 1886.

384 SURFACE WAVES. [CHAP. IX

222. The theory of progressive waves may be investigated, in
a very compact manner, by the method of Art. 172*.

Thus if (/>, -\/r be the velocity- and stream-functions when the
problem has been reduced to one of steady motion, we assume

(</> -I- i^)/c = - (x + iy} + meik(*+iy) + i/3e-ik(x+iy},

whence <£/c = - x - (oLe~ky - /3eky) sin kx,\ , .

^/c = -y + (ae~ky + /3eky) cos kx }

This represents a motion which is periodic in respect to x, super
posed on a uniform current of velocity c. We shall suppose that
ka and k/3 are small quantities ; in other words that the amplitude
of the disturbance is small compared with the wave-length.

The profile of the free surface must be a stream-line; we will
take it to be the line ^r = 0. Its form is then given by (1), viz. to
a first approximation we have

y = (a + /3) cos kx (2),

shewing that the origin is at the mean level of the surface.
Again, at the bottom (y = — h) we must also have ty = const. ; this
requires

•Pe-kh=0.
The equations (1) may therefore be put in the forms

(j)/c = — x + C cosh k (y + h) sin kx, ) ,« ,

Tfr/c = — y + C sinh A; (y + A) cos &# j

The formula for the pressure is

— = const. — gy — i- \ { -2- I + ( -^~
p \\CUB / \ay

C2

= const. — gy — -= {1 — 2kC cosh k(y + h) cos &#},

neglecting A:202. Since the equation to the stream-line ty = 0 is

y = (7 sinh kh cos &# (4),

approximately, we have, along this line,

- = const. + (kcz coth M — a) y.
P

* Lord Kayleigh, I. c, ante p. 279.

222-223] ARTIFICE OF STEADY MOTION. 385

The condition for a free surface is therefore satisfied, provided

, tanhM

This determines the wave-length (2-7T/&) of possible stationary
undulations on a stream of given uniform depth h, and velocity c.
It is easily seen that the value of kh is real or imaginary
according as c is less or greater than (gh)*.

If, on the other hand, we impress on everything the velocity
— c parallel to a?, we get progressive waves on still water, and (5)
is then the formula for the wave-velocity, as in Art. 218.

When the ratio of the depth to the wave-length is sufficiently great, the
formulae (1) become

leading to ^ = const. -gy-c-{\- 2kpe*v cos lex + k^e^} ... ... (ii).

P •

If we neglect F/32, the latter equation may be written

= const. + (k#-g}y + Tccty ......................... (iii).

Hence if c2 = gjk ....................................... (iv),

the pressure will be uniform not only at the upper surface, but along every
stream-line >//• = const.* This point is of some importance ; for it shews that
the solution expressed by (i) and (iv) can be extended to the case of any
number of liquids of different densities, arranged one over the other in
horizontal strata, provided the uppermost surface be free, and the total depth
infinite. And, since there is no limitation to the thinness of the strata, we
may even include the case of a heterogeneous liquid whose density varies
continuously with the depth.

223. The method of the preceding Art. can be readily
adapted to a number of other problems.

For example, to find the velocity of propagation of waves over
the common horizontal boundary of two masses of fluid which are
otherwise unlimited, we may assume

'
where the accent relates to the upper fluid. For these satisfy the

* This conclusion, it must be noted, is limited to the case of infinite depth.
It was first remarked by Poisson, I.e. ante, p. 380.

L. 25

SURFACE WAVES. [CHAP. IX

condition of irrotational motion, V2-^r = 0 ; and they give a uniform
velocity c at a great distance above and below the common surface,
at which we have -fy = ^r' = 0, say, and therefore y = /3 cos kx, ap
proximately.

The pressure-equations are

*• = const. — gy — -~ (1 — 2&/3e^ cos kx),

$ c2

— , = const. — gy — -= (1 + 2kj3e~ky cos lex),

which give, at the common surface,

p/p = const. -(g-kc*)y,
p'lp = const. -(g + kc2) y,

the usual approximations being made. The condition p=p' thus
leads to

a result first obtained by Stokes.

The presence of the upper fluid has therefore the effect of
diminishing the velocity of propagation of waves of any given
length in the ratio {(1 — s)/(l + s)}i, where s is the ratio of the
density of the upper to that of the lower fluid. This diminution
has a two-fold cause ; the potential energy of a given deformation
of the common surface is diminished, whilst the inertia is in
creased. As a numerical example, in the case of water over
mercury (s~1 = 13'6) the above ratio comes out equal to '929.

It is to be noticed, in this problem, that there is a disconti
nuity of motion at the common surface. The normal velocity
(d^rjdx) is of course continuous, but the tangential velocity
(— dty/dy) changes from c(l—k/3 cos Tex) to c(l + k/3 cos kx) as we
cross the surface ; in other words we have (Art. 149) a vortex-sheet
of strength — kc/3 cos kx. This is an extreme illustration of the
remark, made in Art. 18, that the free oscillations of a liquid of
variable density are not necessarily irrotational.

If p < />', the value of c is imaginary. The undisturbed
equilibrium -arrangement is then of course unstable.

223] OSCILLATIONS OF SUPERPOSED FLUIDS. 387

The case where the two fluids are bounded by rigid horizontal planes
y=—h, y = h', is almost equally simple. We have, in place of (1),

sinh k (y + A)

^--y-g™^"^.****
smh kK

leading to c2 = ?. ., .f — ^ — ,. 77>t. ...(ii).

£ p coth M -Hp coth kh'

When M and M' are both very great, this reduces to the form (2). When kh'
is large, and kh small, we have

(iii),

the main effect of the presence of the upper fluid being now the change in
the potential energy of a given deformation.

When the upper surface of the upper fluid is free, we may assume

ABinh£(v+A) 7 \

•bc= -y+p -- . .y.T 'cos&g, I

(iv),
!//•'/<? = — y + (/3 cosh ky + y sinh &y) cos &*? /

and the conditions that ^ = \^', ^>=j0' at the free surface then lead to the
equation

c4 (p coth kh coth M' + p') - c2P (coth kh' + coth M) | + (p - p') ^ = 0 ...... (v).

Since this is a quadratic in c2, there are two possible systems of waves of any
given length (2?r/^). This is as we should expect, since when the wave-length
is prescribed the system has virtually two degrees of freedom, so that there
are two independent modes of oscillation about the state of equilibrium. For
example, in the extreme case where p'/p is small, one mode consists mainly in
an oscillation of the upper fluid which is almost the same as if the lower fluid
were solidified, whilst the other mode may be described as an oscillation of the
lower fluid which is almost the same as if its upper surface were free.

The ratio of the amplitudes at the upper and lower surfaces is found
to be

kc*
kc* cosh kh'-g sinh kh'

Of the various special cases that may be considered, the most interesting
is that in which kh is large ; i. e. the depth of the lower fluid is great compared
with the wave-length. Putting coth kh = 1, we see that one root of (v) is now

c2=#/£ ..................................... (vii),

exactly as in the case of a single fluid of infinite depth, and that the ratio of
the amplitudes is e*A'. This is merely a particular case of the general result
stated at the end of Art. 222 ; it will in fact be found on examination that

25—2

388 SURFACE WAVES. [CHAP. IX

there is now no slipping at the common boundary of the two fluids. The
second root of (v) is

c2- p~p> - $- (viii)'

~' '*" ...ivrnj,

and for this the ratio (vi) assumes the value

»' ............... (ix).

If in (viii) and (ix) we put &A' = oo , we fall back on a former case ; whilst if we
make kh' small, we find

£-(l-0fV (x),

and the ratio of the amplitudes is

.(xi).

These problems were first investigated by Stokes*. The case of any
number of superposed strata of different densities has been treated by Webbf
and Greenhill|. For investigations of the possible rotational oscillations in a
heterogeneous liquid the reader may consult the papers cited below §.

224. As a further example of the method of Art. 222, let us
suppose that two fluids of densities p, p, one beneath the other,
are moving parallel to x with velocities U, U', respectively, the
common surface (when undisturbed) being of course plane and
horizontal. This is virtually a problem of small oscillations about
a state of steady motion.

The fluids being supposed unlimited vertically, we assume, for
the lower fluid

Tjr = - U [y - $eky cos kx\ (1),

and for the upper fluid

(2),

* " On the Theory of Oscillatory Waves," Carnb. Trans, t. viii. (1847) ; Math,
and Phys. Papers, t. i., pp. 212—219.

t Math. Tripos Papers, 1884.

$ "Wave Motion in Hydrodynamics," Amer. Journ. Math., t. ix. (1887).

§ Lord Kayleigh, "Investigation of the Character of the Equilibrium of an
Incompressible Heavy Fluid of Variable Density." Proc. Lond. Math. Soc., t. xiv.,
p. 170 (1883).

Burnside, "On the small Wave-Motions of a Heterogeneous Fluid under Gravity."
Proc. Lond. Math. Soc., t. xx., p. 392 (1889).

Love, "Wave-Motion in a Heterogeneous Heavy Liquid." Proc. Lond. Math.
Soc., t. xxii., p. 307 (1891).

223-224] EFFECT OF WIND ON WAVE-VELOCITY. 389

the origin being at the mean level of the common surface, which
is assumed to be stationary, and to have the form

y — /3cos kx (3).

The pressure-equations give

2 = const. — qy — ^ U2 (1 — 2k8eky cos kx\

P-, = const. — gy — \ U'2 (1 + 2kpe~ky cos kx)
P

whence, at the common surface,

= const.
p

= const. - (k U'2 + g)
P

Since we must have p=p' over this surface, we get

PU* + p'U* = £(P-p') ..................... (6).

This is the condition for stationary waves on the common
surface of the two currents U, V.

If we put U' = U, we fall back on the case of Art. 223. Again
if we put

17= -c, lP = -c + u,

we get the case where the upper fluid has a velocity u relative to
the lower ; c then denotes the velocity (relative to the lower fluid)
of waves on the common surface. An interesting application of
this is to the effect of wind on the velocity of water-waves.

The equation (6) now takes the form

or, if we write s for p'/p, and put c0 for the wave- velocity in the
absence of wind, as given by Art. 223 (2),

2su su2

-,

The roots of this quadratic in c are

su

390 SURFACE WAVES. [CHAP. IX

These are both real, provided

«<^.Co (10),

and they have, moreover, opposite signs, if

c» (ID-

In this latter case waves of the prescribed length (2-7T/&) may
travel with or against the wind, but the velocity is greater with
the wind than against it. If u lie between the limits (10) and (11),
waves of the given length cannot travel against the wind. Finally
when u exceeds the limit (10), the values of c are imaginary. This
indicates that the plane form of the common surface is now un
stable. Any disturbance whose wave-length is less than

(12)

tends to increase indefinitely.

Hence, if there were no modifying circumstances, the slightest
breath of wind would suffice to ruffle the surface of water. We
shall give, later, a more complete investigation of the present
problem, taking account of capillary forces, which act in the
direction of stability.

It appears from (6) that if p = p', or if g = 0, the plane form of
the surface is unstable for all wave-lengths.

These results illustrate the statement, as to the instability of
surfaces of discontinuity in a liquid, made in Art. 80*.

When the currents are confined by fixed horizontal planes y= -h, y = h',
we assume

I S_inlO^+A) I ..................

|f smh kh j

sn — 7

(n).

The condition for stationary waves on the common surface is then found
to be

(iii)f.

* This instability was first remarked by von Helmholtz, I.e. ante, p. 24.
t Greenhill, Lc. ante, p. 388.

224-225] SUKFACES OF DISCONTINUITY. . 391

It appears on examination that the undisturbed motion is stable or
unstable, according as

< p coth kh+p coth kh'

> (pp' coth kh coth kh'}%

where u is the velocity of the upper current relative to the lower, and c0 is
the wave-velocity when there are no currents (Art. 223 (ii)). When h and h'
both exceed half the wave-length, this reduces practically to the former
result (10).

225. These questions of stability are so important that it is
worth while to give the more direct method of treatment*.

If </> be the velocity-potential of a slightly disturbed stream
flowing with the general velocity U parallel to x, we may write

</> = - t/"tf + <k ........................... (1),

where fa is small. The pressure-formula is, to the first order,

and the condition to be satisfied at a bounding surface y — 77, where

7? is small, is

dri rjdrj_ dfa
drUdx~~~fy ......

To apply this to the problem stated at the beginning of Art.
224, we assume, for the lower fluid,

^ = ae«y+*<te+rf) ........................ (4);

for the upper fluid

<j)l' = C'e-ky+Wx+^ ..................... (5);

with, as the equation of the common surface,

The continuity of the pressure at this surface requires, by (2),
p{i(<r + kU)C-ga}=p'{i(<r + kU')C'-ga} ...... (7);

whilst the surface-condition (3) gives

= -kC,\

' ....................

* Sir W. Thomson, " Hydrokinetic Solutions and Observations," Phil. Mag.,
Nov. 1871; Lord Kayleigh, "On the Instability of Jets," Proc. Lond. Math. Soc.t
t. x., p. 4 (1878).

392 SURFACE WAVES. [CHAP. IX

Eliminating a, G, C", we get

p(<r + kU)* + p'(o- + kUy = gk(p-p') ............ (9).

It is obvious that whatever the values of U, V, other than
zero, the values of a will become imaginary when k is sufficiently
great.

Nothing essential is altered in the problem if we impress on
both fluids an arbitrary velocity in the direction of x. Hence,
putting U = 0, U' = u, we get

p<T* + P'(<r + ku)* = gk(p-p') ............... (10),

which is equivalent to Art. 224 (7).

If p = p, it is evident from (9) that a will be imaginary for all
values of k. Putting U' = — U} we get

o-=±ikU ........................... (11).

Hence, taking the real part of (6), we find

<n = ae±kutcoskx ........................ (12).

The upper sign gives a system of standing waves whose height
continually increases with the time, the rate of increase being
greater, the shorter the wave-length.

The case of p~p, with U=U', is of some interest, as illustrating the
flapping of sails and flags*. We may conveniently simplify the question by
putting U=U'=0', any common velocity may be superposed afterwards if
desired.

On the present suppositions, the equation (9) reduces to o-2=0. On
account of the double root the solution has to be completed by the method
explained in books on Differential Equations. In this way we obtain the two
independent solutions

and

The former solution represents a state of equilibrium ; the latter gives a
system of stationary waves with amplitude increasing proportionally to the
time.

In this problem there is no physical surface of separation to begin with ;
but if a slight discontinuity of motion be artificially produced, e.g. by impulses

* Lord Kayleigh, I.e.

225-226] INSTABILITY. 393

applied to a thin membrane which is afterwards dissolved, the discontinuity
will persist, and, as we have seen, the height of the corrugations will continu
ally increase.

The above method, when applied to the case where the fluids are confined
between two rigid horizontal planes y=—h,y=kf, leads to

p(a- + kU'}2cot'hkk+p'(a-+kU')2cothkk'^gk(p-p) (iii),

which is equivalent to Art. 224 (iii).

We may next calculate the effect of an arbitrary, but
steady, application of pressure to the surface of a stream flowing
with uniform velocity c in the direction of x positive*.

It is to be noted that, in the absence of dissipative forces, this
problem is to a certain extent indeterminate, for on any motion
satisfying the prescribed pressure-conditions we may superpose a
train of free waves, of arbitrary amplitude, whose length is such
that their velocity relative to the water is equal and opposite to
that of the stream, arid which therefore maintain a fixed position
in space.

To remove this indeterminateness we may suppose that the
deviation of any particle of the fluid from the state of uniform
flow is resisted by a force proportional to the relative velocity.
This law of resistance is not that which actually obtains in nature,
but it will serve to represent in a rough way the effect of small
dissipative forces ; and it has the great mathematical convenience
that it does not interfere with the irrotational character of the
motion. For if we write, in the equations of Art. 6,

X = —fj,(u — c), Y=—g — fj,v, Z= — p,w ......... (1),

where the axis of y is supposed drawn vertically upwards, and c
denotes the velocity of the stream, the investigation of Art. 34,
when applied to a closed circuit, gives

(2),

whence / (udx + vdy + wdz) = Ce~^ ...... ........... (3).

* The first steps of the following investigation are adapted from a paper by
Lord Rayleigh, "On the Form of Standing Waves on the Surface of Eunning
Water," Proc. Lond. Math. Soc., t. xv., p. 69 (1883), being simplified by the
omission, for the present, of all reference to Capillarity. The definite integrals
involved are treated, however, in a somewhat more general manner, and the
discussion of the results necessarily follows a different course.

394 SURFACE WAVES. [CHAP. IX

Hence the circulation in a circuit moving with the fluid, if once
zero, is always zero.

If (f> be the velocity-potential, the equations of motion have
now the integral

k2 ............... (4),

this being, in fact, the form assumed by Art. 21 (4) when we write
n = gy-p(cx + <l>) ..................... (5),

in accordance with (1) above.

To calculate, in the first place, the effect of a simple-harmonic
distribution of pressure we assume

y sin kx, \

f .................. W'

The equation (4) becomes, on neglecting as usual the square
of k/3,

2 = ... - gy + fi^y (&c2 cos kx + pc sin kx) ......... (7).

This gives for the variable part of the pressure at the upper
surface (i|r = 0)

— £ = (3 [(kd1 — a) cos kx + pc sin kx} ............ (8),

P

which is equal to the real part of

If we equate the coefficient to P, we may say that to the pressure

7 = ^** (9>

corresponds the surface-form

y = kd,_P _i ceikx (io).

Hence taking the real parts, we find that the surface-pressure

A/n

(11)

P

produces the wave-form

p (kc* — g) cos kx — /AC sin kz

226-227] SIMPLE-HARMONIC APPLICATION OF PRESSURE. 395

If we write ic=g/c2, so that ZTT/K is the wave-length of the
free waves which could maintain their position in space against
the flow of the stream, the last formula may be written
P (k — /c) cos kx — /^ sin kx

where /^ = //./c.

This shews that if /u, be small the wave-crests will coincide in position with
the maxima, and the troughs with the minima, of the applied pressure, when
the wave-length is less than 2?r/< ; whilst the reverse holds in the opposite
case. This is in accordance with a general principle. If we impress on
every thing a velocity -c parallel to #, the result obtained by putting /z1 = 0 in
(13) is seen to be merely a special case of Art. 165 (12).

In the critical case of k = K, we have

y= -- . sin&£,

fiC

shewing that the excess of pressure is now on the slopes which face down the
stream. This explains roughly how a system of progressive waves may be
maintained against our assumed dissipative forces by a properly adjusted
distribution of pressure over their slopes.

227. The solution expressed by (13) may be generalized, in
the first place by the addition of an arbitrary constant to xt and
secondly by a summation with respect to k. In this way we may
construct the effect of any arbitrary distribution of pressure, say

using Fourier's expression

f(x) = -\ dk( f(\)cosk(x-\)d\ ........ (15).

7T J o J -oo *

It will be sufficient to consider the case where the imposed
pressure is confined to an infinitely narrow strip of the surface,
since the most general case can be derived from this by in
tegration. We will suppose then that f(\) vanishes for all but
infinitely small values of X, so that (15) becomes

a>)=- dkcoskx.i f(\)d\ ......... (16)*.

TTJo J_oo

teness of this expression may be avoided
a*L» 1 f e-*kdk I f(\)cosk(x-\)d\

* The indeterminateness of this expression may be avoided by the temporary use
of Poisson's formula

in place of (15).

396 SURFACE WAVES. [CHAP. IX

Hence in (13) we must replace P by Q/jr.&k, where

and integrate with respect to k between the limits 0 and oo ; thus

If we put £=k + im, where k, m are taken to be the rectangular coordinates
of a variable point in a plane, the properties of the expression (18) are
contained in those of the complex integral

:«*£.

It is known (Art. 62) that the value of this integral, taken round the
boundary of any area which does not include the singular point (£=c), is zero.
In the present case we have c = K + i^ , where K and /^ are both positive.

Let us first suppose that x is positive, and let us apply the above theorem
to the region which is bounded externally by the line m = 0 and by an infinite
semicircle, described with the origin as centre on the side of this line for
which m is positive, and internally by a small circle surrounding the point
(*, JAJ). The part of the integral due to the infinite semicircle obviously

vanishes, and it is easily seen, putting £— c = reld, that the part due to the small
circle is

if the direction of integration be chosen in accordance with the rule of Art. 33.
We thus obtain

dk _ V«-MI)* = 0

which is equivalent to

rpikx C™ p~ikx

, * . .dk = 27riel{K+t^x+
*-(K+«ft) 70

On the other hand, when x is negative we may take the integral (i) round
the contour made up of the line m = 0 and an infinite semicircle lying on the
side for which m is negative. This gives the same result as before, with the
omission of the term due to the singular point, which is now external to the
contour. Thus, for x negative,

p-ikx

An alternative form of the last term in (ii) may be obtained by integrating
round the contour made up of the negative portion of the axis of &, and the

227] EFFECT OF A LINE OF PRESSURE. 397

pti

/;

positive portion of the axis of m, together with an infinite quadrant. We
thus find

which is equivalent to

p-mx

- \dk= -r-rdm .................. (iv).

-

This is for x positive. In the case of K negative, we must take as our
contour the negative portions of the axes of £, m, and an infinite quadrant.
This leads to

re-ikx /•» emx

, f . .dk = \ — - - -dm .................. (v),
m *+(K-Hpi) J Q m + ^-iK

as the transformation of the second member of (iii).

In the foregoing argument /zx is positive. The corresponding results for
the integral

are not required for our immediate purpose, but it will be convenient to state
them for future reference. For x positive, we find

/•» eikx r e~** r e~mx

I TT-, - — \dk= \ T-r-f - ^dk= \ - -dm ... (vii):
/o *-(«-«*«l) 7o^ + («-^i) Jo m+^ + iK

whilst, for x negative,

= _ 2,^^^+ r . •"*'. N<ft

J0 *+5-53

" ^^ dm ...... (viii).

--

The verification is left to the reader.

If we take the real parts of the formulae (ii), (iv), and (iii), (v), respectively,
we obtain the results which follow.

The formula (18) is equivalent, for x positive, to
^ . y = - **«-*> sin «, + /" (t + *)(™%

~ (19),

'°° (k + fc) cos A;^ — yu,i sin A;a? ,
-7 dk

and, for x negative, to
7TC2

Q 'y~ (20)-

398 SURFACE WAVES. [CHAP. IX

The interpretation of these results is simple. The first term of
(19) represents a train of simple-harmonic waves, on the down
stream side of the origin, of wave-length 2?rc2/^, with amplitudes
gradually diminishing according to the law e~^x. The remaining
part of the deformation of the free-surface, expressed by the
definite integals in (19) and (20), though very great for small
values of #, diminishes very rapidly as x increases, however small
the value of the frictional coefficient /j^.

When /*! is infinitesimal, our results take the simpler forms
7TC2 ~ . f00 cos kx

f °° rfliQ—mx

— — 2vr sin KX + I — 2 c?m (21),

Jo w + #

for a? positive, and

7TC2 pcos&tf 77 f00 raema; , /rtox

T-y = l. ^T7*=J0^+7^m (22)'

for a? negative. The part of the disturbance of level which is
represented by the definite integrals in these expressions is now
symmetrical with respect to the origin, and diminishes constantly
as the distance from the origin increases. It is easily found, by
usual methods, that when KX is moderately large

me~mx d 1 _3J_ 5!

Q /it \ fC /C JU tC JU rC Ju

the series being of the kind known as 'semi-convergent.' It
appears that at a distance of about half a wave-length from the
origin, on the down-stream side, the simple-harmonic wave-system
is fully established.

The definite integrals in (21) and (22) can be reduced to known functions
as follows. If we put (&+K) x=u, we have, for x positive,

r

cos kx 77 /"* cos (KX - u) _
-; -- dk= I - 2 - Lfo

k+* f« u

= - Ci KX cos KX + (far — Si KX} sin KX ...... (ix),

where, in the usual notation,

n. f00 cosu , 0. fu siuu ,

(J1U—-I - du. Siu= I --- du ............... (x).

ju U Jo U '

227] DISCUSSION OF THE INTEGRALS. 399

The functions Ci u and Si u have been tabulated by Glaisher*. It appears
that as u increases from zero they tend very rapidly to their asymptotic values
0 and ^?r, respectively. For small values of u we have

where y is Euler's constant -5772....

The expressions (19), (20) and (21), (22) alike make the eleva
tion infinite at the origin, but this difficulty disappears when the
pressure, which we have supposed concentrated on a mathematical
line of the surface, is diffused over a band of finite breadth. In
fact, to calculate the effect of a distributed pressure, it is only
necessary to write x— x' for x, in (21) and (22), to replace Q by
Ap/p . §x', where Ap/p is any given function of #', and to in
tegrate with respect to of between the proper limits. It follows
from known principles of the Integral Calculus that, if Ap be
finite, the resulting integrals are finite for all values of x.

If we write x (u) = Ciu sin u + (-|7r — Si'w) cos u (xii)j

it is easily found from (19) and (20) that, when ^ is infinitesimal, we have, for
positive values of #,

no T00

— / ydk— — Sroos*r+x(«i?) (xiii).

V J x

and for negative values of x

-n-x-Kx} (xiv).

In particular, the integral depression of the free surface is given by

and is therefore independent of the velocity of the stream.

By means of a rough table of the function x (u\ it is easy to construct the
wave-profile corresponding to a uniform pressure applied over a band of any
given breadth. It may be noticed that if the breadth of the band be an
exact multiple of the wave-length (2»r/ic), we have zero elevation of the surface
at a distance, on the down-stream as well as on the up-stream side of the
seat of disturbance.

* " Tables of the Numerical Values of the Sine-Integral, Cosine-Integral, and
Exponential-Integral," Phil Trans., 1870. The expression of the last integral in
(22) in terms of the sine- and cosine-integrals, was obtained, in a different manner

from the above, by Schlomilch, " Sur l'inte"grale d6finie / dd e~X& " Crelle

'« 6'2 + a*

t. xxxiii. (1846) ; see also De Morgan, Differential and Integral Calculus, London
1842, p. 654.

400

SURFACE WAVES.

[CHAP, ix

227-228] FORM OF THE WAVE-PROFILE. 401

The figure on p. 400 shews, with a somewhat extravagant vertical scale,
the case where the band (AB) has a breadth K~I, or '159 of the length of a
standing wave.

The circumstances in any such case might be realized approximately by
dipping the edge of a slightly inclined board into the surface of a stream,
except that the pressure on the wetted area of the board would not be uniform,
but would diminish from the central parts towards the edges. To secure
a uniform pressure, the board would have to be curved towards the edges, to
the shape of the portion of the wave-profile included between the points
A, B in the figure.

If we impress on everything a velocity — c parallel to x, we
get the case of a pressure-disturbance advancing with constant
velocity c over the surface of otherwise still water. In this form
of the problem it is not difficult to understand, in a general way,
the origin of the train of waves following the disturbance. It is
easily seen from the theory of forced oscillations referred to in Art.
165 that the only motion which can be maintained against small
dissipative forces will consist of a train of waves of the velocity
c, equal to that of the disturbance, and therefore of the wave
length 2-7rc2/(7. And the theory of 'group-velocity' explained in
Art. 221 shews that a train of waves cannot be maintained ahead
of the disturbance, since the supply of energy is insufficient.

228. The main result of the preceding investigation is that a
line of pressure athwart a stream flowing with velocity c produces
a disturbance consisting of a train of waves, of length 27rc2/#,
lying on the down-stream side. To find the effect of a line of
pressure oblique to the stream, making (say) an angle JTT — 6 with
its direction, we have only to replace the velocity of the stream
by its two components, c cos 6 and c sin 6, perpendicular and
parallel to the line. If the former component existed alone, we
should have a train of waves of length 2?rc2/# . cos2 0, and the
superposition of the latter component does not affect the con
figuration. Hence the waves are now shorter, in the ratio cos2 0:1.
It appears also from Art. 227 (21) that, for the same integral
pressure, the amplitude is greater, varying as sec2 6, but against
this must be set the increased dissipation to which the shorter
waves are subject*.

To infer the effect of a pressure localized about a point of the

* On the special hypothesis made above this is indicated by the factor e~Ml<r in
Art. 227 (19), where AC^/A/C . sec0.

I*. 26

402

SURFACE WAVES.

[CHAP, ix

surface, we have only to take the mean result of a series of lines
of pressure whose inclinations 6 are distributed uniformly between
the limits + JTT*. This result is expressed by a definite integral
whose interpretation would be difficult ; but a general idea of the
forms of the wave-ridges may be obtained by a process analogous
to that introduced by Huyghens in Physical Optics, viz. by tracing
the envelopes of the straight lines which represent them in the
component systems. It appears on reference to (21) that the
perpendicular distance p of any particular ridge from the origin
is given by

«p = (2«-i)lT,

where s is integral, and K = g/c* cos2 6. The tangential polar equa
tion of the envelopes in question is therefore

p = acos*0 (1),

where, for consecutive crests or hollows, a differs by 27rc2/#. The

forms of the curves are shewn in the annexed figure, traced from
the equations

x =jocos 6 — -^sin0= \ a (5 cos 6 — cos SB), }

[ (2).

y = p sin 0 + -A cos 6 = — J a (sin 6 + sin 30)

etc/ /

* This artifice is taken from Lord Rayleigh's paper, cited on p. 393.

228]

SHIP-WAVES.

403

this

The values of x and y are both stationary when sin2 9 =
gives a series of cusps lying on the straight lines

y\x = ± 2-» = ± tan-1 19° 28' *.

We have here an explanation, at all events as to the main
features, of the peculiar system of waves which accompanies a ship
in sufficiently rapid motion through the water. To an observer on
board the problem is of course one of steady motion ; and although
the mode of disturbance is somewhat different, the action of the
bows of the ship may be roughly compared to that of a pressure-
point of the kind we have been considering. The preceding
figure accounts clearly for the two systems of transverse and
diverging waves which are in fact observed, and for the specially
conspicuous 'echelon5 waves at the cusps, where these two systems
coalesce. These are well shewn in the annexed drawing f by
Mr. R. E. Froude of the waves produced by a model.

A similar system of waves is generated at the stern of the
ship, which may roughly be regarded as a negative pressure-point.

* Of. Sir W. Thomson, " On Ship Waves," Proc. Inst. Mech. Eng., Aug. 3, 1887,
Popular Lectures, t. iii., p. 482, where a similar drawing is given. The investigation
there referred to, based apparently on the theory of ' group-velocity,' has unfortu
nately not been published. See also E. E. Froude, "On Ship Eesistance," Papers
of the Greenock Phil. Soc., Jan. 19, 1894.

t Copied, by the kind permission of Mr Froude and the Council of the Institute
of Naval Architects, from a paper by the late W. Froude, "On the Effect on the
Wave-Making Kesistance of Ships of Length of Parallel Middle Body," Trans. Inst.
Nov. Arch,, t. xvii. (1877).

26—2

404 SURFACE WAVES. [CHAP. IX

With varying speeds of the ship the stern-waves may tend parti
ally to annul, or to exaggerate, the effect of the bow-waves, and
consequently the wave-resistance to the ship as a whole for a
given speed may fluctuate up and down as the length of the ship
is increased*.

229. If in the problem of Art. 226 we suppose the depth to be
finite and equal to h, there will be, in the absence of dissipation,
indeterminateness or not, according as the velocity c of the stream
is less or greater than (gh)%, the maximum wave-velocity for the
given depth. See Art. 222. The difficulty presented by the former
case can be evaded as before by the introduction of small frictional
forces ; but it may be anticipated from the investigation of Art. 227
that the main effect of these will be to annul the elevation of the
surface at a distance on the up-stream side of the region of
disturbed pressure f, and if we assume this at the outset we need
not complicate our equations by retaining the frictional terms.

For the case of a simple- harmonic distribution of pressure we
assume

<£/c = — x 4- 0 cosh k (y + h) sin lex, ] ,^.

^fr/c = — y + ft sinh k (y 4- h) cos lex}"
as in Art. 222 (3). Hence, at the surface

y = ft sinh kh cos lex ..................... (2),

we have

- = — gy — \ (q- — c2) = /3 (kc~ cosh kh — g sinh Mi) cos kx ... (3),
so that to the imposed pressure

cosfcp., ......... (4),

P
will correspond the surface-form

T-. sinh kh

-,

— , , COS KX

0 , ,

KG- cosh kh—g sinh ten

* See W. Fronde, I.e., and B. E. Froude, " On the Leading Phenomena of the
Wave-Making Eesistance of Ships," Trans. Inst. Nav. Arch., t. xxii. (1881), where
drawings of actual wave-patterns under varied conditions of speed are given, which
are, as to their main features, in striking agreement with the results of the above
theory. Rome of these drawings are reproduced in Lord Kelvin's paper in the Proc.
Inst. Mech. Eng., above cited.

t There is no difficulty in so modifying the investigation as to take the frictional
forces into account, when these are very small,

228-229] CASE OF FINITE DEPTH. 405

As in Art. 226, the pressure is greatest over the troughs, and
least over the crests, of the waves, or vice versa, according as the
wave-length is greater or less than that corresponding to the velo
city c, in accordance with general theory.

The generalization of (5) by Fourier's method gives, with the help and in
the notation of Art. 227 (15) and (17),

' sinh kh cos kx ,,
kc2coshkh-gsmhkh W)

as the representation of the effect of a pressure of integral amount pQ applied
to a narrow band of the surface at the origin. This may be written

7TC2 r00 cos(#w/A) ,

— .11= \ — jrr — L-TTJ du (11).

Q * J Q ucothu-gh/c*

Now consider the complex integral

where £=u + iv. The function under the integral sign has a singular point at
£= + 100 , according as x is positive or negative, and the remaining singular
points are given by the roots of

(iv).

Since (i) is an even function of x, it will be sufficient to take the case of x
positive.

Let us first suppose that c2 > gh. The roots of (iv) are then all pure
imaginaries ; viz. they are of the form + i/3, where /3 is a root of

(v).

The smallest positive root of this lies between 0 and |TT, and the higher roots
approximate with increasing closeness to the values (S + ^)TT, where s is integral.
We will denote these roots in order by /30, /31} /32,.... Let us now take the
integral (iii) round the contour made up of the axis of u, an infinite semicircle
on the positive side of this axis, and a series of small circles surrounding the
singular points C=fc/30> *|8i, ^v--- The part due to the infinite semicircle
obviously vanishes. Again, it is known that if a be a simple root o
the value of the integral

taken in the positive direction round a small circle enclosing the point £=a is
equal to

Forsyth, Theory of Functions, Art. 24.

406 SURFACE WAVES. [CHAP. IX

Now in the case of (iii) we have,

......... (vii),

whence, putting a = 2&, the expression (vi) takes the form

(viii),
where JBS= - -& - -j- ....................... (ix).

The theorem in question then gives

L

Oixu/h /•« &ixulh

- -du+

u coth u — gh/c2 J Q u coth u — gh/c2

If in the former integral we write - u for u, this becomes

o u coth u -
The surface-form is then given by

It appears that the surface-elevation (which is symmetrical with respect
to the origin) is insensible beyond a certain distance from the seat of disturb

When, on the other hand, c2 < gk, the equation (iv) has a pair of real roots
( + a, say), the lowest roots ( + /30) of (v) having now disappeared. The integral
(ii) is then indeterminate, owing to the function under the integral sign
becoming infinite within the range of integration. One of its values, viz. the
' principal value,' in Cauchy's sense, can however be found by the same method
as before, provided we exclude the points £= +a from the contour by drawing
semicircles of small radius e round them, on the side for which v is positive.
The parts of the complex integral (iii) due to these semicircles will be

f(±«Y

where /' (a) is given by (vii) ; and their sum is therefore equal to

where A= , a , ... (xiv).

2_gh(ghL

The equation corresponding to (xi) now takes the form

j (a~e + f 1 ^^...du^-nA *maxlh + n^B8e-Wh. ..(xv),
(Jo Ja+J ucothu-gh/c2 *\ ^ h

229-230] REDUCTION OF THE INTEGRALS. 407

so that, if we take the principal value of the integral in (ii), the surface-form
on the side of .^-positive is

(xvi).

Hence at a distance from the origin the deformation of the surface consists
of the simple-harmonic train of waves indicated by the first term, the wave
length ZTrhja being that corresponding to a velocity of propagation c relative
to still water.

Since the function (ii) is symmetrical with respect to the origin, the
corresponding result for negative values of x is

(xvii).

The general solution of our indeterminate problem is completed by adding
to (xvi) and (xvii) terms of the form

Ccosax/h 4- Dsin axfh ...... ... ............... (xviii).

The practical solution including the effect of infinitely small dissipative
forces is obtained by so adjusting these terms as to make the deformation of
the surface insensible at a distance on the up-stream side. We thus get,
finally, for positive values of x,

(xix),
and, for negative values of #,

For a different method of reducing the definite integral in this problem we
must refer to the paper by Lord Kelvin cited below.

230. The same method can be employed to investigate the
effect on a uniform stream of slight inequalities in the bed.*

Thus, in the case of a simple-harmonic corrugation given by

y = — h-\- 7 cos kx (1),

the origin being as usual in the undisturbed surface, we assume
(f)/c = — % + (& cosh ky -f ft sinh ky) sin kx,

(2)
— — y + (a sinh ky + ft cosh ky) cos kx { "

The condition that (1) should be a stream-line is

7 = — a sinh kh + ft cosh kh (3).

* Sir W. Thomson, " On Stationary Waves in Flowing Water," Phil. Mag., Oct.
Nov. and Dec. 1886, and Jan. 1887.

408 SURFACE WAVES. [CHAP. IX

The pressure-formula is

'D

- = const. —gy + kc2 (a cosh ky + /3 sinh ky) cos kx ... (4),
approximately, and therefore along the stream-line -\Jr = 0
- = const, -f (kcza — g(3) cos kx,

so that the condition for a free surface gives

kc*OL-g@ = 0 ........................... (5).

The equations (3) and (5) determine a and j3. The profile of the
free surface is then given by
y = /3 cos kx

= — T-TT - 77-^ — • u / 1 cos&# ............ (6).

cosh &n — $r/A;c2 . Sinn &/&

If the velocity of the stream be less than that of waves in still
water of uniform depth h, of the same length as the corrugations,
as determined by Art. 218 (4), the denominator is negative, so
that the undulations of the free surface are inverted relatively to
those of the bed. In the opposite case, the undulations of the
surface follow those of the bed, but with a different vertical scale.
When c has precisely the value given by Art. 218 (4), the solution
fails, as we should expect, through the vanishing of the denomi
nator. To obtain an intelligible result in this case we should be
compelled to take special account of dissipative forces.

The above solution may be generalized, by Fourier's Theorem, so as to
apply to the case where the inequalities of the bed follow any arbitrary law.
Thus, if the profile of the bed be given by

^- I dk(

J 0 •* — >o

that of the free surface will be obtained by superposition of terms of the type
6) due to the various elements of the Fourier-integral ; thus

1 r ji r
=-\ dk \
TT J 0 J _

/(A)cos£(#-A)

y=- u/ //9 -T_/i

TT _00 cosh kh - kc* . smh kh

In the case of a single isolated inequality at the point of the bed verti
cally beneath the origin, this reduces to

_ ^

cosh kh - kc* . sinh kh

/;

— r ~ 9-—- u—

0 u cosh u-ghjc*. smh u

,(iii),

2:30-231] INEQUALITIES IN THE BED OF A STREAM. 409

where Q represents the area included by the profile of the inequality above the
general level of the bed. For a depression Q will of course be negative.

The discussion of the integral

...(iv)

I

£ cosh f - cfh/c2 . sinh £

can be conducted exactly as in the last Art. The function to be integrated
differs in fact only by the factor £/sinh £ ; the singular points therefore are the
same as before, and we can at once write down the results.

Thus when c2 > gh we find, for the surface-form,

<v>>

the upper or the lower sign being taken according as x is positive or negative.
When c2 < gh, the ' practical' solution is, for x positive,

*i sm £,
and, for x negative,

The symbols a, /3g, At BB have here exactly the same meanings as in
Art. 230*

Waves of Finite Amplitude.

231. The restriction to ' infinitely small ' motions, in the
investigations of Arts. 216,... implies that the ratio (a/\) of
the maximum elevation to the wave-length must be small. The
determination of the wave- forms which satisfy the conditions of
uniform propagation without change of type, when this restric
tion is abandoned, forms the subject of a classical research by
Sir G, Stokesf.

The problem is, of course, most conveniently treated as one
of steady motion. If we neglect small quantities of the order

* A very interesting drawing of the wave-profile produced by an isolated in
equality in the bed is given in Lord Kelvin's paper, Phil. Mag., Dec. 1886.

+ "On the theory of Oscillatory Waves," Cavrib. Trans., t. viii. (1847); reprinted,
with a "Supplement," Math, and Phys. Papers, t. i., pp. 197, 314.

The outlines of a more general investigation, including the case of permanent
waves on the common surface of two horizontal currents, have been given by
von Helmholtz, " Zur Theorie von Wind und Wellen," Berl. Monatsber., July 25,
1889.

410 SURFACE WAVES. [CHAP. IX

a?l\*, the solution of the problem in the case of infinite depth
is contained in the formulae

. .#
' ' '

The equation of the wave-profile ^ = 0 is found by successive
approximations to be

y = fidw cos ka; = P(I+ky + |-%2 + . . .) cos kx
= i&£2 + £ (1 + f &2/32) cos kx + |&/32 cos Zkx + -f &2£3 cos 3kx + ...

............ (2);

or, if we put 0(1+ &2/32) = a,

y — ^ka2 = a cos kx + ^ka? cos 2kx + f &2a3 cos 3Aj# + ...... (3).

So far as we have developed it, this coincides with the equation of
a trochoid, the circumference of the rolling circle being 2ir/&, or X,
and the length of the arm of the tracing point being a.

We have still to shew that the condition of uniform pressure
along this stream-line can be satisfied by a suitably chosen value
of c. We have, from (1), without approximation

= const. - gy - Jc2 { 1 - 2kp& cos Tex + k^e2ky} ..... (4),
and therefore, at points of the line y = $eky cos kx,
£ = const. + (kc* -g)y-

-const.

Hence the condition for a free surface is satisfied, to the present
order of approximation, provided

c^ + ^c^^a+^a2) .................. (6).

This determines the velocity of progressive waves of per
manent type, and shews that it increases somewhat with the
amplitude a.

For methods of proceeding to a higher approximation, and for
the treatment of the case of finite depth, we must refer to the
original investigations of Stokes.

* Lord Kayleigh, I c. ante p. 279.

231] FINITE WAVES OF PERMANENT TYPE. 411

The figure shews the wave-profile, as given by (3), in the case
of ka = J, or a/\ = '0796.

The approximately trochoidal form gives an outline which is
sharper near the crests, and flatter in the troughs, than in the case
of the simple-harmonic waves of infinitely small amplitude investi
gated in Art. 218, and these features become accentuated as the
amplitude is increased. If the trochoidal form were exact, instead
of merely approximate, the limiting form would have cusps at the
crests, as in the case of Gerstner's waves to be considered presently.
In the actual problem, which is one of irrotational motion, the
extreme form has been shewn by Stokes*, in a very simple manner,
to have sharp angles of 120°.

The problem being still treated as one of steady motion, the motion near
the angle will be given by the formulae of Art. 63 ; viz. if we introduce polar
coordinates r, 6 with the crest as origin, and the initial line of 6 drawn
vertically downwards, we have

(i),

with the condition that \//>--0 when 6= ±a (say), so that ma=^n. This
formula leads to

q = mCrm~l .................................... (ii),

where q is the resultant fluid velocity. But since the velocity vanishes at the
crest, its value at a neighbouring point of the free surface will be given by

(iii),

as in Art. 25 (2). Comparing (ii) and (iii), we see that we must have m=£,
and therefore a = ^7rf.

In the case of progressive waves advancing over still water, the particles
at the crests, when these have their extreme forms, are moving forwards with
exactly the velocity of the wave.

Another point of interest in connection with these waves of permanent
type is that they possess, relatively to the undisturbed water, a certain

* Math, and Phys. Papers, t. i., p. 227.

t The wave-profile has been investigated and traced, for the neighbourhood of
the crest, by Michell, " The Highest Waves in Water," Phil. Mag., Nov. 1893. He
finds that the extreme height is -142 X, and that the wave- velocity is greater than in
the case of infinitely small height in the ratio of 1-2 to 1.

412 SURFACE WAVES. [CHAP. IX

momentum in the direction of wave-propagation. The momentum, per wave
length, of the fluid contained between the free surface and a depth h (beneath
the level of the origin) which we will suppose to be great compared with X, is

since ^ = 0, by hypothesis, at the surface, and —ch, by (1), at the great depth
h. In the absence of waves, the equation to the upper surface would be
y=^£a2, by (3), and the corresponding value of the momentum would there

fore be

A .................................. (v).

The difference of these results is equal to

7rpa2c ....................................... (vi),

which gives therefore the momentum, per wave-length, of a system of
progressive waves of permanent type, moving over water which is at rest at a
great depth.

To find the vertical distribution of this momentum, we remark that the
equation of a stream-line ty = cti is found from (2) by writing y-f- h' for ?/, and
j3e~kh' for /3. The mean-level of this stream-line is therefore given by

y= -h' + ^^e~2kh' ........................... (vii).

Hence the momentum, in the case of undisturbed flow, of the stratum of
fluid included between the surface and the stream-line in question would
be, per wave-length,

pcAtA'+JJ/S* (!-«-»")} ....................... (viii).

The actual momentum being pcA'X, we have, for the momentum of the same
stratum in the case of waves advancing over still water,

7rp«2c(l-e-2fc>0 ................................. (ix).

It appears therefore that the motion of the individual particles, in these
progressive waves of permanent type, is not purely oscillatory, and that there
is, on the whole, a slow but continued advance in the direction of wave-
propagation*. The rate of this flow at a depth h' is found approximately
by differentiating (ix) with respect to A', and dividing by pX, viz. it is

tfatce-™* .................................... (x).

This diminishes rapidly from the surface downwards.

232. A system of exact equations, expressing a possible form
of wave-motion when the depth of the fluid is infinite, was given
so long ago as 1802 by Gerstnerf, and at a later period indepen
dently by Rankine. The circumstance, however, that the motion

1 * Stokes, 1. c. ante, p. 409. Another very simple proof of this statement has
been given by Lord Eayleigh, 1. c. ante, p. 279.

t Professor of Mathematics at Prague, 1789—1823.

231-232] GERSTNER'S WAVES. 413

in these waves is not irrotational detracts somewhat from the
physical interest of the results.

If the axis of x be horizontal, and that of y be drawn vertically
upwards, the formulae in question may be written

so = a + r ekb sin k (a + ct\

= b — Tekb cos k (a + ct)

ox

where the specification is on the Lagrangian plan (Art. 16), viz.,
a, b are two parameters serving to identify a particle, and #, y are
the coordinates of this particle at time t. The constant k deter
mines the wave-length, and c is the velocity of the waves, which
are travelling in the direction of ^-negative.

To verify this solution, and to determine the value of c, we
remark, in the first place, that

_ kb
d(a,b)~

so that the Lagrangian equation of continuity (Art. 16 (2)) is
satisfied. Again, substituting from (1) in the equations of motion
(Art. 13), we find

-j- (- + = e sn

,

%r ( - +

db \p
whence

- = const. — g ] b — y ekb cos k (a + ct) \
P I * )

- c2ekb cos k(a + ct) + $c*e*kb ...... (4).

For a particle on the free surface the pressure must be
constant ; this requires

<? = g/k ............................ (5);

cf. Art. 218. This makes

= const. - gb + |c2e2W> ................... (6).

414

SURFACE WAVES.

[CHAP, ix

It is obvious from (1) that the path of any particle (a, b) is a
circle of radius k~lekb.

The figure shews the forms of the lines of equal pressure
b = const., for a series of equidistant values of b*. These curves
are trochoids, obtained by rolling circles of radii k~l on the under
sides of the lines y = b + &"1, the distances of the tracing points
from the respective centres being k~l^b. Any one of these lines
may be taken as representing the free surface, the extreme
admissible form being that of the cycloid. The dotted lines
represent the successive forms taken by a line of particles which
is vertical when it passes through a crest or a trough.

It has been already stated that the motion of the fluid in these waves is
rotational. To prove this, we remark that

sin Jc (a + ct}} + ce2kb 8a

which is not an exact differential.

* The diagram is very similar to the one given originally by Gerstner, and copied
more or less closely by subsequent writers,

232] ROTATIONAL CHARACTER. 415

The circulation in the boundary of the parallelogram whose vertices
coincide with the particles

(«, 6), (a + 8a, 6), (a, b + 8b), (a

is,by(i), -

and the area of the circuit is

d(a,b)
Hence the angular velocity (&>) of the element (a, b} is

±_6_ (ii).

This is greatest at the surface, and diminishes rapidly with increasing depth.
Its sense is opposite to that of the revolution of the particles in their circular
orbits.

A system of waves of the present type cannot therefore be originated from
rest, or destroyed, by the action of forces of the kind contemplated in the
general theorem of Arts. 18, 34. We may however suppose that by properly
adjusted pressures applied to the surface of the waves the liquid is gradually
reduced to a state of flow in horizontal lines, in which the velocity (u'} is
a function of the ordinate (?/') only*. In this state we shall have af=at
while y' is a function of b determined by the condition

or = l-e2A* .................................... (iv).

m, . , du' du' dii' , dyf

Thismakes ^ = ^, J£= -2<0J =2te™ ..................... (v),

and therefore u'=cezkb .................................... (vi).

Hence, for the genesis of the waves by ordinary forces, we require as a
foundation an initial horizontal motion, in the direction opposite to that of
propagation of the waves ultimately set up, which diminishes rapidly from the
surface downwards, according to the law (vi), where b is a function of y deter
mined by

It is to be noted that these rotational waves, when established, have zero
momentum.

* For a fuller statement of the argument see Stokes, Math, and Phys. Papers,
t. i., p. 222.

416 SURFACE WAVES. [CHAP. IX

233. Rankine's results were obtained by him by a synthetic
process for which we must refer to his paper*.

Gerstner's procedure f, again, is different. He assumed,
erroneously, that when the problem is reduced to one of steady
motion the pressure must be uniform, not only along that par
ticular stream-line which coincides with the free surface, but also
along every other stream-line. Considered, however, as a deter
mination of the only type of steady motion, under gravity, which
possesses this property, his investigation is perfectly valid, and,
especially when regard is had to its date, very remarkable.

The argument, somewhat condensed with the help of the more modern
invention of the stream-function, is as follows.

Fixing our attention at first on any one stream-line, and choosing the origin
on it at a point of minimum altitude, let the axis of x be taken horizontal,
in the general direction of the flow, and let that of y be drawn vertically up
wards. If v be the velocity at any point, and VQ the velocity at the origin, we
have, resolving along the arc s,

dv dv

"5 = -** .................................... «>

on account of the assumed uniformity of pressure. Hence

as in Art. 25. Again, resolving along the normal,

v2 1 d dx

where 8n is an element of the normal, and R is the radius of curvature.

Now v= -d\ls/dn, where \^ is the stream-function, so that if we write & for
dpfpd\lr, which is, by hypothesis, constant along the stream-line, we have

tf dx

Putting ^ »™)

multiplying by dyjds, and making use of (i), we obtain

d*-x do dx _ dy
ds* ds ds~ * ds

d'2x dv dx _ dy

J-9. "• J_ j_ "" "jT" \^/>

whence, on integration,

dx

* "On the Exact Form of Waves near the Surface of Deep Water," Phil. Trans.,
1863.

f "Theorie der Wellen," Abh. der k. b'ohm. Ges. der Witt., 1802; Gilbert's
Annalen der Physik, t. xxxii. (1809).

233] GERSTNER'S INVESTIGATION. 417

which is a formula for the horizontal velocity. Combined with (ii), this gives

) ........ .... (vii),

provided /3 = VQ/O- - gja* ................................. (viii).

Hence, for the vertical velocity, we have

(ix).

If the coordinates x, y of any particle on the stream-line be regarded as
functions of t, we have, then,

)} ..................... 00,

whence x = -t+fB8in<rt, y = /3(l — cos<r£) ..................... (xi),

(T

if the time be reckoned from the instant at which the particle passes through the
origin of coordinates. The equations (xi) determine a trochoid ; the radius of
the rolling circle is g/tr2, and the distance of the tracing point from the centre
is $. The wave-length of the curve is X =

It remains to shew that the trochoidal paths can be so adjusted that the
condition of constancy of volume is satisfied. For this purpose we must
take an origin of y which shall be independent of the particular path considered,
so that the paths are now given by

x = 0- t + /3 sin at, y = b-$ cos at ..................... (xii),

<7

where b is a function of /3, and conversely. It is evident that a- must be an
absolute constant, since it determines the wave-length. Now consider two
particles P, P', on two consecutive stream-lines, which are in the same phase
of their motions. The projections of PP' on the coordinate axes are

8/3sincr£ and db — 8ft cos at.

The flux (Art. 59) across a line fixed in space which coincides with the
instantaneous position of PP' is obtained by multiplying these projections by

dyjdt and —dxldt,
respectively, and adding ; viz. we find

coso-Z ............... (xiii).

27

<r
In order that this may be independent of t, we must have

418 SURFACE WAVES. [CHAP. IX

or $ = C<P ^ (xiv),

where k = <r2/<7 = 2?r/X . . . . '. (xv).

Hence, finally,

(xvi),

where

-*-(&

The addition of a constant to b merely changes the position of the origin and
the value of C \ we may therefore suppose that 6 = 0 for the limiting cycloidal
form of the path. This makes C=k~l.

If the time be reckoned from some instant other than that of passage
through a lowest point, we must in the above formula} write a + ct for ct, where
a is arbitrary. If we further impress on the whole mass a velocity c in the
direction of ^-negative, we obtain the formulae (1) of Art. 232.

234. Scott Russell, in his interesting experimental investiga
tions*, was led to pay great attention to a particular type which
he calls the ' solitary wave.' This is a wave consisting of a single
elevation, of height not necessarily small compared with the depth
of the fluid, which, if properly started, may travel for a consider
able distance along a uniform canal, with little or no change of
type. Waves of depression, of similar relative amplitude, were
found not to possess the same character of permanence, but to
break up into series of shorter waves.

The solitary type may be regarded as an extreme case of
Stokes' oscillatory waves of permanent type, the wave-length
being great compared with the depth of the canal, so that the
widely separated elevations are practically independent of one
another. The methods of approximation employed by Stokes
become, however, unsuitable when the wave-length much exceeds
the depth ; and the only successful investigations of the solitary
wave which have yet been given proceed on different lines.

The first of these was given independently by Boussinesqf and Lord
RayleighJ. The latter writer, treating the problem as one of steady motion,
starts virtually from the formula

where F(x) is real. This is especially appropriate to cases, such as the

* "Keport on Waves," Brit. Ass. Rep., 1844.

t Comptes Eendus, June 19, 1871.

t "On Waves," Phil. Mag., April, 1876.

233-234] THE SOLITARY WAVE. 419

present, where one of the family of stream-lines is straight. We derive
from (i)

.("),

Y=yF' — j— F'" + — FV — ...

where the accents denote differentiations with respect to as. The stream-line
^ = 0 here forms the bed of the canal, whilst at the free surface we have
y = - cA, where c is the uniform velocity, and A the depth, in the parts of the
fluid at a distance from the wave, whether in front or behind.

The condition of uniform pressure along the free surface gives
or, substituting from (ii),
But, from (ii), we have, along the same surface,

(v).

It remains to eliminate /"between (iv) and (v) ; the result will be a differential
equation to determine the ordinate y of the free surface. If (as we will
suppose) the function F' (x} and its differential coefficients vary slowly with
x, so that they change only by a small fraction of their values when x increases
by an amount comparable with the depth A, the terms in (iv) and (v) will be of
gradually diminishing magnitude, and the elimination in question can be
carried out by a process of successive approximation.

Thus, from (v),

and if we retain only terms up to the order last written, the equation (iv)
becomes

or, on reduction,

y^ 3~7~3p = A2 ~&W~ (V11)'

If we multiply by y', and integrate, determining the arbitrary constant so
as to make y' = 0 for y =A, we obtain

_ . _ ,

y 3 y Ii A2 c2A2

or

(viii).

Hence y1 vanishes only for y = h and # = c2/#, and since the last factor must
be positive, it appears that <^Jg is a maximum value of y. Hence the wave is

27—2

420

SURFACE WAVES.

[CHAP, ix

necessarily one of elevation only, and denoting by a the maximum height above
the undisturbed level, we have

<*=ff(h + a) .................................... (ix),

which is exactly the empirical formula for the wave-velocity adopted by
Russell.

The extreme form of the wave will, as in Art. 231, have a sharp crest of
120° ; and since the fluid is there at rest we shall have c'2 = 2#a. If the
formula (ix) were applicable to such an extreme case, it would follow that

If we put, for shortness,

y
we find, from (viii)

the integral of which is

(x),

(xii),

if the origin of x be taken beneath the summit.

There is no definite ' length ' of the wave, but we may note, as a rough in
dication of its extent, that the elevation has one-tenth of its maximum value
when #/6=3'636.

0-5

1-0

1-5

The annexed drawing of the curve

represents the wave-profile in the case a — ^h. For lower waves the scale of y
must be contracted, and that of x enlarged, as indicated by the annexed
table giving the ratio b/h, which determines the horizontal scale, for various
values of a/h.

It will be found, on reviewing the above investigation, that
the approximations consist in neglecting the fourth power of
the ratio (A+a)/26.

If we impress on the fluid a velocity — c parallel to x we
get the case of a progressive wave on still water. It is not
difficult to shew that, when the ratio a/h is small, the path
of each particle is an arc of a parabola having its axis vertical
and apex upwards*.

It might appear, at first sight, that the above theory is
inconsistent with the results of Art. 183, where it was shewn
that a wave whose length is great compared with the depth
* Boussinesq, I. c,

a/h

b/h

•1

1-915

•2

1-414

•3

1-202

•4

1-080

•5

1-000

•6

•943

•7

•900

•8

•866

•9

•839

1-0

•816

234-235] FORM OF PROFILE. 421

must inevitably suffer a continual change of form as it advances, the changes
being the more rapid the greater the elevation above the undisturbed level.
The investigation referred to postulates, however, a length so great that the
vertical acceleration may be neglected, with the result that the horizontal
velocity is sensibly uniform from top to bottom (Art. 169). The numerical
table above given shews, on the other hand, that the longer the 'solitary
wave ' is, the lower it is. In other words, the more nearly it approaches to
the character of a 'long' wave, in the sense of Art. 169, the more easily is
the change of type averted by a slight adjustment of the particle-velocities*.

The motion at the outskirts of the solitary wave can be represented by a
very simple formula f. Considering a progressive wave travelling in the
direction of ^-positive, and taking the origin in the bottom of the canal, at a
point in the front part of the wave, we assume

This satisfies v2</> = 0, and the surface-condition
will also be satisfied for y = h, provided

This will be found to agree approximately with Lord Kayleigh's investigation
if we put m = b~1.

235. The theory of waves of permanent type has been brought
into relation with general dynamical principles by von HelmholtzJ,

If in the equations of motion of a 'gyrostatic' system, Art.
139 (14), we put

dV dV

* Stokes, " On the Highest Wave of Uniform Propagation," Proc. Camb. Phil.
Soc., t. iv., p. 361 (1883).

For another method of investigation see McCowan " On the Solitary Wave,"
Phil. Mag., July 1891; and "On the Highest Wave of Permanent Type," Phil.
Mag., Oct. 1894. The latter paper gives an approximate determination of the
extreme form of the wave, when the crest has a sharp angle of 120°. The limiting
value for the ratio a/ft is found to be -78.

t Kindly communicated by Sir George Stokes.

t " Die Energie der Wogen und des Windes," Berl. Monatsber., July 17, 1890;
Wied. Ann., t. xli., p. 641.

422 SURFACE WAVES. [CHAP. IX

where V is the potential energy, it appears that the conditions for
steady motion, with qlt q2, ... constant, are

where K is the energy of the motion corresponding to any given
values of the coordinates ql} q2, ..., when these are prevented from
varying by the application of suitable extraneous forces.

This energy is here supposed expressed in terms of the constant
momenta C, C',... corresponding to the ignored coordinates
2£, 2£7, ..,, and of the palpable coordinates ql} qz, — It may how
ever also be expressed in terms of the velocities %, ^', ... and
the coordinates qlt #2, ...; in this form we denote it by T0. It
may be shewn, exactly as in Art. 141, that dT0/dqr = — dK/dqr) so
that the conditions (2) are equivalent to

Hence the condition for free steady motion with any assigned
constant values of qlt q2)... is that the corresponding value of
V + K, or of V-T0, should be stationary. Cf. Art. 195.

Further, if in the equations of Art. 139 we write —dV/dqs + Q8
for Q8, so that Qs now denotes a component of extraneous force, we
find, on multiplying by qlt q.2) ... in order, and adding,

Q1<z1+Q2j2+ ............... (4),

where f& is the part of the energy which involves the velocities
qlt fa, .... It follows, by the same argument as in Art. 197, that
the condition for 'secular' stability, when there are dissipative
forces affecting the coordinates qlt qz, ..., but not the ignored
coordinates ^, ^', ..., is that V+K should be a minimum.

In the application to the problem of stationary waves, it will
tend to clearness if we eliminate all infinities from the question
by imagining that the fluid circulates in a ring-shaped canal of
uniform rectangular section (the sides being horizontal and
vertical), of very large radius. The generalized velocity ^ corre-

235] DYNAMICAL CONDITION FOR STATIONARY WAVES. 423

spending to the ignored coordinate may be taken to be the flux
per unit breadth of the channel, and the constant momentum of
the circulation may be replaced by the cyclic constant K. The
coordinates qltqt, ... of the general theory are now represented by
the value of the surface-elevation (T?) considered as a function of
the longitudinal space-coordinate x. The corresponding com
ponents of extraneous force are represented by arbitrary pressures
applied to the surface.

If I denote the whole length of the circuit, then considering
unit breadth of the canal we have

V = \go \ r)2dx (5),

Jo

where 77 is subject to the condition

= 0.. ...(6).

ri

JQ

If we could with the same ease obtain a general expression for
the kinetic energy of the steady motion corresponding to any
prescribed form of the surface, the minimum condition in either of
the forms above given would, by the usual processes of the Calculus
of Variations, lead to a determination of the possible forms, if any,
of stationary waves*.

Practically, this is not feasible, except by methods of successive
approximation, but we may illustrate the question by reproducing,
on the basis of the present theory, the results already obtained
for ' long ' waves of infinitely small amplitude.

If h be the depth of the canal, the velocity in any section when the surface
is maintained at rest, with arbitrary elevation 77, is x/(/i + ??)> where x is the
flux. Hence, for the cyclic constant,

* For some general considerations bearing on the problem of stationary waves
on the common surface of two currents reference may be made to von Helmholtz'
paper. This also contains, at the end, some speculations, based on calculations of
energy and momentum, as to the length of the waves which would be excited
in the first instance by a wind of given velocity. These appear to involve the
assumption that the waves will necessarily be of permanent type, since it is only on
some such hypothesis that we get a determinate value for the momentum of a train
of waves of small amplitude.

424 SURFACE WAVES. [CHAP. IX

approximately, where the term of the first order in 77 has been omitted, in
virtue of (6).

The kinetic energy, ^PK\> may be expressed in terms of either x °r *• We
thus obtain the forms

The variable part of V- T0 is
and that of V+Kis

It is obvious that these are both stationary for rj = 0 ; and that they will
be stationary for any infinitely small values of 77, provided x2=gk?, or
if We put x=Uh, or K=Ul, this condition gives

in agreement with Art. 172.

It appears, moreover, that 77 = 0 makes V-\-K a maximum or a minimum
according as U'2 is greater or less than gli. In other words, the plane
form of the surface is secularly stable if, and only if, U<(gfif. It is to be
remarked, however, that the dissipative forces here contemplated are of a special
character, viz. they affect the vertical motion of the surface, but not (directly)
the flow of the liquid. It is otherwise evident from Art. 172 that if pressures
be applied to maintain any given constant form of the surface, then if
U*>gh these pressures must be greatest over the elevations and least over
the depressions. Hence if the pressures be removed, the inequalities of the
surface will tend to increase.

Standing Waves in Limited Masses of Water.

236. The problem of wave-motion in two horizontal dimensions
(x, y), in the case where the depth is uniform and the fluid is
bounded laterally by vertical walls, can be reduced to the same
analytical form as in Art. 185*.

If the origin be taken in the undisturbed surface, and if f
denote the elevation at time t above this level, the pressure-

* For references to the original investigations of Poisson and Lord Kayleigh
see p. 310. The problem was also treated by Ostrogradsky, " Memoire sur la
propagation des ondes dans un bassin cylindrique," Mem. des Sav. Etrang.,
t. iii. (1832).

235-236] STANDING WAVES : UNIFORM DEPTH. 425

condition to be satisfied at the surface is, in the case of infinitely
small motions,

and the kinematical surface-condition is

d% _ [df\ /o\

dt~ L«kJ«-«"

Hence, for z — 0, we must have

or, in the case of simple- harmonic motion,

if the time-factor be e'i{<rt+e). The proof is the same as in
Art. 216.

The equation of continuity, V-$ = 0, and the condition of zero
vertical motion at the depth z = — h, are both satisfied by

(f) = ^ cosh k (z 4- h) ..................... (5),

where ^ is a function of x, y, provided

The form of fa and the admissible values of k are determined by
this equation, and by the condition that

at the vertical walls. The corresponding values of the ' speed ' (cr)
of the oscillations are then given by the surface-condition (4), viz.
we have

a- = gk tanh kl ........................ (8).

From (2) and (5) we obtain

(7

.(9).

The conditions (6) and (7) are of the same form as in the case
of small depth, and we could therefore at once write down the

426 SURFACE WAVES. [CHAP. IX

results for a rectangular or a circular tank. The values of k, and
the forms of the free surface, in the various fundamental modes,
are the same as in Arts. 186, 187*, but the amplitude of the
oscillation now diminishes with increasing depth below the surface,
according to the law (5) ; whilst the speed of any particular mode
is given by (8).

When kh is small, we have <j* = k*gh, as in the Arts, re
ferred to.

237. The number of cases of motion with a variable depth, of
which the solution has been obtained, is very small.

1°. We may notice, first, the two-dimensional oscillations of water across
a channel whose section consists of two straight lines inclined at 45° to the
vertical f.

The axes of y, z being respectively horizontal and vertical, in the plane of
the cross-section, we assume

(f> + fy = A {cosh £ (y-ft'z)-}- cos £(#+w)} .......(i),

the time-factor e*(a*+e) being understood. This gives

<f> = A (cosh ky cos kz + cos ky cosh kz\ ) . .

ty = A (sinh ky sin kz - sin ky sinh kz) J "

The latter formula shews at once that the lines y=±z constitute the
stream-line ^ = 0, and may therefore be taken as fixed boundaries.

The condition to be satisfied at the free surface is, as in Art. 216,

o*4>=gd<j>ld£....: (iii).

Substituting from (ii) we find, if h denote the height of the surface above the
origin,

<r2 (cosh ky cos kh + cos ky cosh M) =gk ( - cosh ky sin kh + cos ky sinh M).
This will be satisfied for all values of y, provided

o-2 cos kh — —gk sin kht | /JVA

o-2 cosh kh=gk sinh kh )

whence tanh kh = - tan kh (v).

This determines the admissible values of k ; the corresponding values of
o- are then given by either of the equations (iv).

Since (ii) makes <£ an even function of y, the oscillations which it represents
are symmetrical with respect to the medial plane y = 0.

* It may be remarked that either of the two modes figured on pp. 308, 309 may
be easily excited by properly-timed horizontal agitation of a tumbler containing
water.

t Kirchhoff, " Ueber stehende Schwingungen einer schweren Fliissigkeit," Berl.
Honatsber.. May 15, 1879; Ges. Abh., p. 428. Greenhill, 1. c. ante p. 388.

236-237] VARIABLE DEPTH. 427

The asymmetrical oscillations are given by

or (f) = - A (sinh Jcy sin kz + sin ky sinh kz\ }

\js = A (cosh Icy cos kz - cos ky cosh /#) J
The stream-line ^ = 0 consists, as before, of the lines y = ±z; and the surface-
condition (iii) gives

o-2 (sinh ky sin kh + sin ky sinh kh} =gk (sinh % cos kh 4- sin £y cosh kh}.
This requires a2 sin M=^ cos kh, } , ....

<r2smh£A=^coshM f"
whence tanh£A = tanM ................................. (ix).

The equations (v) and (ix) present themselves in the theory of the lateral
vibrations of a bar free at both ends ; viz. they are both included in the
equation

cosmcoshm = l .............................. (x)*>

where m=2M.

The root M = 0, of (ix), which is extraneous in the theory referred to, is
now important ; it corresponds in fact to the slowest mode of oscillation in
the present problem. Putting Ak2 = B, and making k infinitesimal, the
formulae (vii) become, on restoring the time-factor, and taking the real parts,

whilst from (viii)

The corresponding form of the free surface is

i r/7/^n

0 (xiii).

The surface in this mode is therefore always plane.

The annexed figure shews the lines of motion (^ = const.) for a series
of equidistant values of \//-.

* Of. Lord Bayleigh, Theory of Sound, t. i., Art. 170, where the numerical
solution of the equation is fully discussed.

428 SURFACE WAVES. [CHAP. IX

The next gravest mode is symmetrical, and is given by the lowest finite
root of (v), which is M = 2-3650, whence o- = l'5244 (gjKf.

In this mode, the profile of the surface has two nodes, whose positions are
determined by putting 0 = 0, z = k, in (ii) ; whence it is found that

*/&« £-5516*.

The next mode corresponds to the lowest finite root of (ix), and so ont.

2°. Greenhill, in the paper already cited, has investigated the symmetrical
oscillations of the water across a channel whose section consists of two straight
lines inclined at 60° to the vertical. In the (analytically) simplest mode of
this kind we have, omitting the time-factor,

4) + i^ = iA(y + iz)3 + B ........................... (xiv),

or <t> = Az(z*-Zy*) + B, + = Ay(i?-W) ............... (xv),

the latter formula making \//- = 0 along the boundary y=±>j3.z. The
surface-condition (iii) is satisfied for z=k, provided

<r*=g/h, B = 2A/i3 ........................... (xvi).

The corresponding form of the free surface is

a parabolic cylinder, with two nodes at distances of '5774 of the half-breadth
from the centre. Unfortunately, this is not the slowest mode, which must
evidently be of asymmetrical type.

3°. If in any of the above cases we transfer the origin to either edge of
the canal, and then make the breadth infinite, we get a system of standing
waves on a sea bounded by a sloping bank. This may be regarded as made
up of an incident and a reflected system. The reflection is complete, but
there is in general a change of phase.

When the inclination of the bank is 45° the solution is
^ — H{^z (cos ky- sin ky] +e~kv (cos &z + sin kz)} cos

For an inclination of 30° to the horizontal we have

.......... ...(xix).

In each case o-2=#£, as in the case of waves on an unlimited sheet of deep
water.

These results, which may easily be verified ab initio, were given by
Kirchhoff (I. c.}.

* Lord Eayleigh, Theory of Sound, Art. 178.

f An experimental verification of the frequencies, and of the positions of the
loops (places of maximum vertical amplitude), in various fundamental modes, has
been made by Kirchhoff and Hansemann, " Ueber stehende Schwingungen des
Wassers," Wied. Ann., t. x. (1880); Kirchhoff, Ges. Abh., p. 442.

237-239] TRANSVERSE OSCILLATIONS IN A CANAL. 429

238. An interesting problem which presents itself in this
connection is that of the transversal oscillations of water contained
in a canal of circular section. This has not yet been solved, but
it may be worth while to point out that an approximate determi
nation of the frequency of the slowest mode, in the case where the
free surface is at the level of the axis, can be effected by Lord
Rayleigh's method, explained in Art. 165.

If we assume as an ' approximate type ' that in which the free
surface remains always plane, making a small angle Q (say) with
the horizontal, it appears, from Art. 72, 3°, that the kinetic energy
T is given by

aW (1),

where a is the radius, whilst for the potential energy V we have

2V = %gpa>ffi ........................ (2).

If we assume that 6 oc cos (<rt + e), this gives

-
whence a = 1-169 a*.

In the case of a rectangular section of breadth 2a, and depth
a, the speed is given by Art. 236 (8), where we must put k — 7r/2a
from Art. 186, and h — a. This gives

0-2 = ^TJ- tanh £TT . - ........................ (4),

or a = 1'200 (#/«)*. The frequency in the actual problem is less,
since the kinetic energy due to a given motion of the surface
is greater, whilst the potential energy for a given deformation
is the same. Cf. Art. 45.

239. We may next consider the free oscillations of the water
included between two transverse partitions in a uniform horizontal
canal. It will be worth while, before proceeding to particular
cases, to examine for a moment the nature of the analytical
problem, with the view of clearing up some misunderstandings
which have arisen as to the general question of wave-propagation
in a uniform canal of unlimited length.

430 SURFACE WAVES. [CHAP. IX

If the axis of x be parallel to the length, and the origin be
taken in one of the ends, the velocity potential in any one of the
fundamental modes referred to may, by Fourier's theorem, be
supposed expressed in the form

$ = (P0 + pl cos kx + P2 cos 2kx + ...

+ Pg cos skx + . . .) cos (o-t + e) ......... (1),

where k = ir/l, if I denote the length of the compartment. The
coefficients Ps are here functions of y, z. If the axis of z be drawn
vertically upwards, and that of y be therefore horizontal and
transverse to the canal, the forms of these functions, and the
admissible values of <r, are to be determined from the equation
of continuity

V'£ = 0 ............................. (2),

with the conditions that

d<f>ldn = 0 ........................... (3)

at the sides, and that

......................... (4)

at the free surface. Since dfyjdx must vanish for x = 0 and x = Z,
it follows from known principles* that each term in (1) must
satisfy the conditions (2), (3), (4) independently ; viz. we must
have

^2P-0 (5)

with dPK/dn = 0 ........................... (6)

at the lateral boundary, and

a*P8 = gdP./dz ......................... (7)

at the free surface.

The term P0 gives purely transverse oscillations such as have
been discussed in Art. 237. Any other term Ps cos skx gives a
series of fundamental modes with s nodal lines transverse to the
canal, and 0, 1, 2, 3,... nodal lines parallel to the length.

It will be sufficient for our purpose to consider the term
P! cos kx. It is evident that the assumption

<£ = Px cos kx . cos (at + e) .................. (8),

with a proper form of Pj and the corresponding value of cr deter-

* See Stokes, "On the Critical Values of the Sums of Periodic Series," Camb.
Trans., t. viii. (1847) ; Math, and Phys. Paper*, t. i., p. 236,

239] WAVES IN UNIFORM CANAL: GENERAL THEORY. 431

mined as above, gives the velocity-potential of a possible system of
standing waves, of arbitrary wave-length 2-7T/&, in an unlimited
canal of the given form of section. Now, as explained in Art. 218,
by superposition of two properly adjusted systems of standing
waves of this type we can build up a system of progressive waves

<f) = Pl cos (kas + at) (9).

Hence, contrary to what has been sometimes asserted, progressive
waves of simple-harmonic profile, of any assigned wave-length, are
possible in an infinitely long canal of any uniform section.

We might go further, and assert the possibility of an infinite
number of types, of any given wave-length, with wave-velocities
ranging from a certain lowest value to infinity. The types, how
ever, in which there are longitudinal nodes at a distance from the
sides are from the present point of view of subordinate interest.

Two extreme cases call for special notice, viz. where the wave
length is very great or very small compared with the dimensions
of the transverse section.

The most interesting types of the former class have no longi
tudinal nodes, and are covered by the general theory of 'long'
waves given in Arts. 166, 167. The only additional information
we can look for is as to the shapes of the ' wave-ridges ' in the
direction transverse to the canal.

In the case of relatively short waves, the most important type
is one in which the ridges extend across the canal with gradually
varying height, and the wave-velocity is that of free waves on
deep-water as given by Art. 218 (6).

There is another type of short waves which may present itself
when the banks are inclined, and which we may distinguish by
the name of 'edge-waves,' since the amplitude diminishes expo
nentially as the distance from the bank increases. In fact, if the
amplitude at the edges be within the limits imposed by our
approximations, it will become altogether insensible at a distance
whose projection on the slope exceeds a wave-length. The wave-
velocity is less than that of waves of the same length on deep
water. It does not appear that the type of motion here referred
to is of any physical importance.

4-32 SURFACE WAVES. [CHAP. IX

A general formula for these edge-waves has been given by Stokes*.
Taking the origin in one edge, the axis of z vertically upwards, and that of y
transverse to the canal, and treating the breadth as relatively infinite, the
formula in question is

0 = m-^cos^sin^cos£(^-cO (i),

where /3 is the slope of the bank to the horizontal, and

The reader will have no difficulty in verifying this result.

240. We proceed to the consideration of some special cases.
We shall treat the question as one of standing waves in an
infinitely long canal, or in a compartment bounded by two
transverse partitions whose distance apart is a multiple of half the
arbitrary wave-length (27T/&), but the investigations can be easily
modified as above so as to apply to progressive waves, arid we
shall occasionally state results in terms of the wave- velocity.

1°. The solution for the case of a rectangular section, with horizontal bed
and vertical sides, could be written down at once from the results of Arts.
186, 236. The nodal lines are transverse and longitudinal, except in the case
of a coincidence in period between two distinct modes, when more complex
forms are possible. This will happen, for instance, in the case of a square
tank.

2°. In the case of a canal whose section consists of two straight lines in
clined at 45° to the vertical we have, first, the type discovered by Kellandf ;
viz. if the axis of x coincide with the bottom line of the canal,

0 = A cosh -If- cosh -^ cos kx . cos (o-t + t) (i).

This evidently satisfies v2<£ = 0, and makes

d(f)/dy — +d(f>/dz (ii),

for y=+z, respectively. The surface-condition (7) then gives

~/, />./,

(i"),

v/2 "

where h is the height of the free surface above the bottom line. If we put
a = kc, the wave- velocity c is given by

where # = 2ir/A, if X be the wave-length.

* "Report on Recent Researches in Hydrodynamics," Brit. Ass. Rep., 1846;
Math, and Phys. Papers, t. i., p. 167.

f " On Waves," Trans. R. S, Edin., t. xiv. (1839),

239-240] TRIANGULAR SECTION. 433

When k/\ is small, this reduces to

<? = (fe^)* (v),

in agreement with Art. 167 (8), since the mean depth is now denoted by \h.

When, on the other hand, h/\ is moderately large, we have

• (vi).

The formula (i) indicates now a rapid increase of amplitude towards the
sides. We have here, in fact, an instance of « edge- waves,' and the wave-
velocity agrees with that obtained by putting /3=45° in Stokes' formula.

The remaining types of oscillation which are symmetrical with respect to
the medial plane y--=Q are given by the formula

<£ = C (cosh ay cos fjz + cos /3y cosh az) cos kx . cos (trt+c) ( vii),

provided a, /3, a- are properly determined. This evidently satisfies (ii), and the
equation of continuity gives

a2-/32 = P (viii).

The surface-condition, Art. 239 (4), to be satisfied for z = h, requires

o-2coshaA= <7asinhaA,|
a2 cos j3h = — gft sin /3A j "

Hence aA tanh aA+£A tan £A = 0 (x).

The values of a, 0 are determined by (viii) and (x), and the corresponding
values of o- are then given by either of the equations (ix).

If, for a moment, we write

x=ah, y = fih (xi),

the roots are given by the intersections of the curve

x tanh x -f y tan y — 0 (xii),

whose general form can be easily traced, with the hyperbola

(xiii).

There are an infinite number of real solutions, with values of fth lying in
the second, fourth, sixth, ... quadrants. These give respectively 2, 4, 6, ...
longitudinal nodes of the free surface. When h/\ is moderately large, we have
tanh ah = 1, nearly, and /3A is (in the simplest mode of this class) a little greater
than £TT. The two longitudinal nodes in this case approach very closely to
the edges as X is diminished, whilst the wave-velocity becomes practically
equal to that of waves of length X on deep water. As a numerical example,
assuming £A= I'l x £TT, we find

aA = 10-910, M = 1O772, c= 1-0064 x (
The distance of either nodal line from the nearest edge is then -12^.

L, 28

434 SURFACE WAVES. [CHAP. IX

We may next consider the asymmetrical modes. The solution of this type
which is analogous to Kelland's was noticed by Greenhill (I. c.}. It is

0 = A sinh -^ sinh -^ cos kas . cos (<rf+e) ......... (xiv),

V^ V*

with a2 =

When kh is small, this makes o-2=^/A, so that the 'speed' is very great
compared with that given by the theory of 'long' waves. The oscillation
is in fact mainly transversal, with a very gradual variation of phase as we
pass along the canal. The middle line of the surface is of course nodal.

When kh is great, we get ' edge-waves,' as before.
The remaining asymmetrical oscillations are given by

<f> = A (sinh ay sin fiz + sin /3y sinh az) cos ksc . cos (crt+f) ...... (xvi).

This leads in the same manner as before to

a2-/32 = £2 ................................. (xvii),

o-'2 sinh ah = ga cosh ah,}
and ..................... (xvm>'

whence ah coth ah — ^h cot fth ........................... (xix).

There are an infinite number of solutions, with values of /3A in the third,
fifth, seventh, ... quadrants, giving 3, 5, 7, ... longitudinal nodes, one of which
is central.

3°. The case of a canal with plane sides inclined at 60° to the vertical has
been recently treated by Macdonald*. He has discovered a very compre
hensive type, which may be verified as follows.

The assumption

0 = P cos kx . cos (<rt+e) ........................... (xx),

where

P = A cosh kz+B sinh /b+cosh - ccosh +D sinh ..

evidently satisfies the equation of continuity; and it is easily shewn that
it makes

for y= ±*j3z, provided

C=2A, D=-2B ........................... (xxii).

The surface-condition, Art. 239 (4), is then satisfied, provided

•^7 (A cosh kh + B sinh kh) = A sinh kh + B cosh M, }

9* [ ...... (xxiii).

2<r2 / , ,kh D . , M\ 4 . , kh D , kh (

-j- ( A cosh -zr-B sinh — ) = A sinh -5- — /» cosh — \

gk \ 2.2/ L * )

* "Waves in Canals," Proc. Lond. Math. Soc., t. xxv., p. 101 (1894).

240] SPECIAL CASES. 435

The former of these is equivalent to

/ 2 \ \

A = H ( cosh kh T sinh kh ) ,

\ 2 9k / (xxiv),

Bmff(£ cosh kh - sinh kh\
and the latter then leads to

2 | — \ — 3 — coth 3 — j-l = 0 (xxv).

Also, substituting from (xxii) and (xxiv) in (xxi), we find

The equations (xxv) and (xxvi) are those arrived at by Macdonald, by a
different process. The surface- value of P is

...... (xxvii).

The equation (xxv) is a quadratic in <^igk. In the case of a wave whose
length (27r/£) is great compared with A, we have

t, 3M 2
C0th 2~- = 3M'

nearly, and the roots of (xxv) are then

<r*lgk=%kh, and a*lgk=\lkh .................. (xxviii),

approximately. If we put <r = kc, the former result gives c'* = ^gh, in accord
ance with the usual theory of ' long ' waves (Arts. 166, 167). The formula
(xxvii) now makes P=3jST, approximately; this is independent of y, so that
the wave ridges are nearly straight. The second of the roots (xxviii) makes
(T2=g/h, giving a much greater wave-velocity ; but the considerations adduced
above shew that there is nothing paradoxical in this*. It will be found on
examination that the cross-sections of the waves are parabolic in form, and
that there are two nodal lines parallel to the length of the canal. The period
is, in fact, almost exactly that of the symmetrical transverse oscillation
discussed in Art. 237, 2°.

When, on the other hand, the wave-length is short compared with the
transverse dimensions of the canal, kh is large, and coth |M = 1, nearly. The
roots of (xxv) are then

l and ^\gk=\ ........................ (xxix),

approximately. The former result makes P=ff, nearly, so that the wave-
ridges are straight, experiencing only a slight change of altitude towards the

* There is some divergence here, and elsewhere in the text, from the views
maintained by Greenhill and Macdonald in the papers cited.

28—2

436 SURFACE WAVES. [CHAP. IX

sides. The speed, o- = (gk}%, is exactly what we should expect from the general
theory of waves on relatively deep water.

If in this case we transfer the origin to one edge of the water-surface,
writing z+h for zt and y — *J%h for y, and then make h infinite, we get the case
of a system of waves travelling parallel to a shore which slopes downwards at
an angle of 30° to the horizon. The result is

....... (xxx),

where c =(#/£)*. This admits of immediate verification. At a distance of a
wave-length or so from the shore, the value of <£, near the surface, reduces to

</> = H^z cos kx . cos (<rt + e) ........................ (xxxi),

practically, as in Art. 217 *. Near the edge the elevation changes sign, there
being a longitudinal node for which

(xxxii),

The second of the two roots (xxix) gives a system of edge- waves, the results
being equivalent to those obtained by making /3=30° in Stokes' formula.

Oscillations of a Spherical Mass of Liquid.

241. The theory of the gravitational oscillations of a mass of
liquid about the spherical form has been given by Lord Kelvin •(•.

Taking the origin at the centre, and denoting the radius vector
at any point of the surface by a + f, where a is the radius in the
undisturbed state, we assume

r=2"Cn .............................. (1),

where fn is a surface-harmonic of integral order n. The equation
of continuity V2<£ = 0 is satisfied by

where Sn is a surface-harmonic, and the kinematical condition

* The result contained in (xxx) does not appear to have been hitherto noticed.

t Sir W. Thomson, "Dynamical Problems regarding Elastic Spheroidal Shells
and Spheroids of Incompressible Liquid," Phil. Trans., 1863; Math, and Phijs.
Papers, t. iii., p. 384.

240-241] OSCILLATIONS OF A LIQUID GLOBE. 437

to be satisfied when r — a, gives

d£n _ n « ...

~S- a "-

The gravitation-potential at the free surface is, by Art. 192,

47T7pa3 * 4,Trypa (

W Zi 27i+l>"

where 7 is the gravitation-constant. Putting

g=%7rypa, r = a + 2?w,
we find

n = const. +^22(^r^ ................. (6>-

Substituting from (2) and (6) in the pressure equation

(7),

-
p at

we find, since p must be constant over the surface,

Eliminating Sn between (4) and (8), we obtain

This shews that fw oc cos (crnt + e), where

2n(n-l)jr
"n' 2» + l B"

For the same density of liquid, g oc a, and the frequency is
therefore independent of the dimensions of the globe.

The formula makes o^ = 0, as we should expect, since in the
deformation expressed by a surface-harmonic of the first order the
surface remains spherical, and the period is therefore infinitely
long.

" For the case n = 2, or an ellipsoidal deformation, the length of
the isochronous simple pendulum becomes }a, or one and a quarter
times the earth's radius, for a homogeneous liquid globe of the
same mass and diameter as the earth ; and therefore for this case,

438

SURFACE WAVES.

[CHAP, ix

or for any homogeneous liquid globe of about 5 J times the density
of water, the half-period is 47 m. 12s.*"

" A steel globe of the same dimensions, without mutual gravi
tation of its parts, could scarcely oscillate so rapidly, since the
velocity of plane waves of distortion in steel is only about 10,140
feet per second, at which rate a space equal to the earth's diameter
would not be travelled in less than 1 h. 8 m. 40s.f "

When the surface oscillates in the form of a zonal harmonic spheroid of the
second order, the equation of the lines of motion is xw'i = const., where zsr
denotes the distance of any point from the axis of symmetry, which is taken
as axis of x (see Art. 94 (11)). The forms of these lines, for a series of equi
distant values of the constant, are shewn in the annexed figure.

242. This problem may also be treated very compactly by the
method of 'normal coordinates' (Art. 165).

The kinetic energy is given by the formula

* Sir W. Thomson, /.. c.

t Sir W. Thomson. The exact theory of the vibrations of an elastic sphere
gives, for the slowest oscillation of a steel globe of the dimensions of the earth, a
period of 1 h. 18m. Proc. Lond. Math. Soc., t. xiii., p. 212 (1882).

241-242] METHOD OF NORMAL COORDINATES. 439

where 88 is an element of the surface r = a. Hence, when the
surface oscillates in the form r = a + fn, we find, on substitution
from (2) and (4),

(12).

To find the potential energy, we may suppose that the external
surface is constrained to assume in succession the forms r = a + 0?n,
where 6 varies from 0 to 1. At any stage of this process, the
gravitation potential at the surface is, by (6),

fl = const. + ^?n ............... (13).

Hence the work required to add a film of thickness £n&0 is

W.^2gpfKSdS .................. (14).

Integrating this from 6 = 0 to 6 = I, we find

(15).

The results corresponding to the general deformation (1) are
obtained by prefixing the sign 2 of summation with respect to n,
in (12) and (15); since the terms involving products of surface-
harmonics of different orders vanish, by Art. 88.

The fact that the general expressions for T and V thus reduce
to sums of squares shews that any spherical-harmonic deformation
is of a ' normal type.' Also, assuming that fw oc cos (<rnt + e), the
consideration that the total energy T + V must be constant leads
us again to the result (10).

In the case of the forced oscillations due to a disturbing
potential IT cos (at + e) which satisfies the equation V2fy = 0 at
all points of the fluid, we must suppose H' to be expanded in
a series of solid harmonics. If fn be the equilibrium -elevation
corresponding to the term of order n, we have, by Art. 165 (12),
for the forced oscillation,

where a- is the imposed speed, and <rn that of the free oscillations
of the same type, as given by (10).

440 SURFACE WAVES. [CHAP. IX

The numerical results given above for the case n = 2 shew
that, in a non-rotating liquid globe of the same dimensions and
mean density as the earth, forced oscillations having the cha
racters and periods of the actual lunar and solar tides, would
practically have the amplitudes assigned by the equilibrium-theory.

243. The investigation is easily extended to the case of an
ocean of any uniform depth, covering a symmetrical spherical
nucleus.

Let b be the radius of the nucleus, a that of the external surface. The
surface-form being

we assume, for the velocity-potential,

{rn /,«, + 11

f*+1>P+*^}* ........................... (ii),

where the coefficients have been adjusted so as to make d<j)/dr = 0 for r=b.
The condition that

for r=a, gives

t

For the gravitation -potential at the free surface (i) we have

where p0 is the mean density of the whole mass. Hence, putting # =
we find

The pressure-condition at the free surface then gives

-A

:» (vii).

The elimination of Sn between (iv) and (vii) then leads to

j:r + 0na£n=0 (viii),

242-243] OCEAN OF UNIFORM DEPTH. 441

If p=p0, we have o-1 = 0 as we should expect. When p>p0 the value of o-j
is imaginary ; the equilibrium configuration in which the external surface
of the fluid is concentric with the nucleus is then unstable. (Of. Art. 192.)

If in (ix) we put & = 0, we reproduce the result of the preceding Art. If,
on the other hand, the depth of the ocean be small compared with the radius,
we find, putting 6 = a-A, and neglecting the square of A/a,

3 P\ ffh t,\

provided n be small compared with a/A. This agrees with Laplace's result,
obtained in a more direct manner in Art. 192.

But if n be comparable with a/A, we have, putting n = ka,

so that (ix) reduces to (ra=#£tanh£A ................................. (xi),

as in Art. 217. Moreover, the expression (ii; for the velocity-potential
becomes, if we write r=a + z,

+ h) .............................. (xii),

where <px is a function of the coordinates in the surface, which may now be
treated as plane. Cf. Art. 236.

The formulae for the kinetic and potential energies, in the general case, are
easily found by the same method as in the preceding Art. to be

and

The latter result shews, again, that the equilibrium configuration is one of
minimum potential energy, and therefore thoroughly stable, provided p < p0 .

In the case where the depth is relatively small, whilst n is finite, we obtain,
putting 6 = a -A,

whilst the expression for V is of course unaltered.

If the amplitudes of the harmonics £,t be regarded as generalized co
ordinates (Art. 165), the formula (xv) shews that for relatively small depths
the ' inertia-coefficients ' vary inversely as the depth. We have had frequent
illustrations of this principle in our discussions of tidal waves.

442 SURFACE WAVES. [CHAP. IX

Capillarity.

244. The part played by Cohesion in certain cases of fluid
motion has long been recognized in a general way, but it is only
within recent years that the question has been subjected to exact
mathematical treatment. We proceed to give some account of
the remarkable investigations of Lord Kelvin and Lord Rayleigh
in this field.

It is, of course, beyond our province to discuss the physical
theory of the matter*. It is sufficient, for our purpose, to know
that the free surface of a liquid, or, more generally, the common
surface of two fluids which do not mix, behaves as if it were in a
state of uniform tension, the stress between two adjacent portions
of the surface, estimated at per unit length of the common
boundary-line, depending only on the nature of the two fluids
and on the temperature. We shall denote this ' surface-tension/
as it is called, by the symbol T1. Its value in c.G.s. units (dynes
per linear centimetre) appears to be about 74 for a water-air
surface at 20°C.f ; it diminishes somewhat with rise of temperature.
The corresponding value for a mercury-air surface is about 540.

An equivalent statement is that the potential energy of any
system, of which the surface in question forms part, contains a
term proportional to the area of the surface, the amount of this
'superficial energy' per unit area being equal to T^. Since the
condition of stable equilibrium is that the energy should be a
minimum, the surface tends to contract as much as is consistent
with the other conditions of the problem.

The chief modification which the consideration of surface-
tension will introduce into our previous methods is contained in
the theorem that the fluid pressure is now discontinuous at a
surface of separation, viz. we have

* For this, see Maxwell, Encyc. Britann., Art. "Capillary Action"; Scientific
Papers, Cambridge, 1890, t. ii., p. 541, where references to the older writers are
given. Also, Lord Kayleigh, "On the Theory of Surface Forces," Phil. Mag.,
Oct. and Dec., 1890, and Feb. and May, 1892.

f Lord Kayleigh " On the Tension of Water-Surfaces, Clean and Contaminated,
investigated by the method of Hippies," Phil. Mag. Nov. 1890.

§ See Maxwell, Theory of Heat, London, 1871, c. xx.

244-245] EFFECT OF SURFACE TENSION. 443

where p, p' are the pressures close to the surface on the two sides,
and Rl, RZ are the principal radii of curvature of the surface, to be
reckoned negative when the corresponding centres of curvature lie
on the side to which the accent refers. This formula is readily
obtained by resolving along the normal the forces acting on a rect
angular element of a superficial film, bounded by lines of curvature ;
but it seems unnecessary to give here the proof, which may be found
in most recent treatises on Hydrostatics.

245. The simplest problem we can take, to begin with, is that
of waves on a plane surface forming the common boundary of two
fluids at rest.

If the origin be taken in this plane, and the axis of y normal
to it, the velocity-potentials corresponding to a simple-harmonic
deformation of the common surface may be assumed to be

<f) = Ceky cos Tex . cos (at + e), j

<£' = 0V^cosA?a?.cos(<r*+€)) ................ "*

where the former equation relates to the side on which y is
negative, and the latter to that on which y is positive. For these
values satisfy V2<£ = 0, V2<£' = 0, and make the velocity zero for
y = qp oo , respectively.

The corresponding displacement of the surface in the direction
of y will be of the type

rj = a cos kx . sin (at + e) .................. (2) ;

and the conditions that

drj/dt = — d(f>/dy = — dfy'jdy,
for y = 0, give

<ra = -kC = kC' ........................ (3).

If, for the moment, we ignore gravity, the variable part of the
pressure is given by

p dd> 0-a . , . , ^ .
- — —rr = -- eky cos kx . sin (<rt 4- e),

, (4).

' cos ^x • s^n (fft + 6)

To find the pressure-condition at the common surface, we may
calculate the forces which act in the direction of y on a strip of

444 SURFACE WAVES. [CHAP. IX

breadth Boo. The fluid pressures on the two sides have a resultant
(p'—p)Sx, and the difference of the tensions parallel to y on
the two edges gives S (f^ifty/tfe). We thus get the equation

to be satisfied when y — 0 approximately. This might have been
written down at once as a particular case of the general surface-
condition (Art. 244 (1)). Substituting in (5) from (2) and (4), we
find

T le*

<T2 = ff-7 ........................... (6),

P + P

which determines the speed of the oscillations of wave-length 2tr/k.

The energy of motion, per wave-length, of the fluid included between two
planes parallel to xyt at unit distance apart, is

o .-0 o V .y=o

If we assume 77 = a cos &e ....................................... (ii),

where a depends on t only, and therefore, having regard to the kinematical
conditions,

0 = - 1<rl a(*v cos lex, $' = k~l ae~*v cos kx ............... (iii),

we find T=±(p+p')k-l&.\ ........................... (iv).

Again, the energy of extension of the surface of separation is

Substituting from (ii), this gives

.\ ................................. (vi).

To find the mean energy, of either kind, per unit length of the axis of #,
we must omit the factor X.

If we assume that a <x cos (a-t + e), where o- is determined by (6), we verify
that the total energy T+ V is constant.

Conversely, if we assume that

T) = 2 (a cos kx -f /3 sin kx] ........................... ( vii),

it is easily seen that the expressions for T and V will reduce to sums of
squares of d, /3 and a, 0, respectively, with constant coefficients, so that the
quantities a, /3 are 'normal coordinates.' The general theory of Art. 165
then leads independently to the formula (6) for the speed.

CAPILLARY WAVES.

445

245-246]

By compounding two systems of standing waves, as in Art. 218,
we obtain a progressive wave-system

?; = a cos (kso + crt) (7),

travelling with the velocity

cr

c=k =

or, in terms of the wave-length,

.(9).

The contrast with Art. 218 is noteworthy; as the wave-length
is diminished, the period diminishes in a more rapid ratio, so that
the wave-velocity increases.

Since c varies as X~*, the group-velocity, Art. 221 (2), is in the
present case

c-xj = |o (10).

The fact that the group-velocity for capillary waves exceeds the
wave- velocity helps to explain some interesting phenomena to be
referred to later (Art. 249).

For numerical illustration we may take the case of a free
water-surface; thus, putting /> = !, /o' = 0, 2^ = 74, we have the
following results, the units being the centimetre and second *.

Wave-length.

Wave-velocity.

Frequency.

•50
•10
•05

30
68
96

61
680
1930

246. When gravity is to be taken into account, the common
surface, in equilibrium, will of course be horizontal. Taking the

* Cf. Sir W. Thomson, Math, and Phys. Papers, t. iii., p. 520.

The above theory gives the explanation of the 'crispations' observed on the
surface of water contained in a finger-bowl set into vibration by stroking the rim
with a wetted finger. It is to be observed, however, that the frequency of the
capillary waves in this experiment is double that of the vibrations of the bowl ; see
Lord Kayleigh, " On Maintained Vibrations," Phil. Mag., April, 1883.

446 SURFACE WAVES. [CHAP. IX

positive direction of y upwards, the pressure at the disturbed
surface will be given by

, e),

P, ,

-, = -^ - — gy = — I ?=r + g J a cos kx . sin (at + e)

approximately. Substituting in Art. 245 (5), we find

at^^gk+W .................... (2).

P+> /> + ?

Putting a = kc, we find, for the velocity of a train of progressive
waves,

where we have written

P') = T' ................ (4).

In the particular cases of Tl = 0 and g = 0, respectively, we fall
back on the results of Arts. 223, 245.

There are several points to be noticed with respect to the
formula (3). In the first place, although, as the wave-length
(2ir/&) diminishes from oo to 0, the speed (a-) continually increases,
the wave-velocity, after falling to a certain minimum, begins to
increase again. This minimum value (cm, say) is given by

(5),

and corresponds to a wave-length

Xm = 2^m = 27r(r/Sr)i .................. (6)*.

In terms of \m and cm the formula (3) may be written

«>•

* The theory of the minimum wave -velocity, together with most of the substance
of Arts. 245, 246, was given by Sir W. Thomson, " Hydrokinetic Solutions and
Observations," Phil. Mag., Nov. 1871; see also Nature, t. v., p. 1 (1871),

246] MINIMUM WAVE-VELOCITY. 447

shewing that for any prescribed value of c, greater than cm, there
are two admissible values (reciprocals) of X/Xm. For example,
corresponding to

c/cm = T2 1-4 1-6 1-8 2-0

we have

, f 2-476 3-646 4-917 6*322 7'873
1 m~{ -404 -274 -203 -158 -127,

to which we add, for future reference,

sin-1cni/c = 56°26/ 45° 35' 38° 41' 33° 45' 30°.

For sufficiently large values of X the first term in the formula
(3) for c2 is large compared with the second ; the force governing
the motion of the waves being mainly that of gravity. On the
other hand, when X is very small, the second term preponderates,
and the motion is mainly governed by cohesion, as in Art. 245. As
an indication of the actual magnitudes here in question, we may
note that if X/\m>10, the influence of cohesion on the wave-
velocity amounts only to about 5 per cent., whilst gravity becomes
relatively ineffective to a like degree if X/Xm < -fa.

It has been proposed by Lord Kelvin to distinguish by the
name of ' ripples' waves whose length is less than Xm.

The relative importance of gravity and cohesion, as depending on the
value of X, may be traced to the form of the expression for the potential
energy of a deformation of the type

The part of this energy due to the extension of the bounding surface is, per
unit area,

Tr^o'/X8 .................................... (ii),

whilst the part due to gravity is

p')«2 .................................... (iii).

As X diminishes, the former becomes more and more important compared with
the latter.

For a water-surface, using the same data as before, with # = 981, we find
from (5) and (6),

Xm = l-73, cm = 23'2,

the units being the centimetre and the second. That is to say, roughly, the
minimum wave-velocity is about nine inches per second, or -45 sea-miles per

448 SURFACE WAVES. [CHAP. IX

hour, with a wave-length of two-thirds of an inch. Combined with the
numerical results already obtained, this gives,

for c= 27-8 32-5 37'1 41-8 46 '4

in centimetres and seconds.

If we substitute from (7) in the general formula (Art. 221 (2))
for the group-velocity, we find

Hence the group-velocity is greater or less than the wave- velocity,
according as X > Xm. For sufficiently long waves the group- velocity

is practically equal to \c, whilst for very short waves it tends to
the value fc*.

A further consequence of (2) is to be noted. We have hitherto tacitly
supposed that the lower fluid is the denser (i.e. p>p')> as is indeed necessary
for stability when Tl is neglected. The formula referred to shews, however,
that there is stability even when p<p', provided

i.e. provided X be less than the wave-length Xm of minimum velocity when the
denser fluid is below. Hence in the case of water above and air below the
maximum wave-length consistent with stability is 1'73 cm. If the fluids
be included between two parallel vertical walls, this imposes a superior limit
to the admissible wave-length, and we learn that there is stability (in the two-
dimensional problem) provided the interval between the walls does not exceed
•86 cm. We have here an explanation, in principle, of a familiar experiment in
which water is retained by atmospheric pressure in an inverted tumbler, or
other vessel, whose mouth is covered by a gauze with sufficiently fine meshes t-

247. We next consider the case of waves on a horizontal
surface forming the common boundary of two parallel currents
U, Uf.

* Cf. Lord Rayleigh, L c. ante p. 383.

t The case where the fluids are contained in a cylindrical tube has been solved
by Maxwell, Encyc. Britann., Art. " Capillary Action," t. v., p. 69, Scientific Papers,
t. ii., p. 585, and compared with some experiments of Duprez. The agreement is
better than might have been expected when we consider that the special condition
to be satisfied at the line of contact of the surface with the wall of the tube has
been left out of account.

246-247] WAVES ON THE BOUNDARY OF TWO CURRENTS. 449

If we apply the method of Art. 224, we find without difficulty
that the condition for stationary waves is now

the last term being due to the altered form of the pressure
condition which has to be satisfied at the surface. Putting

U — — c, U' = - c + u, p'/p = s,
we get

where

i.e. c0 is the velocity of waves of the given length (Zir/k) when
there are no currents.

The various inferences to be drawn from (2) are much as in
Art. 224, with the important qualification that, since c0 has now a
minimum value, viz. the cm of Art. 246 (5), the equilibrium of
the surface when plane is stable for disturbances of all wave
lengths so long as

u<LJf.c»1 ........................ (4).

When the relative velocity u of the two currents exceeds this
value, c becomes imaginary for wave-lengths lying between certain
limits. It is evident that in the alternative method of Art. 225
the time-factor ei<Tt will now take the form e*=*t+Wt where

& — "\ T~: rr U — CA f K3 /3 — z /t'U ( O ).

J/llo\2 ' 1lr» \/

The real part of the exponential indicates the possibility of a
disturbance of continually increasing amplitude.

For the case of air over water we have s = -00129, cm = 23'2 (c.s.), whence
the maximum value of u consistent with stability is about 646 centimetres
per second, or (roughly) 12*5 sea-miles per hour*. For slightly greater values
of u the instability will manifest itself by the formation, in the first in-

* The wind-velocity at which the surface of water actually begins to be ruffled
by the formation of capillary waves, so as to lose the power of distinct reflection, is
much less than this, and is determined by other causes. We shall revert to this
point later (Art. 302).

L. 29

450 SURFACE WAVES. [CHAP. IX

stance, of wavelets of about two-thirds of an inch in length, which will
continually increase in amplitude until they transcend the limits implied
in our approximation.

248. We resume the investigation of the effect of a steady
pressure-disturbance on the surface of a running stream, by the
method of Arts. 226, 227, including now the effect of capillary
forces. This will give, in addition to the former results, the ex
planation (in principle) of the fringe of ripples which is seen in
advance of a solid moving at a moderate speed through still
water, or on the up-stream side of any disturbance in a uniform
current.

Beginning with a simple-harmonic distribution of pressure, we
assume

<f>/c = - x + ftefr sin lex, \ .

^/c«-y+£tf**coefcr J '

the upper surface coinciding with the stream-line -^ = 0, whose
equation is

y = ft cos kx ........................... (2),

approximately. At a point just beneath this surface we find, as
in Art. 226 (8), for the variable part of the pressure,

— = ft {(kc* - g) cos kx + yuc sin kx} ............ (3),

where //, is the frictional coefficient. At an adjacent point just
above the surface we must have

p p x*

= ft {(kcz -g- &T) cos kx + pc sin kx] ......... (4),

where T' is now written for TJp. This is equal to the real part of

ft (kc* -g- k*T' - ific) eikx.
Writing P for the coefficient, we find that to the imposed pressure

will correspond the surface-form

_ (k^ ~ 9 ~ k*Tf) cos kx — nc sin kx

(5)

247-249] SURFACE DISTURBANCE OF A STREAM. 451

Let us first suppose that the velocity c of the stream exceeds
the minimum wave-velocity (cm) investigated in Art. 246. We may
then write

K,)(K2-k) .................. (7),

where K^ tc2 are the two values of k corresponding to the wave-
velocity c on still water; in other words, ZTT/KH 27r//c2 are the lengths
of the two systems of free waves which could maintain a stationary
position in space, on the surface of the flowing stream. We will
suppose that K<>> K^.

In terms of these quantities, the formula (6) may be written

_ P (k — K!) (K2 — k) cos kx — /// sin kx , .
y = T" (k-Kjfa-ky + p'*

where /// = pc/T '. This shews that if p! be small the pressure is
least over the crests, and greatest over the troughs of the waves
when k is greater than K.2 or less than KI} whilst the reverse is
the case when k is intermediate to telt K2. In the case of a pro
gressive disturbance advancing over still water, these results are
seen to be in accordance with Art. 165 (12).

249. From (8) we can infer as in Art. 227 the effect of
a pressure of integral amount Q concentrated on a line of the
surface at the origin, viz. we find

Q r (k- ^(Kz-k) cos kx - fjf sin kx „ ( .

y - *TI - Jo -^tf&i-ky+jfi-

This definite integral is the real part of

e**dk

r

J o

The dissipation-coefficient // has been introduced solely for the purpose of
making the problem determinate; we may therefore avail ourselves of the
slight gain in simplicity obtained by supposing pf to be infinitesimal. In this
case the two roots of the denominator in (i) are

k =

where v=p.'/(ic.2-Kl).

Since *2 > Ki> " is positive. The integral (i) is therefore equivalent to

i f r <p*dk r _jttatifc__i

K2-Kl-2iv\J0 A-fa + fr) Jo k-(K2-iv)f~

29—2

452 SURFACE WAVES. [CHAP. IX

These integrals are of the forms discussed in Art. 227. It appears that
when x is positive the former integral is equal to

r e~ilcx 7

27rieiKlX+ I TT— «* (1U)>

Jo &-*-<!

and the latter to

dk

o

On the other hand, when x is negative, the former reduces to

~ dk... ....(v),

and the latter to

0-ikx

r°° p-ikx

+j.jS£f (vi)-

We have here simplified the formulae by putting v = 0 after the transfor
mations.

If we now discard the imaginary parts of our expressions, we obtain the
results which immediately follow.

When fjf is infinitesimal, the equation (9) gives, for x positive,
^.y = - *?—sanc& + F(x) (10),

V #2 ~ Kl

and, for x negative,

2?T . -r., *. /i-i\

^) ............... (11),

where

' 1 ([""coskxjj [xcoskx,j} .

=~ \\ -j --- dk- ,- dk> ....... (12).

K2- K! (Jo k + K! Jo k + K2 J

This function F(x) can be expressed in terms of the known func
tions Ci «!#, Si #!#, Ci K&, Si K.&, by Art. 227 (ix). The disturb
ance of level represented by it is very small for values of x,
whether positive or negative, which exceed, say, half the greater
wave-length (^TT/KJ).

Hence, beyond some such distance, the surface is covered on
the down-stream side by a regular train of simple-harmonic waves
of length 27r/Klt and on the up-stream side by a train of the
shorter wave-length 27r/«;2. It appears from the numerical results
of Art. 246 that when the velocity c of the stream much exceeds
the minimum wave- velocity (cm) the former system of waves is
governed mainly by gravity, and the latter by cohesion.

249]

WAVES AND RIPPLES.

453

It is worth notice that, in contrast with
the case of Art. 227, the elevation is now
finite when x = 0, viz. we have

(^ K2 Kl KI

This follows easily from (10).

The figure shews the transition between
the two sets of waves, in the case of K2 = 5^.

The general explanation of the effects
of an isolated pressure-disturbance advanc
ing over still water, indicated near the end
of Art. 227, is now modified by the fact
that there are tivo wave-lengths correspond
ing to the given velocity c. For one of
these (the shorter) the group-velocity is
greater, whilst for the other it is less, than
c. We can thus understand why the waves
of shorter wave-length should be found
ahead, and those of longer wave-length in
the rear, of the disturbing pressure.

It will be noticed that the formulge (10),
(11) make the height of the up-stream
capillary waves the same as that of the
down-stream gravity waves ; but this result
will be greatly modified when the pressure
is diffused over a band of finite breadth,
instead of being concentrated on a mathe
matical line. If, for example, the breadth
of the band do not exceed one-fourth of the
wave-length on the down-stream side, whilst
it considerably exceeds the wave-length of
the up-stream ripples, as may happen with
a very moderate velocity, the different parts
of the breadth will on the whole reinforce
one another as regards their action on the
down-stream side, whilst on the up-stream
side we shall have ' interference,' with a
comparatively small residual amplitude.

454 SURFACE WAVES. [CHAP. IX

When the velocity c of the stream is less than the minimum
wave- velocity, the factors of

are imaginary. There is now no indeterminateness caused by
putting fju = 0 ab initio. The surface-form is given by

Q r coskx „

The integral might be transformed by the previous method, but it
is evident a priori that its value tends rapidly, with increasing x,
to zero, on account of the more and more rapid fluctuations in
sign of cos kx. The disturbance of level is now confined to the
neighbourhood of the origin. For x = 0 we find

Finally we have the critical case where c is exactly equal to
the minimum wave- velocity, and therefore K2 = /q. The first term
in (10) or (11) is now infinite, whilst the remainder of the expres
sion, when evaluated, is finite. To get an intelligible result in
this case it is necessary to retain the frictional coefficient //.

If we put // = 2sr2) we have

(k-^ + i^f={k-(< + ^-iw}}{k-(<-w-\-iw}} ............ (vii),

so that the integral (i) may now be equated to

l-M' r r» <p* r» j** \ .

- — \ I j — -. -- .—.dk- I -j — r—i - !~v<fifcV ...... (vm).

4tzr \J9 k-^-w+iw] J Q k-(K. + TZ-iw} J

The formulae of Art. 227 shew that when or is small the most important
part of this expression, for points at a distance from the origin on either side,

is

It appears that the surface-elevation is now given by

7T ,

/i/>\

(16).

The examination of the effect of inequalities in the bed of a
stream, by the method of Art. 230, must be left to the reader.

249-250] EFFECT OF A PRESSURE-POINT. 455

250. The investigation by Lord Rayleigh*, from which the
foregoing differs principally in the manner of treating the definite
integrals, was undertaken with a view to explaining more fully
some phenomena described by Scott Russell f and Lord Kelvin J.

"When a small obstacle, such as a fishing line, is moved
forward slowly through still water, or (which of course cornes to
the same thing) is held stationary in moving water, the surface is
covered with a beautiful wave-pattern, fixed relatively to the
obstacle. On the up-stream side the wave-length is short, and, as
Thomson has shewn, the force governing the vibrations is prin
cipally cohesion. On the down-stream side the waves are longer,
and are governed principally by gravity. Both sets of waves move
with the same velocity relatively to the water ; namely, that
required in order that they may maintain a fixed position relatively
to the obstacle. The same condition governs the velocity, and
therefore the wave-length, of those parts of the pattern where the
fronts are oblique to the direction of motion. If the angle between
this direction and the normal to the wave-front be called 0, the
velocity of propagation of the waves must be equal to VQ cos 0,
where v0 represents the velocity of the water relatively to the fixed
obstacle.

" Thomson has shewn that, whatever the wave-length may be,
the velocity of propagation of waves on the surface of water cannot
be less than about 23 centimetres per second. The water must
run somewhat faster than this in order that the wave-pattern may
be formed. Even then the angle 6 is subject to a limit defined by
v0 cos 6 = 23, and the curved wave-front has a corresponding
asymptote.

" The immersed portion of the obstacle disturbs the flow of the
liquid independently of the deformation of the surface, and renders
the problem in its original form one of great difficulty. We may
however, without altering the essence of the matter, suppose that
the disturbance is produced by the application to one point of the
surface of a slightly abnormal pressure, such as might be produced
by electrical attraction, or by the impact of a small jet of air.

* I. e. ante p. 393. + « On Waves," Brit. Ass. Rep., 1844.

£ 1. c. ante p. 446.

456 SURFACE WAVES. [CHAP. IX

Indeed, either of these methods — the latter especially — gives very
beautiful wave-patterns*."

The solution of the problem here stated is to be derived from
the results of the last Art. in the manner explained in Art. 228.

For a line of pressure making an angle JTT — 0 with the
direction of the stream, the distances (p) of the successive wave-
ridges from the origin are given by

kp = (2m - %) TT,
where m is an integer, and the values of k are determined by

l&T -&c2cos20 + # = 0 (1).

If we put cm = (40r)* (2),

and cosa = cm/c, a = (m - J) 7rc2/# (3),

this gives £_- 2^cos*0+ cos4a = 0 (4),

whence p/a = cos2 9 ± (cos4 6 — cos4 a)* (5).

The greater of these two values of p corresponds to the down
stream and the smaller to the up-stream side of the seat of
disturbance.

The general form of the wave-ridges due to a pressure-point at
the origin is then given, on Huyghens' principle, by (5), considered
as a ' tangential-polar ' equation between p and 6. The four lines
for which 6 = ± a. are asymptotes. The values of ^TT — a for several
values of c/cm have been tabulated in Art. 246.

The figure opposite shews the wave-system thus obtained,
in the particular case where the ratio of the wave-lengths in the
line of symmetry is 4 : 1. This corresponds to a= 26° 34'*h

In the outlying parts of the wave-pattern, where the ridges
are nearly straight, the wave-lengths of the two systems are
nearly equal, and we have then the abnormal amplitude indicated
by equation (16) of the preceding Art.

When the ratio c/cm is at all considerable, a is nearly equal to |TT, and
the asymptotes make a very acute angle with the axis. The wave-envelope

* Lord Eayleigh, I. c.

t The figure may be compared with the drawing, from observation, given by
Scott Russell, I c.

250-251]

WAVE-PATTERN.

457

on the down-stream side then approximates to the form investigated in Art.
228, except that the curve, after approaching the axis of x near the origin,
runs back along the asymptotes. On the up-stream side we have approxi
mately

j» = &sec2# (i),

where b = \a cos4 a. This gives

251. Another problem of great interest is the determination
of the nature of the equilibrium of a cylindrical column of liquid,
of circular section. This contains the theory of the well-known
experiments of Bidone, Savart, and others, on the behaviour of a
jet issuing under pressure from a small orifice in the wall of a
containing vessel. It is obvious that the uniform velocity in the

458 SURFACE WAVES. [CHAP. IX

direction of the axis of the jet does not affect the dynamics of the
question, and may be disregarded in the analytical treatment.

We will take first the two-dimensional vibrations of the
column, the motion being supposed to be the same in each
section. Using polar coordinates r, 0 in the plane of a section,
with the origin in the axis, we may write, in accordance with
Art. 63,

rs

(/> = J. — cos s# . cos (oi + e) ............... (1),

Cl>

where a is the mean radius. The equation of the boundary at
any instant will then be

r-o+f .............................. (2),

where £ = -- cos s6 . sin (at + e) ............... (3),

<T&

the relation between the coefficients being determined by

dt ~ dr ' '

for r = a. For the variable part of the pressure inside the column,
close to the surface, we have

- = -~- = — aA cos sO . sin (at + e) ............ (5).

p ctt

The curvature of a curve which differs infinitely little from a
circle having its centre at the origin is found by elementary
methods to be

1_ = 1_ 1 &r

R ~ r r* dQ* '
or, in the notation of (2),

Hence the surface condition

p = T,/R + const., ..................... (7),

gives, on substitution from (5),

* For the original investigation, by the method of energy, see Lord Eayleigh,
"On the Instability of Jets," Proc. Lond. Math. Soc., t. x., p. 4 (1878); "On the
Capillary Phenomena of Jets," Proc. Boy. Soc., May 5, 1879. The latter paper
contains a comparison of the theory with experiment.

251-252] VIBRATIONS OF A CYLINDRICAL JET. 459

For s=l, we have cr = 0; to our order of approximation the
section remains circular, being merely displaced, so that the
equilibrium is neutral. For all other integral values of s, o-2 is
positive, so that the equilibrium is thoroughly stable for two-
dimensional deformations. This is evident a priori, since the
circle is the form of least perimeter, and therefore least potential
energy, for given sectional area.

In the case of a jet issuing from an orifice in the shape of an
ellipse, an equilateral triangle, or a square, prominence is given to
the disturbance of the type 5 = 2, 3, or 4, respectively. The motion
being steady, the jet exhibits a system of stationary waves, whose
length is equal to the velocity of the jet multiplied by the period

252. Abandoning now the restriction to two dimensions,
we assume that

<f> = facoskz . cos(<rt + e) ..................... (9),

where the axis of z coincides with that of the cylinder, and fa is a
function of the remaining coordinates x, y. Substituting in the
equation of continuity, V20 = 0, we get

07^-^)^ = 0 ..................... (10),

where Vi2 = d^jdx^ + d2/dy'2. If we put x — r cos 6, y = r sin 0, this
may be written

SKt^S-*- ............ <»>•

This equation is of the form considered in Art. 187, except for the
sign of & ; the solutions which are finite for r = 0 are therefore of
the type

fa=BIt(kr)™\80 ..................... (12),

bill \

where

Zs

Hence, writing

<f) = BIS (kr) cos s6 cos Icz . cos (at + e) ......... (14),

we have, by (4),

s ......... (15).

o~ft

460 SURFACE WAVES. [CHAP. IX

To find the sum of the principal curvatures, we remark that, as an
obvious consequence of Euler's and Meunier's theorems on curva
ture of surfaces, the curvature of any section differing infinitely
little from a principal normal section is, to the first order of small
quantities, the same as that of the principal section itself. It is
sufficient therefore in the present problem to calculate the curva
tures of a transverse section of the cylinder, and of a section
through the axis. These are the principal sections in the
undisturbed state, and the principal sections of the deformed
surface will make infinitely small angles with them. For the
transverse section the formula (6) applies, whilst for the axial
section the curvature is — d2 Qdz* ; so that the required sum of
the principal curvatures is

._

R R a a*V:dPi dz*

Also, at the surface,

- = -./= — crB Is (ka) cos s6 cos kz . sin (at + e) . . . (17).
p at

The surface-condition Art. 244 (1) then gives

^2 = 2_ ............ ^

l,(ka) pas

For s > 0, o-2 is positive ; but in the case (s = 0) of symmetry about
the axis a2 will be negative if ka < 1 ; that is, the equilibrium is
unstable for disturbances whose wave-length (2ir/k) exceeds the
circumference of the jet. To ascertain the type of disturbance for
which the instability is greatest, we require to know the value of
ka which makes

kaIQ'(ka)
I9(ka) (k l)'

a maximum. For this Lord Rayleigh finds k2a2 = '4858, whence,
for the wave-length of maximum instability,

2-7T/& = 4-508 x 2a.
There is a tendency therefore to the production of bead-like

252-253] INSTABILITY OF A JET. 461

swellings and contractions of this wave-length, with continually
increasing amplitude, until finally the jet breaks up into detached
drops*.

253. This leads naturally to the discussion of the small
oscillations of a drop of liquid about the spherical form-f. We
will slightly generalize the question by supposing that we have a
sphere of liquid, of density p, surrounded by an infinite mass
of other liquid of density p.

Taking the origin at the centre, let the shape of the common
surface at any instant be given by

?' = a + f =a+ 8n.sin(at + e) (1),

where a is the mean radius, and Sn is a surface-harmonic of order
n. The corresponding values of the velocity-potential will be,
at internal points,

fTCt W'v

= -- -Sn. cos (at + e) (2),

7i a

and at external points

(T(i a

n+i

(3),

Id T JL /

since these make

~di==~~dr= ~~dr'

for r = a. The variable parts of the internal and external pressures
at the surface are then given by

To find the sum of the curvatures we make use of the theorem

* The argument here is that if we have a series of possible types of disturbance,
with time-factors ettl , e°2<, e"3*, ..., where a1>a2>a3> ..., and if these be excited
simultaneously, the amplitude of the first will increase relatively to those of the

other components in the ratios a'*1'***, e(otl ~ a3^, . . . . The component with the
greatest a will therefore ultimately predominate.

The instability of a cylindrical jet surrounded by other fluid has been discussed
by Lord Eayleigh, " On the Instability of Cylindrical Fluid Surfaces," Phil. Mag.,
Aug. 1892. For a jet of air in water the wave-length of maximum instability is
found to be 6-48 x 2a.

t Lord Eayleigh, I. c. ante p. 458; Webb, Mess, of Math., t. ix. p. 177 (1880).

462

SURFACE WAVES.

[CHAP. IX

of Solid Geometry, that if X, p, v be the direction-cosines of the
normal at any point of a surface F(x, y, z) = 0, viz.

1 1 d\ diji dv
then ~p~"*~fcr = ;7~"*~77+;7~ ( )'

Since the square of ? is to be neglected, the equation (1) of the
harmonic spheroid may also be written

r = o + ft,, (6),

rn

where f?l= — Sn . sin (at + e) (7),

CL

i.e. %n is a solid harmonic of degree n. We thus find

X d%n X y

• ' f dii w& ' V /'

Z Cv^n ^ ^

~r~dzlr2^n
whence

~~ a a2

Substituting from (4) and (9) in the general surface-condition of
Art. 244, we find

If we put p = 0, this gives

The most important mode of vibration is the ellipsoidal one,
for which n = 2 ; we then have

253] VIBRATIONS OF A GLOBULE. 463

Hence for a drop of water, putting Tl = 74, p = 1, we find, for the
frequency,

<r/27r = 3'87a-§ seconds,

if a be the radius in centimetres. The radius of the sphere
which would vibrate seconds is a = 2*47 cm. or a little less than an
inch.

The case of a spherical bubble of air, surrounded by liquid,
is obtained by putting p = 0 in (10), viz. we have

l)(» + 2)j ............ (12).

For the same density of the liquid, the frequency of any given
mode is greater than in the case represented by (11), on account
of the diminished inertia ; cf. Art. 90 (6), (7).

CHAPTER X.

WAVES OF EXPANSION.

254. A TREATISE on Hydrodynamics would hardly be complete
without some reference to this subject, if merely for the reason
that all actual fluids are more or less compressible, and that it is
only when we recognize this compressibility that we escape such
apparently paradoxical results as that of Art. 21, where a change
of pressure was found to be propagated instantaneously through a
liquid mass.

We shall accordingly investigate in this Chapter the general
laws of propagation of small disturbances, passing over, however,
for the most part, such details as belong more properly to the
Theory of Sound.

In most cases which we shall consider, the changes of pressure
are small, and may be taken to be proportional to the changes
in density, thus

Aj> = *> (1),

where K (=p dp/dp) is a certain coefficient, called the 'elasticity of
volume.1 For a given liquid the value of K varies with the
temperature, and (very slightly) with the pressure. For water at
15° C., K = 2'22 x 1010 dynes per square centimetre; for mercury
at the same temperature tc= 5*42 x 1011. The case of gases will
be considered presently.

Plane Waves.

255. We take first the case of plane waves in a uniform
medium.

The motion being in one dimension (x), the dynamical equation
is, in the absence of extraneous forces,

du du _ 1 dp _ 1 dp dp X-.X

dt dx p dx p dp dx"

254-255] PLANE WAVES. 465

whilst the equation of continuity, Art. 8 (4), reduces to

If we put

p=pQ(I+s) ........................... (3),

where pQ is the density in the undisturbed state, s may be called
the 'condensation' in the plane x. Substituting in (1) and (2),
we find, on the supposition that the motion is infinitely small,

du K ds ,..

Tt = ~p,dx'"

ds du /K,

and -j- = — --j- .......................... (5),

dt dx

if K = [pdp/dp]p=p(} ........................ (6),

as above. Eliminating s we have

#u = c,#u

dt2 dx2'"

where c2 = K/p0 = [dp/dp]p=po ..................... (8).

The equation (7) is of the form treated in Art. 107, and the
complete solution is

u = F(ct-x)+f(ct + x) .................. (9),

representing two systems of waves travelling with the constant
velocity c, one in the positive and the other in the negative
direction of x. It appears from (5) that the corresponding value
of s is given by

............... (10).

For a single wave we have

u = ± cs ..................... ...... (11),

since one or other of the functions F,f is zero. The upper or the
lower sign is to be taken according as the wave is travelling in
the positive or the negative direction.

There is an exact correspondence between the above approximate theory
and that of ' long ' gravity- waves on water. If we write 77 /A for s, and gh for
ic/Po, the equations (4) and (5), above, become identical with Art. 166 (3), (5).

I,. 30

466 WAVES OF EXPANSION. [CHAP. X

256. With the value of K given in Art. 254, we find for water

at 15° C.

c = 1490 metres per second.

The number obtained directly by Colladon and Sturm in their
experiments on the lake of Geneva was 1437, at a temperature
of 8° C *

In the case of a gas, if we assume that the temperature is
constant, the value of K is determined by Boyle's Law

viz. K=PO .............................. (2),

so that c = O>/po)* ........................ (<S)-

This is known as the ' Newtonian ' velocity of sound •(*. If we
denote by H the height of a 'homogeneous atmosphere' of the
gas, we have pQ = gp0H, and therefore

c=foH)» ........................... (4),

which may be compared with the formula (8) of Art. 167 for
the velocity of ' long ' gravity-waves in liquids. For air at 0° C.
we have as corresponding values J

p0 = 76 x 13-60 x 981, p0 = -00129,
in absolute C.G.S. units; whence

c = 280 metres per second.
This is considerably below the value found by direct observation.

The reconciliation of theory and fact is due to Laplace §.
When a gas is suddenly compressed, its temperature rises, so
that the pressure is increased more than in proportion to the
diminution of volume ; and a similar statement applies of course
to the case of a sudden expansion. The formula (1) is appro
priate only to the case where the expansions and rarefactions are
so gradual that there is ample time for equalization of temperature
by thermal conduction and radiation. In most cases of interest,
the alternations of density are exceedingly rapid ; the flow of heat

* Ann. de Chim. et de Phys., t. xxxvi. (1827).
t Principia, Lib. ii., Sect, viii., Prop. 48.
J Everett, Units and Physical Constants.

% " Sur la vitesse du son dans 1'air et dans 1'eau, Ann. de Chim. et de Phys.,
t. iii. (1816); Mecanique Celeste, Livre 12me, c. iii. (1823).

256] VELOCITY OF SOUND. 467

from one element to another has hardly set in before its direction
is reversed, so that practically each element behaves as if it
neither gained nor lost heat.

On this view we have, in place of (1), the ' adiabatic ' law

where, as explained in books on Thermodynamics, 7 is the ratio of
the two specific heats of the gas. This makes

K = ypo .............................. ..... (6),

and therefore c = (WO/PO)* ........................... (7).

If we put 7=1*410*, the former result is to be multiplied by
1-187, whence

c = 332 metres per second,

which agrees very closely with the best direct determinations.

The confidence felt by physicists in the soundness of Laplace's view is so
complete that it is now usual to apply the formula (7) in the inverse manner,
and to infer the values of y for various gases and vapours from observation
of wave-velocities in them.

In strictness, a similar distinction should be made between the ' adiaba
tic ' and ' isothermal ' coefficients of elasticity of a liquid or a solid, but
practically the difference is unimportant. Thus in the case of water the
ratio of the two volume-elasticities is calculated to be l'0012f.

The effects of thermal radiation and conduction on air-waves have been
studied theoretically by Stokes J and Lord Rayleigh§. When the oscillations
are too rapid for equalization of temperature, but not so rapid as to exclude
communication of heat between adjacent elements, the waves diminish in
amplitude as they advance, owing to the dissipation of energy which
takes place in the thermal processes.

According to the law of Charles and Gay Lussac
p0/po oc 1 + -00366 0,

where 6 is the temperature Centigrade. Hence the velocity of
sound will vary as the square root of the absolute temperature.
For several of the more permanent gases, which have sensibly the
same value of 7, the formula (7) shews that the velocity varies

* The value found by direct experiment.
t Everett, Units and Physical Constants.

J " An Examination of the possible effect of the Radiation of Heat on the Pro
pagation of Sound," Phil. Mag., April, 1851.
§ Theory of Sound, Art. 247.

30—2

468 WAVES OF EXPANSION. [CHAP. X

inversely as the square root of the density, provided the relative
densities be determined under the same conditions of pressure and
temperature.

257. The theory of plane waves can also be treated very
simply by the Lagrangian method.

If f denote the displacement at time t of the particles whose
undisturbed abscissa is xt the stratum of matter originally in
cluded between the planes x and as + &e is at the time t + 8t
bounded by the planes

and *+?+

so that the equation of continuity is

where p0 is the density in the undisturbed state. Hence if ,9
denote the 'condensation' (p — /OO)/PO> we nave

dx

The dynamical equation obtained by considering the forces
acting on unit area of the above stratum is

o**- dp (S)

*"~;-

These equations are exact, but in the case of small motions
we may write

........................... (4),

_ .

Substituting in (3) we find

where c2 = K/p0 ........................... (7).

The interpretation of (6) is the same as in Arts. 167, 255.

256-258] ENERGY OF SOUND WAVES. 469

258. The kinetic energy of a system of plane waves is given

by

T-faff&tadydM (1),

where u is the velocity at the point (x, y, z] at time t.

The calculation of the intrinsic energy requires a little care.
If v be the volume of unit mass, the work which this gives out in
expanding from its actual volume to the normal volume v0 is

I pdv

J V

(2).

Putting v = Vu/(l+s), p=po+KS, we find, for the intrinsic energy
(E) of unit mass

^=boS + (i*-po)s2K (3),

if we neglect terms of higher order. Hence, for the intrinsic
energy of the fluid which in the disturbed condition occupies any
given region, we have the expression

W=jtfEpdxdydz = p9fffE (1 + s) dxdydz

= S!5(pQs + %KS>}dxdydz (4),

since p0v0 = I. If we consider a region so great that the con
densations and rarefactions balance, we have

fjfsdxdydz = 0 (5),

and therefore W — \ KJjjs^dxdydz (6).

In a progressive plane wave we have cs = ± u, and therefore
T = W. The equality of the two kinds of energy, in this case, may
also be inferred from the more general line of argument given in
Art. 171.

In the theory of Sound special interest attaches, of course, to
the case of simple-harmonic vibrations. If a be the amplitude
of a progressive wave of period 27r/cr, we may assume, in con
formity with Art. 257 (6),

f = a cos (kx — at + e) (7),

where k = <r/c. The formulae (1) and (6) then give, for the energy
contained in a prismatic space of sectional area unity and length
\ (in the direction #),

T+ W = ^p»<rW\ (8),

470 WAVES OF EXPANSION. [CHAP. X

the same as the kinetic energy of the whole mass when animated
with the maximum velocity <ra.

The rate of transmission of energy across unit area of a plane moving
with the particles situate in it is

p ~ =p<ra sin (kx -a-t+e) , (i).

The work done by the constant part of the pressure in a complete period is
zero. For the variable part we have

Ajo = KS= — K-~

CkOG

Substituting in (i), we find, for the mean rate of transmission of energy,

W£a2 = £Po<72a2xc (iii).

Hence the energy transmitted in any number of complete periods is exactly
that corresponding to the waves which pass the plane in the same time.
This is in accordance with the general theory of Art. 221, since, c being
independent of X, the group-velocity is identical with the wave- velocity.

Waves of Finite Amplitude.

259. If p be a function of p only, the equations (1) and (3) of
Art. 257, give, without approximation,

d^ = £dp d^
dt* pfdp'da?"

On the ' isothermal ' hypothesis that

this becomes -7-~ = —

dt PQ

In the same way, the ' adiabatic ' relation

These exact equations (3) and (4) may be compared with the similar
equation for 'long' waves in a uniform canal, Art. 170 (3).

258-260] WAVES OF FINITE AMPLITUDE. 471

It appears from (1) that the equation (6) of Art. 257 could be regarded as
exact if the relation between p and p were such that

Hence plane waves of finite amplitude can be propagated without change of
type if, and only if,

A relation of this form does not hold for any known substance, whether at
constant temperature or when free from gain or loss of heat by conduction
and radiation. Hence sound-waves of finite amplitude must inevitably under
go a change of type as they proceed.

260. The laws of propagation of waves of finite amplitude
have been investigated, independently and by different methods,
by Earnshaw and Riemann. It is proposed to give here a brief
account of these investigations, referring for further details to the
original papers, and to the very full discussion of the matter
by Lord Rayleigh *.

Riemann's method •(• has already been applied in this treatise
to the discussion of the corresponding question in the theory
of 'long' gravity-waves on liquids (Art. 183). He starts from
the Eulerian equations of Art. 255, which may be written

du du 1 dp dp

-j2 + u -j- = -- j ^ .................. (I)-

at dec p dp dx

dp dp du

-f,+u~T = -p-r ........................ (2).

dt dx r dx

If we put

P=f(p)+u, Q=f(p)-u ............... (3),

where f(p) is as yet undetermined, we find, multiplying (2) by
/' (p), and adding to (1),

dP udP_=_±dpdp_ ff( }du
dt dx p dp dx ™ ^' dx '

If we now determine f(p) so that

* Theory of Sound, c. xi.

t " Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite,"
Gott. Abh., t. viii. (1860) ; Werke, Leipzig, 1876, p. 145.

472 WAVES OF EXPANSION. [CHAP. X

this may be written

dp dp .,. dp

dt+ud^=-tf^d* .................. <°>-

In the same way we obtain

The condition (4) is satisfied by

Substituting in (5) and (6), we find

\dpj j J dx

Hence dP = 0, or P is constant, for a geometrical point moving
with the velocity

*-$)'+« ........................ »

whilst Q is constant for a point whose velocity is

.(10).

Hence, any given value of P moves forward, and any value of Q
moves backward, with the velocity given by (9) or (10), as
the case may be.

These results enable us to understand, in a general way, the
nature of the motion in any given case. Thus if the initial
disturbance be confined to the space between the two planes
x = a, x = b, we may suppose that P and Q both vanish for x > a
and for x < b. The region within which P is variable will
advance, and that within which Q is variable will recede, until
after a time these regions separate and leave between them a
space for which P = 0, Q = 0, and in which the fluid is therefore
at rest. The original disturbance has thus been split up into two
progressive waves travelling in opposite directions. In the
advancing wave we have Q = 0, and therefore

260-261] RIEMANN'S THEORY. 473

so that both the density and the particle-velocity are propagated
forwards at the rate given by (9). Whether we adopt the isother
mal or the adiabatic law of expansion, this velocity of propagation
will be found to be greater, the greater the value of p. The law
of progress of the wave may be illustrated by drawing a curve
with x as abscissa and p as ordinate, and making each point
of this curve move forward with the appropriate velocity, as
given by (9) and (11). Since those parts move faster which
have the greater ordinates, the curve will eventually become at
some point perpendicular to x. The quantities dujdx, dpjdx are
then infinite ; and the preceding method fails to yield any infor
mation as to the subsequent course of the motion. Cf. Art. 183.

261. Similar results can be deduced from Earnshaw's investi
gation*, which is, however, somewhat less general in that it
applies only to a progressive wave supposed already established.

For simplicity we will suppose p and p to be connected by Boyle's Law

P=c*p ....................................... (i).

If we write y=x-\-l~-> so that y denotes the absolute coordinate at time t of
the particle whose undisturbed abscissa is #, the equation (3) of Art. 259
becomes

da?dae

This is satisfied by

Hence a first integral of (ii) is

To obtain the ' general integral ' of (v) we must eliminate a between the
equations

-

where <£ is arbitrary. Now

dy/dx=p0/p,

* " On the Mathematical Theory of Sound," Phil. Trans., 1860.
t See Forsyth, Differential Equations, c. ix.

474 WAVES OF EXPANSION. [CHAP. X

so that, if u be the velocity of the particle #, we have

(vii).

On the outskirts of the wave we shall have u=0, p=p0- It follows that
(7=0, and therefore

P=P^ulc ................................. (viii).

Hence in a progressive wave p and u must be connected by this relation.
If this be satisfied initially, the function $ which occurs in (vi) is to be
determined from the conditions at time t = 0 by the equation

To obtain results independent of the particular form of the wave, consider
two particles (which we will distinguish by suffixes) so related that the value
of p which obtains for the first particle at time ^ is found at the second
particle at time t.2.

The value of a ( = p0/p) is the same for both, and therefore by (vi), with
(7=0,

#2-#i = « (#2 - #1) ±c &- *i) lo§ «>1 ( }

Q = a(xz-x1)±c(tt-t]) J '

The latter equation may be written

A# . p / -N

-TT=+C- .................................... xi),

A* po

shewing that the value of p is propagated from particle to particle at the rate
p/p0 . c. The rate of propagation in space is given by

= +c+u .................................... (xii).

This is in agreement with Eiemann's results, since on the 'isothermal'
hypothesis (dp/dp)*=c.

For a wave travelling in the positive direction we must take the lower
signs. If it be one of condensation (p>p0), u is positive, by (viii). It follows
that the denser parts of the wave are continually gaining on the rarer, and
at length overtake them ; the subsequent motion is then beyond the scope of
our analysis.

Eliminating x between the equations (vi), and writing for c log a its value
— u, we find for a wave travelling in the positive direction,

y = (c + u)t+F(a) ........................... (xiii).

In virtue of (viii) this is equivalent to

261-262] EARNSHAW'S. THEORY. 475

This formula is due to Poisson*. Its interpretation, leading to the same
results as above, for the mode of alteration of the wave as it proceeds, forms
the subject of a paper by Stokes f.

262. The conditions for a wave of permanent type have been
investigated in a very simple manner by Rankine J.

If, as in Art. 172, we impress on everything a velocity c equal
and opposite to that of the waves, we reduce the problem to one of
steady motion.

Let A, B be two points of an ideal tube of unit section drawn
in the direction of propagation, and let the values of the pressure,
the density, and the particle-velocity at A and B be denoted
by plt p1} K! and£>2> p2, u2, respectively.

Since the same amount of matter now crosses in unit time
each section of the tube, we have

Pi(c-ul) = pa(c-ua\ = mt (1),

say ; where m denotes the mass swept past in unit time by a plane
moving with the wave, in the original form of the problem. This
quantity m is called by Rankine the ' mass- velocity ' of the wave.

Again, the total force acting on the mass included between A
and B is p.2—pi, and the rate at which this mass is gaining
momentum is

m(c — tbi) — m(c — ut).

Hence p2—pl = m(t^-u1) (2).

Combined with (1) this gives

p1-\-m2/pl=p2 + m-/p2 (3).

Hence a wave of finite amplitude cannot be propagated un
changed except in a medium such that

p + mz/p = const (4).

This conclusion has already been arrived at, in a different manner,
in Art. 259.

* " Memoire sur la Theorie du Son," Journ. de VEcole Polytechn., t. vii., p. 319
(1808).

t " On a Difficulty in the Theory of Sound," Phil. Mag., NOT. 1848; Math, and
Phys. Papers, t. ii., p. 51.

% "On the Thermodynamic Theory of Waves of Finite Longitudinal Disturb
ance," Phil. Trans., 1870.

476 WAVES OF EXPANSION. [CHAP. X

If the variation of density be slight, the relation (4) may,
however, be regarded as holding approximately for actual fluids,
provided m have the proper value. Putting

0c ............ (5),

we find c2 = /c/p0 .............................. (6),

as in Art. 255.

The fact that in actual fluids a progressive wave of finite
amplitude continually alters its type, so that the variations of
density towards the front become more and more abrupt, has led
various writers to speculate on the possibility of a wave of dis
continuity, analogous to a ' bore ' in water-waves.

It has been shewn, first by Stokes*, and afterwards by several
other writers, that the conditions of constancy of mass and of
constancy of momentum can both be satisfied for such a wave.
The simplest case is when there is no variation in the values of
p and u except at the plane of discontinuity. If, in Rankine's
argument, the sections A, B be taken, one in front of, and the
other behind this plane, we find

m

and, if we further suppose that u2 = 0, so that the medium is at
rest in front of the wave,

(8),

p2 \pi - P*

m -L /(ffl - ff2) (Pi - P2)V /QX

and u, = c = 11 ("/•

ft \

The upper or the lower sign is to be taken according as pl is
greater or less than p2, i.e. according as the wave is one of
condensation or of rarefaction.

These results have, however, lost some of their interest since
it has been pointed out by Lord Rayleigh -f* that the equation of
energy cannot be satisfied consistently with (1) and (2). Con
sidering the excess of the work done on the fluid entering the

* I. c. ante p. 475.

t Theory of Sound, Art. 253.

262-263] CONDITION FOR PERMANENCY OF TYPE. 477

space AB at B over that done by the fluid leaving at A, we find

p2 (c - Us) -pi(c- MJ) = Jm {(c - ?O2 - (c - Ma)2}

+ m(El-E,} ...... (10),

where the first term on the right-hand represents the gain of
kinetic, and the second that of intrinsic energy ; cf. Art. 23. As
in Art. 11 (7), we have

It is easily shewn that (10) is inconsistent with (2) unless

' ^-E^—^-pf) .................. (12),

which is only satisfied provided the relation between p and
p be that given by (4). In words, the conditions for a wave
of discontinuity can only be satisfied in the case of a medium
whose intrinsic energy varies as the square of the pressure.

In the above investigation no account has been taken of
dissipative forces, such as viscosity and thermal conduction and
radiation. Practically, a wave such as we have been considering
would imply a finite difference of temperature between the
portions of the fluid on the two sides of the plane of discontinuity,
so that, to say nothing of viscosity, there would necessarily be a
dissipation of energy due to thermal action at the junction.
Whether this dissipation would be of such an amount as to be
consistent, approximately, with the relation (12) is a physical
question, involving considerations which lie outside the province
of theoretical Hydrodynamics.

Spherical Waves.

263. Let us next suppose that the disturbance is symmetrical
with respect to a fixed point, which we take as origin. The
motion is necessarily irrotational, so that a velocity-potential
(p exists, which is here a function of r, the distance from the
origin, and t, only. If as before we neglect the squares of
small quantities, we have by Art. 21 (3)

J p 'dt '

478 WAVES OF EXPANSION. [CHAP. X

In the notation of Arts. 254, 255 we may write

/> fKCfc
J P J Po

whence tfs = -~ (1).

(MI

To form the equation of continuity we remark that, owing to
the difference of flux across the inner and outer surfaces, the
space included between the spheres r and r + $r is gaining mass
at the rate

dH ^dr,

Since the same rate is also expressed by dp/dt . 4?rr2 Br we have

„ dp d

This might also have been arrived at by direct transformation of
the general equation of continuity, Art. 8 (4). In the case of
infinitely small motions, (2) gives

_ _

dt~r*dr\r dr

whence, substituting from (1),

^-^A(r
d? r* dr V dr

This may be put into the more convenient form

so that the solution is

(6).

Hence the motion is made up of two systems of spherical waves,
travelling, one outwards, the other inwards, with velocity c.
Considering for a moment the first system alone, we have

= --F'(r-ct),

which shews that a condensation is propagated outwards with
velocity c, but diminishes as it proceeds, its amount varying

263-264] SPHERICAL WAVES. 479

inversely as the distance from the origin. The velocity due to
the same train of waves is

As r increases the second term becomes less and less important
compared with the first, so that ultimately the velocity is pro
pagated according to the same law as the condensation.

264. The determination of the functions F and / in terms of
the initial conditions, for an unlimited space, can be effected
as follows. Let us suppose that the initial distributions of velocity
and condensation are determined by the formulae

where i/r, % are arbitrary functions, of which the former must
fulfil the condition ^r'(0) = 0, since otherwise the equation of
continuity would not be satisfied at the origin. Both functions
are given, primd facie, only for positive values of the variable ;
but all our equations are consistent with the view that r changes
sign as the point to which it refers passes through the origin. On
this understanding we have, on account of the symmetry of the
circumstances with respect to the origin,

+ (-r) = + (r), x(-r) = x(r) ............... (8),

that is, -\|r and % are even functions. From (6) and (7) we have

If we put

J%X(r)<fr = Xl(r) ..................... (10),

the latter equation may be written

= iXl(r) .................. (11),

the constant of integration being omitted, as it will disappear
from the final result. We notice that

%i(-'')=%0) ..................... (12).

480 WAVES OF EXPANSION. [CHAP. X

Hence, we have

1_

2c (13).

The complete value of c/> is then given by (6), viz.

As a very simple example, we may suppose that the air is initially at rest,
and that the disturbance consists of a uniform condensation SQ extending
through a sphere of radius a. The formula) then shew that after a certain
time the disturbance is confined to a spherical shell whose internal and
external radii are ct-a and ct + a, and that the condensation at any point
within the thickness of this shell is given by

The condensation is therefore positive through the outer half, and negative
through the inner half, of the thickness. This is a particular case of a
general result stated long ago by Stokes*, according to which a diverging
spherical wave must necessarily contain both condensed and rarefied portions.

We shall require shortly the form which the general value
(14) of $ assumes at the origin. This is found most simply by
differentiating both sides of (14) with respect to r and then
making r = 0. The result is, if we take account of the relations
(8), (10), (12),

(15).

General Equation of Round Waves.

265. We proceed to the general case of propagation of ex
pansion-waves. We neglect, as before, the squares of small
quantities, so that the dynamical equation is as in Art. 263,

r8?-^ m

" dt '

* " On Some Points in the Keceived Theory of Sound," Phil. Mag., Jan. 1849 ;
Math, and Pliy». Papers, t. ii., p. 82.

264-265] GENERAL EQUATION. 481

Also, writing p = pQ (1 + s) in the general equation of continuity,
Art. 8 (4), we have, with the same approximation,

dt dx~ dy~ dzz"
The elimination of s between (1) and (2) gives

or, in our former notation,

Since this equation is linear, it will be satisfied by the arith
metic mean of any number of separate solutions </>ly <£2, </>3,....
As in Art. 39, let us imagine an infinite number of systems of
rectangular axes to be arranged uniformly about any point P as .
origin, and let <f>1, <f>», <£3, ... be the velocity-potentials of motions
which are the same with respect to these systems as the original
motion </> is with respect to the system x, y, z. In this case the
arithmetic mean (</>, say), of the functions <f>l} </>2, </>3,... will be
the velocity-potential of a motion symmetrical with respect to
the point P, and will therefore come under the investigation of
Art. 264, provided r denote the distance of any point from P. In
other words, if $ be a function of r and t, defined by the equation

(5),

where <£ is any solution of (4), and Biz- is the solid angle subtended
at P by an element of the surface of a sphere of radius r having
this point as centre, then

dt* dr*

Hence r$ = F(r-ct)+f(r + ct) .................. (7).

The mean value of $ over a sphere having any point P of
the medium as centre is therefore subject to the same laws as the

* This result was obtained, in a different manner, by Poisson, " Me'moire sur la
the'orie du son," Journ. de VEcole Polytechn., t. vii. (1807), pp. 334—338. The remark
that it leads at once to the complete solution of (4) is due to Liouville, Journ. de
Math., 1856, pp. 1—6.

L. 31

482 WAVES OF EXPANSION. [CHAP. X

velocity-potential of a symmetrical spherical disturbance. We
see at once that the value of <£ at P at the time t depends on
the mean initial values of (/> and d(f)/dt over a sphere of radius ct
described about P as centre, so that the disturbance is propagated
in all directions with uniform velocity c. Thus if the original
disturbance extend only through a finite portion S of space, the
disturbance at any point P external to 2 will begin after a time
r-i/c, will last for a time (r2 — ?*i)/c, and will then cease altogether ;
r-L , r2 denoting the radii of two spheres described with P as centre,
the one just excluding, the other just including 2.

To express the solution of (4), already virtually obtained, in
an analytical form, let the values of (f> and d(f>/dt, when t = 0, be

<j> = ty(x,y,z\ -* = x(a;,y,*) ............... (8).

The mean initial values of these quantities over a sphere of radius
r described about (#, y, z) as centre are

</> = -^ 1 1 ty (x + Ir, y + mr, z + nr) dvr,
=

x + r> y + mr> z + nr ™>

where I, m, n denote the direction- cosines of any radius of this
sphere, and £57 the corresponding elementary solid angle. If we
put

I = sin 6 cos ft>, m = sin 6 sin o>, n — cos 6,

we shall have

Stzr =sin

Hence, comparing with Art. 264 (15), we see that the value of
(f> at the point {x, y, z), at any subsequent time t, is

<f> —T~ -J-. • t 1 1 ty (® + ct sin 6 cos &>, y -f ct sin 6 sin &>,

z -\- ct cos 6) sin OdOda)
+ — 1 1 ^ (x + ct sin 6 cos o>, y + ct sin # sin o>,

s + cZ cos 6) sin 0d0d&> . . . (9),
which is the form given by Poisson*.

* " M6moire sur I'int6gration de quelques Equations lineaires aux differences
partielles, et particulierement de 1'equation generale du mouvement des fluides
elastiques," N.6m. de VAcad. des Sciences, t. iii., 1818-19.

265-266] ARBITRARY INITIAL CONDITIONS. 483

266. In the case of simple-harmonic motion, the time-factor
being ei(Tt, the equation (4) of Art. 265 takes the form

(V* + #)0 = 0 ........................ (1),

where k = a/c .............................. (2).

It appears on comparison with Art. 258 (7) that 2-7T/& is the wave
length of plane waves of the same period (27T/0-).

There is little excuse for trespassing further on the domain of
Acoustics; but we may briefly notice the solutions of (1) which
are appropriate when the boundary- conditions have reference to
spherical surfaces, as this will introduce us to some results of
analysis which will be of service in the next Chapter.

In the case of symmetry with respect to the origin, we have
by Art. 263 (5), or by direct transformation of (1),

%*+"•*-<> ..................... <3>>

the solution of which is

A sin&r ncoskr ...

+ = A +8- .................... (4).

When the motion is finite at the origin we must have B = 0.

1°. This may be applied to the radial vibrations of air contained in a
spherical cavity. The condition that d(f)/dr = 0 at the surface r = a gives

tan ka = Tca .................................... (i),

which determines the frequencies of the normal modes. The roots of this
equation, which presents itself in various physical problems, can be cal
culated without much difficulty, either by means of a series*, or by a
method devised by Fourierf. The values obtained by Schwerdt for the
first few roots are

£a/7r = 1-4303, 2-4590, 3-4709, 4'4774, 5-4818, 6'4844 ......... (ii),

approximating to the form ra + £, where m is integral. These numbers give
the ratio (2a/X) of the diameter of the sphere to the wave-length. Taking
the reciprocals we find

X/2a=-6992, -4067, -2881, -2233, -1824, -1542 ............... (iii).

In the case of the second and higher roots of (i) the roots of lower order give

* Euler, . Introductio in Analysin Infinitorum, Lausannse, 1748, t. ii., p. 319 ;
Rayleigh, Theory of Sound, Art. 207.

t Theorie analytique de la Chaleur, Paris, 1822, Art. 286.
% Quoted by Verdet, Optique Physique, t. i., p. 266.

31—2

484 WAVES OF EXPANSION. [CHAP. X

the positions of the spherical nodes (c?$/dr=0). Thus in the second mode
there is a spherical node whose radius is given by

r/a = (l'4303)/(2-4590)= '5817.

2°. In the case of waves propagated outwards into infinite space from a
spherical surface, it is more convenient to use the solution of (3), including
the time-factor, in the form

(iv).

If the motion of the gas be due to a prescribed radial motion

r=ae^ ....................................... (v)

of a sphere of radius a, C is determined by the condition that r= -dfyjdr for
r = a. This gives

whence, taking the real parts, we have, corresponding to a prescribed normal
motion

(vh),

- a)}

When lea is infinitesimal, this reduces to

where A = knot a. We have here the conception of the 'simple source' of
sound, which plays so great a part in the modern treatment of Acoustics.

The rate of emission of energy may be calculated from the result of Art. 258.
At a great distance r from the origin, the waves are approximately plane, of
amplitude A/4^rcr. Putting this value of a in the expression £p0o-2a2c for the
energy transmitted across unit area, and multiplying by 4nr2, we obtain for
the energy emitted per second

&7TC

267. When the restriction as to symmetry is abandoned, we
may suppose the value of </> over any sphere of radius r, having
its centre at the origin, to be expanded in a series of surface-
harmonics whose coefficients are functions of r. We therefore
assume

n (5),

266-267] SPHERICAL BOUNDARY. 485

where <f>n is a solid harmonic of degree n, and Rn is a function of r
only. Now

dy dy dz dz

dx dy dz

(6).

And, by the definition of a solid harmonic, we have

Hence

If we substitute in (1), the terms in </>n must satisfy the
equation independently, whence

which is the differential equation in Rn.

This can be integrated by series. Thus, assuming that

En = 2Am(kr)™}

the relation between consecutive coefficients is found to be
m (2n +1+ ra) Am + Am_^ = 0.

This gives two ascending series, one beginning with m = 0, and the
other with m = — 2n — 1 ; thus

-
2(l-2n)2.4(l-27i)(3-2n) "V'

where A, B are arbitrary constants. Hence putting <£n = rw<8fn, so
that Sn is a surface-harmonic of order n, the general solution of (1)
may be written

^)J r»& ............ (9),

486 WAVES OF EXPANSION. [CHAP. X

where

---

2(l-2w) 2.4(l-2n)(3-2n) '"
......... (10)*.

The first term of (9) is alone to be retained when the motion
is finite at the origin.

The functions tyn (f), ^n (f) can also be expressed in finite
terms, as follows :

' d sin? ^

-.

(11).

These are readily identified with (10) by expanding sin f, cos f, and
performing the differentiations. As particular cases we have

sn

3 IV . 3 cos?

The formulae (9) and (11) shew that the general solution of the
equation

2 (n + 1) d& + Rn = 0 (12

which is obtained by writing f for AT in (8), is

d nAe*+Ber*

This is easily verified ; for if ,ftn be any solution of (12), we find that the
corresponding equation for Rn + 1 is satisfied by

* There is a slight deviation here from the notation adopted by Heine, Kugel-
functionen, p. 82.

267-268] SOLUTION IN SURFACE HARMONICS. 487

and by repeated applications of this result it appears that (12) is satisfied by

where Itn is the solution of

that is

268. A simple application of the foregoing analysis is to the
vibrations of air contained in a spherical envelope.

1°. Let us first consider the free vibrations when the envelope is rigid.
Since the motion is finite at the origin, we have, by (9),

t^A^W^SH.e™ ................................. (i),

with the boundary-condition

ka^(ka} + n^n(ka}=0 .............................. (ii),

a being the radius. This determines the admissible values of k and thence of

It is evident from Art. 267 (11) that this equation reduces always to the
form

tan £&«(&») ................................. (iii),

where f(ka) is a rational algebraic function. The roots can then be calculated
without difficulty by Fourier's method, referred to in Art. 266.

In the case n = l, if we take the axis of x coincident with that of the
harmonic Slt and write .v = rcos 6, we have

kr cos kr

and the equation (ii) becomes

(v).

The zero root is irrelevant. The next root gives, for the ratio of the
diameter to the wave-length,

and the higher values of this ratio approximate to the successive integers
2, 3, 4.... In the case of the lowest root, we have, inverting,

X/2«= 1-509.

* The above analysis, which has a wide application in mathematical physics,
has been given, in one form or another, by various writers, from Poisson (Tlieorie
matheinatique de la Chaleur, Paris, 1835) downwards. For references to the history
of the matter, considered as a problem in Differential Equations, see Glaisher, "On
Riccati's Equation and its Transformations," PhiL Trans., 1881.

488 WAVES OF EXPANSION. [CHAP. X

In this, the gravest of all the normal modes, the air sways to and fro much
in the same manner as in a doubly-closed pipe. In the case of any one of
the higher roots, the roots of lower order give the positions of the spherical
nodes (d<f)/dr = Q). For the further discussion of the problem we must refer to
the investigation by Lord Kayleigh*.

2°. To find the motion of the enclosed air due to a prescribed normal
motion of the boundary, say

we have, +»4f»(ir}f'^.:/>v .............................. (ix),

with the condition A (ka^ (&*) + n^n (ka)}an~ * = 1 ,

and therefore 0- ^"(^ , n . . a (-Y Sn. ei<Tt ...... (x).

This expression becomes infinite, as we should expect, whenever ka is a root
of (ii) ; i.e. whenever the period of the imposed vibration coincides with that
of one of the natural periods, of the same spherical-harmonic type.

By putting ka = Q we pass to the case of an incompressible fluid. The
formula (x) then reduces to

as in Art. 90. It is important to notice that the same result holds approximately,
even in the case of a gas, whenever ka is small, i.e. whenever the wave-length
(27r/£) corresponding to the actual period is large compared with the circum
ference of the sphere. This is otherwise evident from the mere form of the
fundamental equation, Art. 266 (1), since as k diminishes the equation tends
more and more to the form v2<£ = 0 appropriate to an incompressible fluid t.

* " On the Vibrations of a Gas contained within a Rigid Spherical Envelope,"
Proc. Lond. Math. Soc., t. iv., p. 93 (1872) ; Theory of Sound, Art. 331.

t In the transverse oscillations of the air contained in a cylindrica vessel we

have

where Ylz=d2/dx2 + d'2ldy~. In the case of a circular section, transforming to polar
coordinates r, 6, we have, for the free oscillations,

with k determined by

J.'(fca) = 0,

a being the radius. The nature of the results will be understood from Art. 187,
where the mathematical problem is identical. The figures on pp. 308, 309 shew the
forms of the lines of equal pressure, to which the motion of the particle is ortho
gonal, in two of the more important modes. The problem is fully discussed in
Lord Eayleigh's Theory of Sound, Art. 339.

268-270] VIBRATIONS OF AIR IN SPHERICAL CAVITY. 489

269. To determine the motion of a gas within a space
bounded by two concentric spheres, we require the complete
formula (9) of Art. 267. The only interesting case, however, is
where the two radii are nearly equal ; and this can be solved more
easily by an independent process*.

In terms of polar coordinates r, 6, o>, the equation (v>2 + &2) <£ = 0 becomes

-

dr2 r dr r2 [_dfi (" ' dp J 1 - p.2 da>2

If, now, d(f)/dr = 0 for r=a and »• = &, where a and b are nearly equal, we may
neglect the radial motion altogether, so that the equation reduces to

It appears, exactly as in Art. 191, that the only solutions which are finite over
the whole spherical surface arc of the type

0*S,t ....................................... (iii),

where Sn is a surface-harmonic of integral order n, and that the corresponding
values of k are given by

In the gravest mode (n = l), the gas sways to and fro across the equator
of the harmonic Slt being, in the extreme phases of the oscillation, condensed
at one pole and rarefied at the other. Since ka = >jZ in this case, we have
for the equivalent wave-length A/2a= 2*221.

In the next mode (?i = 2), the type of the vibration depends on that of
the harmonic 8.2. If this be zonal, the equator is a node. The frequency is
determined by tca=j6,or X/2a = 1'283.

270. We may next consider the propagation of waves outivards
from a spherical surface into an unlimited medium.

If at the surface (r—a) we have a prescribed normal velocity

the appropriate solution of (v2+£2) 0=0 is

/ ^

^Cknrn\krd(kr}]

for this is included in the general formula (13) of Art. 207, and evidently
represents a system of waves travelling outwards f.

* Lord Rayleigh, Theory of Sound, Art. 333.

t This problem was solved, in a somewhat different manner, by Stokes, "On
the Communication of Vibrations from a Vibrating Body to a surrounding
Gas," Phil. Trans., 1868.

490 WAVES OF EXPANSION. [CHAP. X

We shall here only follow out in detail the case of ?i = l, which corresponds
to an oscillation of the sphere, as rigid, to and fro in a straight line. Putting

Sl = a cos 8 .................................... (iii),

where 6 is the angle which r makes with the line in which the centre
oscillates, the formula (ii) reduces to

(iv).
The value of C is determined by the surface-condition

-^Loe^cos* ........ , ..................... (v),

for r=a. This gives

0_ _

The resultant pressure on the sphere is

X= - I Ap cos 6 . 27ra2 sin 6dB ..................... (vii),

J o
where A^ = c2p0s = p0c^/c& = iVp0</> ........................ (viii).

Substituting from (iv) and (vi), and performing the integration, we find

„ 2 + Fa2-iFa3 . iat .. .

X= --|7rPoa3. —j-j-p-p— ivae1** .................. (ix).

This may be written in the form

du

where u(=ael<rt) denotes the velocity of the sphere.

The first term of this expression is the same as if the inertia of the
sphere were increased by the amount

_ , .,

whilst the second is the same as if the sphere were subject to a frictional force
varying as the velocity, the coefficient being

In the case of an incompressible fluid, and, more generally, whenever the
wave-length 2?r/^ is large compared with the circumference of the sphere, we
may put ka = Q. The addition to the inertia is then half that of the fluid
displaced ; whilst the frictional coefficient vanishes f. Cf. Art. 91.

The frictional coefficient is in any case of high order in ka^ so that the
vibrations of a sphere whose circumference is moderately small compared with

* This formula is given by Lord Bayleigh, Theory of Sound, Art. 325. For another
treatment of the problem of the vibrating sphere, see Poisson, " Sur les mouvements
simultan6s d'un pendule et de 1'air environnant," Mem. de VAcad. des Sciences,
t. xi. (1832), and Kirchhoff, Mecluinik, c. xxiii.

f Poisson, I.e.

270-271] VIBRATING SPHERE. 491

the wave-length are only slightly affected in this way. To mid the energy
expended per unit time in generating waves in the surrounding medium, we
must multiply the frictional term in (x), now regarded as an equation in real
quantities, by u, and take the mean value ; this is found to be

In other words, if pl be the mean density of the sphere, the fraction of its
energy which is expended in one period is

It has been tacitly assumed in the foregoing investigation that the ampli
tude of vibration of the sphere is small compared with the radius. This
restriction may however be removed, if we suppose the symbols u, v, w to
represent the component velocities of the fluid, not at a fixed point of space,
but at a point whose coordinates relative to a system of axes originating at
the centre of the sphere, and moving with it, are #, y, z. The only change
which this will involve is that we must, in our fundamental equations, replace

<L b d- *-

The additional terms thus introduced are of the second order in the velocities,
and may consistently be neglected*.

271. The theory of such questions as the large-scale oscilla
tions of the earth's atmosphere, where the equilibrium-density
cannot be taken to be uniform, has received little attention at the
hands of mathematicians.

Let us suppose that we have a gas in equilibrium under certain
constant forces having a potential XI, and let us denote by p0 and
PO the values of p and p in this state, these quantities being in
general functions of the coordinates x, y, z. We have, then,

The equations of small motion, under the influence (it may be) of
disturbing forces having a potential XI', may therefore be written
du _ dp p dp0

dx

dv _ dp p dp0
dt" dy p.dy ^

dw _ dp p dp0
» = -']-~

(2).

* Cf. Stokes, Camb. Trans., t. ix., p. [50]. The assumption is that the maximum
velocity of the sphere is small compared with the velocity of sound.

492 WAVES OF EXPANSION. [CHAP. X

The case that lends itself most readily to mathematical treat
ment is where the equilibrium-temperature is uniform*, and
the expansions and contractions are assumed to follow the 'iso
thermal' law, so that

c denoting the Newtonian velocity of sound. If we write

p = p0(l+s), p=p0(l+s),
the equations (2) reduce to the forms

du 0 d , _x

dv d , _,

*=-cV~s)'
„*

^

(4)'
_-*.-/.»

where

e (V/^2 /K\

s- — M/C (&;,

that is, 5 denotes the ' equilibrium- value ' of the condensation due
to the disturbing-potential H'.

The general equation of continuity, Art. 8 (4), gives, with the
same approximation,

ds _ d d , . d

We find, by elimination of it, v, w between (5) and (6),

d2s 2_2. , c2 (dp0 d dp0 d dp0 d\, _. ,^.
d& ~ p0 \dx dx dy dy dz dz) *

272. If we neglect the curvature of the earth, and suppose the
axis of z to be drawn vertically upwards, p0 will be a function of z
only, determined by

^? = - (1)

On the present hypothesis of uniform temperature, we have, by
Boyle's Law,

* The motion is in this case irrotational, and might have been investigated in
terms of the velocity-potential.

271-272] WAVES IN HETEROGENEOUS MEDIUM. 493

where H denotes as in Art. 256 the height of a 'homogeneous
atmosphere' at the given temperature. Hence

y/H /Q\

po°c e~z H v»>

Substituting in Art. 271 (7), and putting s = 0, we find, in
the case of no disturbing forces,

For plane waves travelling in a vertical direction, s will be a function of z
only, and therefore

If we assume a time-factor er<Tt, this is satisfied by

<j> = Ae™ ....................................... (ii),

provided m2-m/IL + o*/<?=0 .............................. (iii),

or wi = l/2H±a-' ................................. (iv),

where k' = ^l~ *

Y

The lower sign gives the case of waves propagated upwards. Expressed in
real form the solution for this case is

The wave-velocity (o-/k") varies with the frequency, but so long as o- is large
compared with c/2H it is approximately constant, differing from c by a small
quantity of the second order. The main effect of the variation of density is
on the amplitude, which increases as the waves ascend upwards into the rarer
regions, according to the law indicated by the exponential factor. This
increase might have been foreseen without calculation ; for when the variation
of density within the limits of a wave-length is small, there is no sensible
reflection, and the energy per wave-length, which varies as «2p0 (a being the
amplitude), must therefore remain unaltered as the waves proceed. Since
pQ oc e~z/K, this shews that a oc ezl2K.

When <r<c/2H, the form of the solution is changed, viz. we have

where m1} m2 (the two roots of (iii)) are real and positive. This represents a
standing oscillation, with one nodal and one ' loop ' plane. For example, if
the nodal plane be that of z — Q, we have m^-f m2.42 = 0, and the position of
the loop (s = 0) is given by

z=^r—^ los? (viii)-

494 WAVES OF EXPANSION. [CHAP. X

For plane waves travelling horizontally, the equation (4) takes the form

The waves are therefore propagated unchanged with velocity c, as we should
expect, since on the present hypothesis of uniform equilibrium-temperature
the wave-velocity is independent of the altitude*.

273. We may next consider the large-scale oscillations of an
atmosphere of uniform temperature covering a globe at rest.

If we introduce angular coordinates 0, w as in Art. 190, and
denote by ut v the velocities along and perpendicular to the
meridian, the equations (4) of Art. 271 give

du c2 d , dv c2 d ,

s— 5»fc-ft s--iiE"=<*-i> (1)>

where a is the radius. If we assume that the vertical motion (w)
is zero, the equation of continuity, Art. 271 (6), becomes

ds 1 (d(usm6) dv

dt a sin 6 ( dd o

The equations (1) and (2) shew that u, v, s may be regarded as
independent of the altitude. The formulae are in fact the same as in
Art. 190, except that s takes the place of %/h, and c2 of gh. Since,
in our present notation, we have c2 = #H, it appears that the free
and the forced oscillations will follow exactly the same laws as
those of a liquid of uniform depth H covering the same globe.

Thus for the free oscillations we shall have

Q n ( 4- \ \ ( *\\

where Sn is a surface-harmonic of integral order n, and

„ (4).

As a numerical example, putting c=2'80 x 104, 2vra = 4 x 109 [c. s.],
we find, in the cases n— l,n = 2, periods of 28*1 and 16'2 hours,
respectively.

* The substance of this Art. is from a paper by Lord Kayleigh, u On Vibrations
of an Atmosphere," Phil. Mag., Feb. 1890. For a discussion of the effects of
upward variation of temperature on propagation of sound-waves, see the same
author's Theory of Sound, Art. 288.

272-274] ATMOSPHERIC TIDES. 495

The tidal variations of pressure due to the gravitational action
of the sun and moon are very minute. It appears from the above
analogy that the equilibrium value s of the condensation will be
comparable with HJJ-L, where His the quantity defined in Art. 177.
Taking H=l'SO ft. (for the lunar tide), and H = 25000 feet, this
gives for the amplitude of s the value 7'2 x 10~5. If the normal
height of the barometer be 30 inches, this means an oscillation of
only -00216 of an inch.

It will be seen on reference to Art. 206 that the analogy with
the oscillations of a liquid of depth H is not disturbed when we
proceed to the tidal oscillations on a rotating globe. The height
H of the homogeneous atmosphere does not fall very far short of
one of the values (29040 ft.) of the depth of the ocean for which
the semi-diurnal tides were calculated by Laplace*. The tides in
this case were found to be direct, and to have at the equator
11*267 times their equilibrium value. Even with this factor the
corresponding barometric oscillation would only amount to '0243
of an inchf.

274. The most regular oscillations of the barometer have solar diurnal
and semi-diurnal periods, and cannot be due to gravitational action, since in
that case the corresponding lunar tides would be 2*28 times as great, whereas
they are practically insensible.

The observed oscillations must be ascribed to the daily variation in tem
perature, which, when analyzed into simple-harmonic constituents, will have
components whose periods are respectively 1, J, J, J, ... of a solar day. It
is very remarkable that the second (viz. the semi-diurnal) component has a
considerably greater amplitude than the first. It has been suggested by
Lord Kelvin that the explanation of this peculiarity is to be sought for in
the closer agreement of the period of the semi-diurnal component with a free
period of the earth's atmosphere than is the case with the diurnal component.
This question has been made the subject of an elaborate investigation by
Margules J, taking into account the earth's rotation. The further consideration
of atmospheric problems is, however, outside our province.

* See the table on p. 361, above.

t Cf. Laplace, "Recherches sur plusieurs points du systeme du monde," Mtm.
de VAcad. roy. des Sciences, 1776 [1779], Oeuvres, t. ix., p. 283. Also Mecanique
Celeste, Livre 4me, chap. v.

J This paper, with several others cited in the course of this work, is included in
a very useful collection edited and (where necessary) translated by Prof. Cleveland
Abbe, under the title : " Mechanics of the Earth's Atmosphere," Smithsonian Miscel
laneous Collections, Washington, 1891.

CHAPTER XL

VISCOSITY.

275. THE main theme of this Chapter is the resistance to
distortion, known as ' viscosity' or ' internal friction,' which is
exhibited more or less by all real fluids, but which we have
hitherto neglected.

It will be convenient, following a plan already adopted on
several occasions, to recall briefly the outlines of the general theory
of a dynamical system subject to dissipative forces which are
linear functions of the generalized velocities*. This will not only
be useful as tending to bring under one point of view most of
the special investigations which follow; it will sometimes indicate
the general character of the results to be expected in cases which
are as yet beyond our powers of calculation.

We begin with the case of one degree of freedom. The equa
tion of motion is of the type

aq + bq + cq — Q ........................ (1).

Here q is a generalized coordinate specifying the deviation from a
position of equilibrium ; a is the coefficient of inertia, and is
necessarily positive ; c is the coefficient of stability, and is positive
in the applications which we shall consider; b is a coefficient of
friction, and is positive.

If we put

T=laf, V=\vf, F=W ............... (2),

the equation may be written

q ..................... (3).

* For a fuller account of the theory reference may be made to Lord Kayleigh,
Theory of Sound, cc. iv., v. ; Thomson and Tait, Nattiral Philosophy (2nd ed.)
Arts. 340-345; Eouth, Advanced Rigid Dynamics, cc. vi., vii.

275] ONE DEGREE OF FREEDOM. 497

This shews that the energy T + V is increasing at a rate less
than that at which the extraneous force is doing work on the
system. The difference 2F represents the rate at which energy is
being dissipated ; this is always positive.

In free motion we have

aq + bq + cq = 0 (4).

If we assume that q oc eu, the solution takes different forms
according to the relative importance of the factional term. If
b" < 4tac, we have

or, say, X = — r~l ± icr (6).

Hence the full solution, expressed in real form, is

q = A e-VT cos (at + e) (7 ),

where A, e are arbitrary. The type of motion which this represents
may be described as a simple-harmonic vibration, with amplitude
diminishing asymptotically to zero, according to the law e~t/T. The
time r in which the amplitude sinks to l/e of its original value is
sometimes called the 'modulus of decay' of the oscillations.

If b/2a be small compared with (c/a)*, b'2/4>ac is a small quantity
of the second order, and the ' speed' a is then practically unaffected
by the friction. This is the case whenever the time (27rr) in
which the amplitude sinks to e~27r (= ^) of its initial value is large
compared with the period (27r/<r).

When, on the other hand, b'2 > 4<ac, the values of X are real and
negative. Denoting them by — «1} — «2, we have

q = A1fT'** + Aser**t (8).

This represents 'aperiodic motion'; viz. the system never passes
more than once through its equilibrium position, towards which it
finally creeps asymptotically.

In the critical case 62 = 4ac, the two values of X are equal ; we
then find by usual methods

q = (A+Bt)er* (9),

which may be similarly interpreted.

L. 32

498 VISCOSITY. [CHAP, xi

As the frictional coefficient b is increased, the two quantities
«!, «2 become more and more unequal; viz. one of them («2, say)
tends to the value b/a, and the other to the value c/b. The
effect of the second term in (8) then rapidly disappears, and the
residual motion is the same as if the inertia-coefficient (a) were
zero.

276. We consider next the effect of a periodic extraneous

force. Assuming that

QoceW+e> (10),

the equation (1) gives

If we put

-* , .

•'

where ex lies between 0 and 180°, we have

Taking real parts, we may say that the force

Q=Ccoa(<rt + e) ........................ (14)

will maintain the oscillation

C

€l) ................... (15).

Since

(16),

it is easily found that if 62 < 4ac the amplitude is greatest when

its value being then

<r= -) . M_i_ (17),

W V 2ac/ v '

b \ ~ I \*-~±~~l (-^)-

In the case of relatively small friction, where 62/4ac may be
treated as of the second order, the amplitude is greatest when the
period of the imposed force coincides with that of the free
oscillation (cf. Art. 165). The formula (18) then shews that the
amplitude when a maximum bears to its 'equilibrium-value'
(C/c) the ratio (arf/b, which is by hypothesis large.

275-277] EFFECT OF FRICTION ON PHASE. 499

On the other hand, when 62 > 4ac the amplitude continually
increases as the speed cr diminishes, tending ultimately to the
' equilibrium-value' C/c.

It also appears from (15) and (12) that the maximum dis
placement follows the maximum of the force at an interval of
phase equal to e1? where

(19).

If the period be longer than the free-period in the absence of
friction this difference of phase lies between 0 and 90° ; in the
opposite case it lies between 90° and 180°. If the frictional
coefficient b be relatively small, the interval differs very little from
0 or 180°, as the case may be.

The rate of dissipation is bq2, the mean value of which is easily

found to be

bC2

This is greatest when <r = (c/a)*.

As in Art. 165, when the oscillations are very rapid the formula
(11) gives

q = - Q/o*a (21),

approximately ; the inertia only of the system being operative.

On the other hand when a is small, the displacement has very
nearly the equilibrium -value

n _ n/r f99\

(/ — V/v» \L&).

277. An interesting example is furnished by the tides in an
equatorial canal*.

The equation of motion, as modified by the introduction of a
frictional term, is

where the notation is as in Art. 178-f.

* Airy, " Tides and Waves," Arts. 315...

f In particular, c2 now stands for gh, where li is the depth.

32—2

500 VISCOSITY. [CHAP, xi

In the case of free waves, putting X = 0 and assuming that

we find X2 + yuX + &2c2 = 0,

whence

X = -i/i±i(^c2-^2) .................. (3).

If we neglect the square of p/hc, this gives, in real form,

% = Ae-^cos{k(ct±x) + 6} .................. (4).

The modulus of decay is 2/ur1, and the wave-velocity is (to the
first order) unaffected by the friction.

To find the forced waves due to the attraction of the moon we
write, in conformity with Art. 178,

where n is the angular velocity of the moon relative to a fixed
point on the canal, and a is the earth's radius. We find, assuming
the same time-factor,

Hence, for the surface-elevation, we have

™ *

( *7\

where H/a =f/g, as in Art. 177.

To put these expressions in real form, we write

where 0 < ^ < 90°. We thus find that to the tidal disturbing force
X = -fsm2(rit+^ + e^ ..................... (9)

corresponds the horizontal displacement

and the surface-elevation

x

'= °* cos

277]

TIDAL FRICTION.

501

Since in these expressions n't + x/a + e measures the hour-angle
of the moon past the meridian of any point (x) on the canal, it
appears that high-water will follow the moon's transit at an in
terval ^ given by n't-^ = %.

If c2 < w'2a2, or h/a < n'2a/g, we should in the case of infinitesimal
friction have ^ = 90°, i.e. the tides would be inverted (cf. Art. 178).
With sensible friction, ^ will lie between 90° and 45°, and the
time of high-water is accelerated by the time-equivalent of the
angle 90° — %.

On the other hand, when hi a > n'2a/g, so that in the absence of
friction the tides would be direct, the value of ^ lies between 0°
and 45°, and the time of high-water is retarded by the time-
equivalent of this angle.

The figures shew the two cases. The letters J/, M' indicate the positions
of the moon and ' anti-moon ' (see p. 365) supposed situate in the plane of the
equator, and the curved arrows shew the direction of the earth's rotation.

It is evident that in each case the attraction of the disturbing

502 VISCOSITY. [CHAP, xi

system on the elevated water is equivalent to a couple tending to
diminish the angular momentum of the system composed of the
earth and sea.

In the present problem the amount of the couple can be easily
calculated. We find, from (9) and (11), for the integral tangential
force on the elevated water

r

J

2y ............... (12),

where h is the vertical amplitude. Since the positive direction of
X is eastwards, this shews that there is on the whole a balance of
westward force. If we multiply by a we get the amount of the
retarding couple, per unit breadth of the canal*.

Another more obvious phenomenon, viz. the retardation of the time of
spring tides behind the days of new and full moon, can be illustrated on the
same principles. The composition of two simple-harmonic oscillations of
nearly equal speed gives

77 = A COS (<rt + f) + Ar COS (tr't + e')
= (A+A' cos(f))cos(ort + f} + A' sin<f>sii\((rt + €) ............... (i),

where <£ = (<r — o-')£ + e-e' .............................. (ii).

If the first term in the second member of (i) represents the lunar, and the
second the solar tide, we shall have <r<o-', and A>A'. If we write

A + A' cos 0 = C cos a, A ' sin <£ = C sin a ............... (iii),

we get 7) = Ccos(<rt + €-a) .............................. (iv),

where C=(A* + 2AA' cos<j>+ A'rf ........................ (v),

This may be described as a simple-harmonic vibration of slowly varying
amplitude and phase. The amplitude ranges between the limits A±A'9
whilst a lies always between +^ir. The 'speed' must also be regarded as
variable, viz. we find

da _ a-A2 + (o- + cr) A A' cos <ft + v'A"* , ...

~~ ~~~r cos ~f+A '*

* Cf. Delaunay, " Sur 1'existence d'une cause nouvelle ayant une influence
sensible sur la valeur de 1'equation seeulaire de la Lune," Comptes llendus, t. Ixi.
(1865) ; Sir W. Thomson, " On the Observations and Calculations required to find
the Tidal Ketardation of the Earth's Rotation," Phil. Mag., June (supplement)
1866, Math, and Pliys. Papers, t. iii., p. 337. The first direct numerical estimate of
tidal retardation appears to have been made by Ferrel, in 1853.

277-278] RETARDATION OF SPRING TIDES. 503

m, . , , AfT + A'v , A(T — A'v . ......

This ranges between — . , and — - -- r,- ........................ ( vin)*.

A-f A A — A

The above is the well-known explanation of the phenomena of the spring-
and neap-tides f ; but we are now concerned further with the question of phase.
In the absence of friction, the maxima of the amplitude C would occur whenever

(<rr -<r)t = 2mir + e - e',

where in is integral. Owing to the friction, e and e' are replaced by quantities
of the form e - f x and e' - e/, and the corresponding times of maximum are
given by

(a-' — <r)t — (e/ - ej) = 2wi7r + e - e',

i. e. they occur later by an interval (e/ - e1)/(o-' - o-). If the difference between
a-' and a- were infinitesimal, this would be equal to dejdo:

In the case of the semi-diurnal tides in an equatorial canal we have <r = 2ft',
f1 = 2x, whence

_
do- dri

278. Returning to the general theory, let q^ q.2,... be the
coordinates of a dynamical system, which we will suppose subject
to conservative forces depending on its configuration, to 'motional'
forces varying as the velocities, and to given extraneous forces.
The equations of small motion of such a system, on the most
general assumptions we can make, will be of the type

where the kinetic and potential energies T, V are given by ex
pressions of the forms

2g^2+ ............ (2),

2g1^2 + ............ (3).

It is to be remembered that

drs = dsr , Cr8 = Csr ........................ \*)>

but we do not assume the equality of Brs and Bsr.

If we now write

..................... (5),

* Helmholtz, Lehre von den, Tommpjindunyen (2° Aufl.), Braunschweig, 1870,
p. 622.

t Cf. Thomson and Tait, Natural Philosophy, Art. 60.
$ Cf. Airy, "Tides and Waves," Arts. 328...

504 VISCOSITY. [CHAP. XI

and &a = -&r = i (•#»•« -#«•) .................. (C),

the typical equation (1) takes the form
d dT dF

provided

2F=buqi' + b2,q/+... +26lag,ga+ ............... (8).

From the equations in this form we derive

%-t(T+V) + 2F=2Qrqr .................... (9).

The right-hand side expresses the rate at which the extraneous
forces are doing work. Part of this work goes to increase the total
energy T-f V of the system; the remainder is dissipated, at the
rate %F. In the application to natural problems the function F is
essentially positive : it is called by Lord Rayleigh *, by whom it
was first formally employed, the ' Dissipation-Function.'

The terms in (7) which are due to F may be distinguished as
the 'frictional terms.' The remaining terms in qly (J2, ..., with
coefficients subject to the relation /3rg = — ftfr, are of the type we
have already met with in the general equations of a 'gyrostatic'
system (Art. 139); they may therefore be referred to as the
' gyrostatic terms.'

279. When the gyrostatic terms are absent, the equation (7)
reduces to

d dT dF dV „

-T. -j-r- + -T^ +-F- =Q, .................... (10).

dt dqr dq, dqr

As in Art. 165, we may suppose that by transformation of
coordinates the expressions for T and V are reduced to sums of
squares, thus :

2r = o1g12 + a2gaa + ..................... (11),

27=^+0^' + ..................... (12).

It frequently, but not necessarily, happens that the same
transformation also reduces F to this form, say

2F =1^ + 14?+ ..................... (13).

* " Some General Theorems relating to Vibrations," Proc. Lond. Math. Soc.,
t. iv., p. 363 (1873) ; Theory of Sound, Art. 81.

278-279] D1SSIPATIVE SYSTEMS IN GENERAL. , 505

The typical equation (10) then assumes the simple form

Qi ...................... (14),

which has been discussed in Art. 275. Each coordinate qr now
varies independently of the rest.

When F is not reduced by the same transformation as T and F, the
equations of small motion are

where bft =

f

The motion is now more complicated ; for example, in the case of free
oscillations about stable equilibrium, each particle executes (in any fun
damental type) an elliptic-harmonic vibration, with the axes of the orbit
contracting according to the law e~°^.

The question becomes somewhat simpler when the frictional coefficients
brg are small, since the modes of motion will then be almost the same as in the
case of no friction. Thus it appears from (i) that a mode of free motion is
possible in which the main variation is in one coordinate, say qr. The rth
equation then reduces to

a,.£V + &n-j»- + <V2,. = 0 .............................. (ii),

where we have omitted terms in which the relatively small quantities y1} y2, ...
(other than qr) are multiplied by the small coefficients brl, 6r2,... We have
seen in Art. 275 that if brr be small the solution of (ii) is of the type

(iii),
where T-i=$brr/ar, <r = (c>,)* ........................... (iv).

The relatively small variations of the remaining coordinates are then given
by the remaining equations of the system (i). For example, with the same
approximations,

whence qs = ~<r~~— Ae~t/T sin (at + e) .

-*

Except in the case of approximate equality of period between two funda
mental modes, the elliptic orbits of the particles will on the present supposi
tions be very flat.

If we were to assume that

3V=acos((r* + e) .............................. (vii),

where a- has the same value as in the case of no friction, whilst a varies slowly

506 VISCOSITY. [CHAP, xi

with the time, and that the variations of the other coordinates are relatively
small, we should find

T+V=$arqr* + &rqr* = ^ara* .................. (viii),

nearly. Again, the dissipation is

the mean value of which is ^o-2brra2 .................................... (ix),

approximately. Hence equating the rate of decay of the energy to the mean
value of the dissipation, we get

da , br

di=-* ar

rr
a

whence a = a0e~^T .................................... (xi),

if r-^i&^/a, .................................. (xii),

as in Art. 275. This method of ascertaining the rate of decay of the oscilla
tions is sometimes useful when the complete determination of the character
of the motion, as affected by friction, would be difficult.

When the frictional coefficients are relatively great, the inertia of the
system becomes ineffective ; and the most appropriate system of coordinates
is that which reduces F and V simultaneously to sums of squares, say

The equations of free-motion are then of the type

brqr + crqr = Q ................................. (xiv),

whence qr — Ce~t'r .................................... (xv)j

if r = br/ct ..................................... (xvi).

280. When gyrostatic as well as frictional terms are present
in the fundamental equations, the theory is naturally more com
plicated. It will be sufficient here to consider the case of two
degrees of freedom, by way of supplement to the investigation of
Art. 198.

The equations of motion are now of the types

%&+*nft+(*is+0)fs+«i?r-$i,) (i)

«ift+(*it-£)&+*si+tyft«<?! J '

To determine the modes of free motion we put Ql = 0) Q2 = 0> and assume that
ql and q.z vary as e*1. This leads to the biquadratic in X :

t 622) X3 + («2ct + a^+ft* + bn 62.2 - 6122) X2

c2 = 0 ...... (ii),

279-280] GYROSTATIC SYSTEM WITH FRICTION. 507

There is no difficulty in shewing, with the help of criteria given by Routh*,
that if, as in our case, the quantities

are all positive, the necessary and sufficient conditions that this biquadratic
should have the real parts of its roots all negative are that cx, c2 should both
be positive.

If we neglect terms of the second order in the frictional coefficients, the
same conclusion may be attained more directly as follows. On this hypothesis
the roots of (ii) are, approximately,

X = - ax + i(rl , - a2 ± io-2 ........................ (iii),

where <TI} <r2 are, to the first order, the same as in the case of no friction, viz.
they are the roots of

a1a2(r^-(a2c1 + a1c2+^2}(r2 + clc2 = 0 .................. (iv),

whilst a1} a2 are determined by

It is evident that, if o-j and o-2 are to be real, clt c.2 must have the same sign,
and that if ax, a2 are to be positive, this sign must be + . Conversely, if clt c2
are both positive, the values of o^2, o-22 are real and positive, and the quantities
ci/ai> c2/^2 b°tn lie ifl the interval between them. It then easily follows from
(v) that al5 a2 are both positivet.

If one of the coefficients clt c2 (say c2) be zero, one of the values of a (say
o-2) is zero, indicating a free mode of infinitely long period. We then have

* Advanced Rigid Dynamics, Art. 287.

t A simple example of the above theory is supplied by the case of a particle in
an ellipsoidal bowl rotating about a principal axis, which is vertical. If the bowl
be frictionless, the equilibrium of the particle when at the lowest point will be stable
unless the period of the rotation lie between the periods of the two fundamental
modes of oscillation (one in each principal plane) of the particle when the bowl is at
rest. But if there be friction of motion between the particle and the bowl, there will
be « secular ' stability only so long as the speed of the rotation is less than that of the
slower of the two modes referred to. If the rotation be more rapid, the particle
will gradually work its way outwards into a position of relative equilibrium in which
it rotates with the bowl like the bob of a conical pendulum. In this state the system
made up of the particle and the bowl has Zt'«s,s energy for the same angular momentum
than when the particle was at the bottom. Cf. Art. 235.

508 VISCOSITY. [CHAP. XI

As in Art. 198 we could easily write down the expressions for the forced
oscillations in the general case where Qlt $2 vary as ei<T\ but we shall here
consider more particularly the case where <?2 = 0 and $2 = 0. The equations (i)
then give

ia (619 — 8) <?i + (i<ra9 + b99) 69 = 0

TT- Vl^r'/ii'x Z v «/ 29

Hence

n ,.

A'2) °"2 2 2

This may also be written

a,a2 {(iV + a^ + oY2} (tcr + a,)

11

Our main object is to examine the case of a disturbing force of long period,
for the sake of its bearing on Laplace's argument as to the fortnightly tide
(Art. 210). We will therefore suppose that the ratio o-j/tr, as well as o^/c^, is
large. The formula then reduces to

i<ra.2+b22

Everything now turns on the values of the ratios o-/a2 and aa.Jb^. If o- be so
small that these may be both neglected, we have

in agreement with the equilibrium theory. The assumption here made
is that the period of the imposed force is long compared with the time in
which free motions would, owing to friction, fall to e~ of their initial
amplitudes. This condition is evidently far from being fulfilled in the case of
the fortnightly tide. If, as is more in agreement with the actual state of
things, we assume o-/a2 and o-a2/622 to be large, we obtain

as in Art. 198 (vii).

Viscosity.

281. sAVe proceed to consider the special kind of resistance
which is met with in fluids. The methods we shall employ are of
necessity the same as are applicable to the resistance to distortion,
known as ' elasticity,' which is characteristic of solid bodies.
The two classes of phenomena are of course physically distinct,
the latter depending on the actual changes of shape produced,
the former on the rate of change of shape, but the mathema
tical methods appropriate to them are to a great extent identical.

If we imagine three planes to be drawn through any point P

280-282] OBLIQUE STRESSES. 509

perpendicular to the axes of x, y, z, respectively, the three com
ponents of the stress, per unit area, exerted across the first of
these planes may be denoted by pxx, pxy,pxz> respectively; those
of the stress across the second plane by pyx, pyy,pyz\ and those of
the stress across the third plane by pZx> Pzy> Pzz*- If we ^x our
attention on an element SxSySz having its centre at P, we find,
on taking moments, and dividing by $x§y§z,

Pyz = Pzy > Pzx == PXZ> Pxy == Pyx ( 1 )>

the extraneous forces and the kinetic reactions being omitted,
since they are of a higher order of small quantities than the
surface tractions. These equalities reduce the nine components
of stress to six ; in the case of a viscous fluid they will also follow
independently from the expressions for pyz,pzx>pxy in terms of the
rates of distortion, to be given presently (Art. 283).

v/282. It appears from Arts. 1, 2 that in a fluid the deviation
of the state of stress denoted by pXX) pxy,... from one of pressure
uniform in all directions depends entirely on the motion of
distortion in the neighbourhood of P, i.e. on the six quantities
a, 6, c,f, g, h by which this distortion was in Art. 31 shewn to be
specified. Before endeavouring to express pxx, pxy) ... as functions
of these quantities, it will be convenient to establish certain for
mulae of transformation.

Let us draw Px't Pyf, Pz' in the directions of the principal
axes of distortion at P, arid let a', &', c' be the
rates of extension along these lines. Further
let the mutual configuration of the two sets of ^
axes, x, y, z and x't y', z, be specified in the y £2, ra<,
usual manner by the annexed scheme of direc- z' ! 13, m3, n3.
tion-cosines. We have, then,

+ *• J?)

••.

dx dy dz

* In conformity with the usual practice in the theory of Elasticity, we reckon
a tension as positive, a pressure as negative. Thus in the case of a frictionless fluid
we have

510 VISCOSITY. [CHAP, xi

Hence a = l*a' + LrV + l*c'9

(1),

the last two relations being written down from symmetry. We
notice that

a + & + c = a' + &' + c' ..................... (2),

an invariant, as it should be, by Art. 7.
Again
dw dv / d d d

/ d \ , ,

= r • &> + «•» *? + m- d?) (n'M + "»« + «'w

and this, with the two corresponding formula?, gives
f= m^af + m^nj)' +

; (3).

liO,' + LmM +

283. From the symmetry of the circumstances it is plain
that the stresses exerted at P across the planes y'z, z'x' , x'y' must
be wholly perpendicular to these planes. Let us denote them by
PI, P-2, P* respectively. In the figure of Art. 2 let ABC now
represent a plane drawn perpendicular to cc, infinitely close to P,
meeting the axes of #', y', z' in A, B, C, respectively ; and let A
denote the area ABC. The areas of the remaining faces of the
tetrahedron PA BC will then be ^A, /2A, £3A. Resolving parallel
to os the forces acting on the tetrahedron, we find

paa-A =p1ll A . h +p2l2 A . 4 +p.Als& . 13 ;

the external impressed forces and the resistances to acceleration
being omitted for the same reason as before. Hence, and by
similar reasoning,

We notice that

(2).

282-284] TRANSFORMATION FORMULA. 511

Hence the arithmetic mean of the normal pressures on any three
mutually perpendicular planes through the point P is the same.
We shall denote this mean pressure by p.

Again, resolving parallel to y, we obtain the third of the
following symmetrical system of equations:

Pyz =

These shew that

Pyz = Pzy > PZX ~ PXZ > pxy — Pyx >

as in Art. 281.

y '

If in the same figure we suppose PA, PB, PC to be drawn
parallel to x, y, z respectively, whilst ABC is any plane drawn near
P, whose direction -cosines are I, m, n, we find in the same way
that the components (phx, phy, Phz) of the stress exerted across this
plane are

Phx = Ipxx + mpxy + npxz ,

(4).

284. Now PI, p2, ps differ from — p by quantities depending
on the motion of distortion, which must therefore be functions of
a', b', c', only. The simplest hypothesis we can frame on this
point is that these functions are linear. We write therefore

Pi = -P + X (a' + b' 4- c') + 2/Aa', |

[ ............ (1),

where X, //, are constants depending on the nature of the fluid,
and on its physical state, this being the most general assumption
consistent with the above suppositions, and with symmetry. Sub
stituting these values of^,^,^ in (1) and (3) of Art. 283, and
making use of the results of Art. 282, we find

Pxx = -P + X (a + 6 + c) + 2yaa,|

^ = -^ + Ma + & + c)+2/^,l ............... (2),

Pzz = - p + X (a -f b + c) + 2/xc }

xy = 2tdi ............... (3).

512 VISCOSITY. [CHAP. XI

The definition of p adopted in Art. 283 implies the relation

3\ + 2fji = 0,
whence, finally, introducing the values of a, b, c, /, g, h, from

i dv dw\ . du

Art. 31,

PZZ — ~~ P

dy
du dv

+ ^Ty

dw
" dz

dw dv

du dw

dw\ _
~dx) =Pxz'

.(5).

dv du

The constant yu, is called the 'coefficient of viscosity.' Its physi
cal meaning may be illustrated by reference to the case of a fluid
in what is called 'laminar' motion (Art. 31); i.e. the fluid moves
in a system of parallel planes, the velocity being in direction
everywhere the same, and in magnitude proportional to the
distance from some fixed plane of the system. Each stratum of
fluid will then exert on the one next to it a tangential traction,
opposing the relative motion, whose amount per unit area is /*
times the variation of velocity per unit distance perpendicular to
the planes. In symbols, if u = ay, v = 0, w — 0, we have

Pxx = Pyy =Pzz:=- P, Pyz = ®> Pzx = ®, pxy = «•

If [M], [L], [T] denote the units of mass, length, and time, the
dimensions of the £>'s are [ML~l T~*], and those of the rates of
distortion (a, b, c, ...) are [T~l], so that the dimensions of p are

The stresses in different fluids, under similar circumstances of
motion, will be proportional to the corresponding values of //, ; but
if we wish to compare their effects in modifying the existing
motion we have to take account of the ratio of these stresses to
the inertia of the fluid. From this point of view, the determining

284] EXPRESSIONS FOR THE STRESSES. 513

quantity is the ratio p/p ; it is therefore usual to denote this by a
special symbol v, called by Maxwell the 'kinematic coefficient' of
viscosity. The dimensions of v are [L2T~1].

The hypothesis made above that the stresses pxx, pxy,. . . are linear functions
of the rates of strain a, b, c,... is of a purely tentative character, and although
there is considerable a priori probability that it will represent the facts
accurately for the case of infinitely small motions, we have so far no assurance
that it will hold generally. It has however been pointed out by Prof. Osborne
Keynolds* that the equations based on this hypothesis have been put to a
very severe test in the experiments of Poiseuille and others, to be referred to
presently (Art. 289). Considering the very wide range of values of the rates of
distortion over which these experiments extend, we can hardly hesitate to
accept the equations in question as a complete statement of the laws of
viscosity. In the case of gases we have additional grounds for this assump
tion in the investigations of the kinetic theory by Maxwell f.

The practical determination of /u (or v) is a matter of some difficulty.
Without entering into the details of experimental methods, we quote a few of
the best-established results. The calculations of von HelmholtzJ, based on
Poiseuille's observations, give for water

•0178
M~l + -0337<9+-000221<92'

in c. G. s. units, where B is the temperature Centigrade. The viscosity, as in
the case of all liquids as yet investigated, diminishes rapidly as the temperature
rises ; thus at 17° C. the value is

^ = •0109.
For mercury Koch § found

^ = •01697, and pw = '01633,
respectively.

In gases, the value of /z is found to be sensibly independent of the pressure,
within very wide limits, but to increase somewhat with rise of temperature.
Maxwell found as the result of his experiments ||,

/u = -0001878 (1 + -00366 6} ;

this makes p proportional to the absolute temperature as measured by the
air-therrnometer. Subsequent observers have found a somewhat smaller
value for the first factor, and a less rapid increase with temperature.
We may take perhaps as a fairly established value

/*0=-000170

* " On the Theory of Lubrication, &c.," Phil. Trans., 1886, Pt. I., p. 165.

t "On the Dynamical Theory of Gases," Phil. Trans., 1867; Scientific Papers,
t. ii., p. 26.

J "Ueber Reibung tropfbarer Fliissigkeiten," Wien. Sitzungsber., t. xl. (I860);
Ges. Abh., t. i., p. 172.

§ Wied. Ann., t. xiv. (1881).

|| " On the Viscosity or Internal Friction of Air and other Gases," Phil. Trans.,
1866; Scientific Papers, t. ii., p. 1.

L. 33

514 VISCOSITY. [CHAP, xi

for the temperature 0° C. For air at atmospheric pressure, assuming p = '00129
this gives

The value of v varies inversely as the pressure*.

285. We have still to inquire into the dynamical conditions
to be satisfied at the boundaries.

At a free surface, or at the surface of contact of two dissimilar
fluids, the three components of stress across the surface must be
continuous •)-. The resulting conditions can easily be written down
with the help of Art. 283 (4).

A more difficult question arises as to the state of things at the
surface of contact of a fluid with a solid. It appears probable that
in all ordinary cases there is no motion, relative to the solid, of the
fluid immediately in contact with it. The contrary supposition
would imply an infinitely greater resistance to the sliding of one
portion of the fluid past another than to the sliding of the fluid
over a solid §.

If however we wish, temporarily, to leave this point open, the most
natural supposition to make is that the slipping is resisted by a tangential
force proportional to the relative velocity. If we consider the motion of a
small film of fluid, of thickness infinitely small compared with its lateral
dimensions, in contact with the solid, it is evident that the tangential traction
on its inner surface must ultimately balance the force exerted on its outer
surface by the solid. The former force may be calculated from Art. 283 (4) ;
the latter is in a direction opposite to the relative velocity, and proportional
to it. The constant (/3, say) which expresses the ratio of the tangential force
to the relative velocity may be called the ' coefficient of sliding friction.'

286. The equations of motion of a viscous fluid are obtained
by considering, as in Art. 6, a rectangular element Sx&ySz having
its centre at P. Taking, for instance, the resolution parallel
to #, the difference of the normal tractions on the two yz- faces
gives (dpxxjdx) §x . Sy Sz. The tangential tractions on the two
£#-faces contribute (dpyx/dy) Sy . 8z$x, and the two xy-f&ces give

* A very full account of the results obtained by various experimenters is
given in Wmkelmann's Handbuch der Physik, t. i., Art. ' Keibung.'

t This statement requires an obvious modification when capillarity is taken into
account. Cf. Art. 302.

§ Stokes, 1. c. p. 518.

284-286] DYNAMICAL EQUATIONS. 515

in like manner (dpzxjdz) Sz . 8x8y. Hence, with our usual
notation,

Du _ _ dpxx dpyx dpzx
pDt~p2 ~ dx "~" dz '

_ y yy

pDt~p dx ~ dy 4 d5 ' '

p n^: = pZ + -%= -f ^p- -f 5=

Dt dx dy dz

Substituting the values of pxx, pxy, ... from Art. 284 (4), (5),
we find

Du_ „ dp, __«W

Z)w „ dp d6

p — p ± — — \~ 4-//< —

jji dy d/ ?/

Dw „ dp dO
where

and V2 has its usual meaning.

When the fluid is incompressible, these reduce to

Dw

These dynamical equations were first obtained by Navier*
and Poisson-f- on various considerations as to the mutual action of
the ultimate molecules of fluids. The method above adopted,
which is free from all hypothesis of this kind, appears to be due
in principle to de Saint-Venant and Stokes f.

* " M6moire sur les Lois du Mouvement des Fluides," Mem. de VAcad. des
Sciences, t. vi. (1822).

t "Memoire sur les Equations g<§n<§rales de l'£quilibre et du Mouvement des
Corps solides elastiques et des Fluides," Journ. de VEcoJe Polytechn., t. xiii. (1829).

J " On the Theories of the Internal Friction of Fluids in Motion, &c.," Camb.
Trans., t. viii. (1845); Math, and Phys. Papers, t. i., p. 88.

33—2

516

VISCOSITY.

[CHAP, xi

The equations (4) admit of an interesting interpretation. The first of
them, for example, may be written

Du „ I dp ...

-=-=X -- -£ + vV*u .............................. (i).

Dt p dx

The first two terms on the right hand express the rate of variation of u in
consequence of the external forces and of the instantaneous distribution of
pressure, and have the same forms as in the case of a frictionless liquid. The
remaining term i>vX due to viscosity, gives an additional variation following
the same law as that of temperature in Thermal Conduction, or of density in
the theory of Diffusion. This variation is in fact proportional to the (positive
or negative) excess of the mean value of u through a small sphere of given
radius surrounding the point (#, y, z] over its value at that point*. In
connection with this analogy it is interesting to note that the value of v for
water is of the same order of magnitude as that (-01249) found by I)r Everett
for the thermometric conductivity of the Greenwich gravel.

When the forces X, F, Z have a potential G, the equations (4) may be
written

where

/== - + i<72
P

q denoting the resultant velocity, and £, 77, £ the components of the angular
velocity of the fluid. If we eliminate x by cross-differentiation, we find,

du

D{ .dw dw dw

(iv).

The first three terms on the right hand of each of these equations express, as
in Art. 143, the rates at which £, ?/, £ vary for a particle, when the vortex-lines
move with the fluid, and the strengths of the vortices remain constant. The
additional variation of these quantities, due to viscosity, is given by the last
terms, and follows the law of conduction of heat. It is evident from this
analogy that vortex-motion cannot originate in the interior of a viscous liquid,
but must be diffused inwards from the boundary.

* Maxwell, Proc. Lond. Math. Soc., t. iii., p. 230; Electricity and Magnetism,
Art. 26.

286-287] INTERPRETATION. 517

287. To compute the rate of dissipation of energy, due to
viscosity, we consider first the portion of fluid which at time t
occupies a rectangular element §x§y§z having its centre at (x, y, z).
Calculating the differences of the rates at which work is being done
by the tractions on the pairs of opposite faces, we obtain

\dx

dx xx xy xz dy (pyxU +PyyV + P

(1).
The terms
\(dpxx dpyx dpzx\ fdpxy dpyy dpzy\

\\ 7 T ~~7 -- r ~j — I w i \ T~~ "1 -- 7 "1 -- 7 I 0

(\ dx dy dz J \ dx dy dz )

/^ dft ^_A ]
V dx dy dz J )

express, by Art. 286 (1), the rate at which the tractions on the faces
are doing work on the element as a whole, in increasing its kinetic
energy and in compensating the work done against the extraneous
forces JT, Y, Z. The remaining terms express the rate at which
work is being done in changing the volume and shape of the
element. They may be written

(Pxxtt + Pyyb + pzzG + %pyzf+ %pzxg + tyxyh) &*% &Z- • '(3)>

where a, b, c, f, g, h have the same meanings as in Arts. 31, 284.
Substituting from Art. 284 (2), (3), we get

— p(a + b + c)
+ }- J/A (a + b

...... (4).

If p be a function of p only, the first line of this is equal to

provided

(5),

i.e. E denotes, as in Art. 11, the intrinsic energy per unit mass.
Hence the second line of (4) represents the rate at which energy
is being dissipated. On the principles established by Joule, the
mechanical energy thus lost takes the form of heat, developed in
the element.

518 VISCOSITY. [CHAP, xi

If we integrate over the whole volume of the fluid, we find, for
the total rate of dissipation,

(6),

where

fdu dv dw^

.dz)

fdw dv\- (du dw \~ (dv duY2}
dy dz) \dz dx) \dx dy) }

If we write this in the form

it appears that F cannot vanish unless

a = 6 = c, and f=g = h=0,

at every point of the fluid. In the case of an incompressible fluid it is
necessary that the quantities «, 6, c, /, y, h should all vanish. It easily
follows, on reference to Art. 31, that the only condition under which a liquid
can be in motion without dissipation of energy by viscosity is that there must
be nowhere any extension or contraction of linear elements ; in other words,
the motion must be composed of a translation and a pure rotation, as in the
case of a rigid body. In the case of a gas there may be superposed on this an
expansion or contraction which is the same in all directions.

We now consider specially the case when the fluid is incompressible, so that

If we subtract from this the expression

^Y*+*4

r \JBUS dy

which is zero, we obtain

dv\2 du dw\2 dv du

=

(dv dw dv dw dw du dw du du dv du dv\
^ \dy dz dz dy dz dx dx dz dx dy dy dx) '" ^

* Stokes, " On the Effect of the Internal Friction of Fluids on the Motion of
Pendulums," Camb. Trans., t. ix., p. [58] (1851).

287-288] DISSIPATION OF ENERGY. 519

If we integrate this over a region such that u, v, w vanish at every point of
the boundary, as in the case of a liquid filling a closed vessel, on the hypothesis
of no slipping, the terms due to the second line vanish (after a partial integra
tion), and we obtain

............ (iv)*

In the general case, when no limitation is made as to the boundary
conditions, the formula (iii) leads to

2^=4^ I I ntf + rf + ffdxdydz- n I l-^dtS

f'fffr >w» *>

+ 4^1/1 \u, v, iv, dS (v),

I £> *?> £» |

where, in the former of the two surface-integrals, dn denotes an element of the
normal, and, in the latter, I, m, n are the direction-cosines of the normal,
drawn inwards in each case from the surface-element dS.

When the motion considered is irrotational, this formula reduces to

simply. In the particular case of a spherical boundary this expression follows
independently from Art. 44 (i).

Problems of Steady Motion.

288. The first application which we shall consider is to the
steady motion of liquid, under pressure, between two fixed parallel
planes, the flow being supposed to take place in parallel lines.

Let the origin be taken half-way between the planes, and the
axis of y perpendicular to them. We assume that u is a function
of y only, and that v, w = 0. Since the traction parallel to x on any
plane perpendicular to y is equal to pdujdy, the difference of the
tractions on the two faces of a stratum of unit area and thickness
% gives a resultant fj,d2u/dy*. 8y. This must be balanced by the
normal pressures, which give a resultant — dpjdx per unit volume

of the stratum. Hence

d2u dp

* Bobyleff, "Einige Betrachtungen tiber die Gleichungen der Hydrodynamik,"
Math. Ann., t. vi. (1873); Forsyth, "On the Motion of a Viscous Incompressible
Fluid," Mess, of Math., t. ix. (1880).

520 VISCOSITY. [CHAP, xi

Also, since there is no motion parallel to y, dpjdy must vanish.
These results might of course have been obtained immediately
from the general equations of Art. 286.

It follows that the pressure-gradient dpjdx is an absolute
constant. Hence (1) gives

and determining the constants so as to make u = 0 for y = ± h, we
find

Zh* dp
Hence y=__ ..................... (4).

289. The investigation of the steady flow of a liquid through
a straight pipe of uniform circular section is equally simple, and
physically more important.

If we take the axis of z coincident with the axis of the tube,
and assume that the velocity is everywhere parallel to z, and a
function of the distance (r) from this axis, the tangential stress
across a plane perpendicular to r will be pdw/dr. Hence, con
sidering a cylindrical shell of fluid, whose bounding radii are r
and r + £r, and whose length is I, the difference of the tangential
tractions on the two curved surfaces gives a retarding force

d [' dw _ A ?

— -J- /* -j- - 2™ °r-
dr \ dr J

On account of the steady character of the motion, this must be
balanced by the normal pressures on the ends of the shell. Since
dwjdz = 0, the difference of these normal pressures is equal to

where pr , p2 are the values of p (the mean pressure) at the two
ends. Hence

dr\dr

Again, if we resolve along the radius the forces acting on a
rectangular element, we find dp/dr = 0, so that the mean pressure
is uniform over each section of the pipe.

288-289] PROBLEMS OF STEADY MOTION. 521

The equation (1) might have been obtained from Art. 286
(4) by direct transformation of coordinates, putting

r = (#2 + 2/2)i
The integral of (1) is

~ (2).

Since the velocity must be finite at the axis, we must have A = 0;
and if we determine B on the hypothesis that there is no slipping
at the wall of the pipe (r = a, say), we obtain

This gives, for the flux across any section,

'sr-*1?* (4)-

It has been assumed, for shortness, that the flow takes place
under pressure only. If we have an extraneous force X acting
parallel to the length of the pipe, the flux will be

In practice, X is the component of gravity in the direction of the
length.

The formula (4) contains exactly the laws found experimentally
by Poiseuille* in his researches on the flow of water through
capillary tubes ; viz. that the time of efflux of a given volume of
water is directly as the length of the tube, inversely as the
difference of pressure at the two ends, and inversely as the fourth
power of the diameter.

This last result is of great importance as furnishing a conclusive proof that
there is in these experiments no appreciable slipping of the fluid in contact
with the wall. If we were to assume a slipping-coefficient /3, as explained in
Art. 285, the surface-condition would be

-p.dw/dr=fiiv,
or w— -\diojdr ..............................

* "Recherches experimentales sur le mouvement des liquides dans les tubes de
tr&s petits diametres," Comptes Rendus, tt. xi., xii. (1840-1), Mem. des Sav.
Etrangers, t. ix. (1846).

522 VISCOSITY. [CHAP, xi

if X = /i//3. This determines B, in (2), so that

(ii).

If X/a be small, this gives sensibly the same law of velocity as in a tube of
radius a + \, on the hypothesis of no slipping. The corresponding value
of the flux is

If X were more than a very minute fraction of a in the narrowest tubes
employed by Poiseuille [a ='001 5 cm.J a deviation from the law of the fourth
power of the diameter, which was found to hold very exactly, would become
apparent. This is sufficient to exclude the possibility of values of X such as
•235 cm., which were inferred by Helmholtz and Piotrowski from their
experiments on the torsional oscillations of a metal globe filled with water,
described in the paper already cited*.

The assumption of no slipping being thus justified, the comparison of the
formula (4) with experiment gives a very direct means of determining the
value of the coefficient p, for various fluids.

It is easily found from (3) and (4) that the rate of shear close
to the wall of the tube is equal to 4w0/a, where w0 is the mean
velocity over the cross-section. As a numerical example, we may
take a case given by Poiseuille, where a mean velocity of 126 '6 c. s.
was obtained in a tube of '01134 cm. diameter. This makes
4>w0/a— 89300 radians per second of time.

290. Some theoretical results for sections other than circular
may be briefly noticed.

1°. The solution for a channel of annular section is readily deduced from
equation (2) of the preceding Art., with A retained. Thus if the boundary-
conditions be that w = Q for r = a and r=b, we find

giving a flux

2°. It has been pointed out by Greenhillt that the analytical conditions
of the present problem are similar to those which determine the motion of a
Motionless liquid in a rotating prismatic vessel of the same form of section

* For a fuller discussion of this point see Whetham, " On the Alleged Slipping
at the Boundary of a Liquid in Motion," Phil. Trans., 1890, A.

f " On the Flow of a Viscous Liquid in a Pipe or Channel," Proc. Lond. Math.
Soc., t. xiii. p. 43 (1881).

289-291] FLOW THROUGH A PIPE. 523

(Art. 72). If the axis of z be parallel to the length of the pipe, and if we
assume that w is a function of #, y only, then in the case of steady motion
the equations reduce to

where Vi2=d2/d^2+d2/d^2. Hence, denoting by P the constant pressure-
gradient ( - dp/dz), we have

Vl*w=-P/fj. ................................. (iv),

with the condition that w=0 at the boundary. If we write ^ — £ o>

for wt and 2o> for P/p, we reproduce the conditions of the Art. referred to.

This proves the analogy in question.

In the case of an elliptic section of semi-axes a, 6, we assume

which will satisfy (iv) provided

The discharge per second is therefore

P

~'

f f
I I
]J

P naW
wdxdy = — . ~5 — TT ........................ (vn)

' ~

This bears to the discharge through a circular pipe of the same sectional
area the ratio 2«6/(a2 -f- 62). For small values of the eccentricity (e) this
fraction differs from unity by a quantity of the order e*. Hence considerable
variations may exist in the shape of the section without seriously affecting
the discharge, provided the sectional area be unaltered. Even when a : b = 8 : 7,
the discharge is diminished by less than one per cent.

291. We consider next some simple cases of steady rotatory
motion.

The first is that of two-dimensional rotation about the axis of
z, the angular velocity being a function of the distance (r) from
this axis. Writing

u = — wy, v = cox, ........................ (1)

we find that the rates of extension along and perpendicular to the
radius vector are zero, whilst the rate of shear in the plane xy is
rdco/dr. Hence the moment, about the origin, of the tangential
forces on a cylindrical surface of radius r, is per unit length of
the axis, = prdcdjdr . 2?rr . r. On account of the steady motion,
the fluid included between two coaxial cylinders is neither gaining

* This, with corresponding results for other forms of section, appears to have
been obtained by Boussinesq in 1868 ; see Hicks, Brit. Ass. Rep., 1882, p. 63.

524 VISCOSITY. [CHAP, xi

nor losing angular momentum, so that the above expression must
be independent of r. This gives

(2).

If the fluid extend to infinity, while the internal boundary is that
of a solid cylinder of radius a, whose angular velocity is o>0, we
have

a> = co0a2/r* .......................... (3).

The frictional couple on the cylinder is therefore

— 47TyU.a4ft)0 ........................... (4).

If the fluid were bounded externally by a fixed coaxial cylin
drical surface of radius b we should find

which gives a frictional couple

£*-,•».• -(6)*-

292. A similar solution, restricted however to the case of
infinitely small motions, can be obtained for the steady motion of
a fluid surrounding a solid sphere which is made to rotate
uniformly about a diameter. Taking the centre as origin, and the
axis of rotation as axis of a?, we assume

u = — coy, v = wx) w = 0 (1),

where o> is a function of the radius vector r, only. If we put

P=fa>rdr (2),

these equations may be written

u = -dP/dy, v = dP/d.x, w = Q (3);

and it appears on substitution in Art. 286 (4) that, provided we
neglect the terms of the second order in the velocities, the
equations are satisfied by

p =• const., V2P = const (4).

* This problem was first treated, not quite accurately, by Newton, Principia,
Lib. ii., Prop. 51. The above results were given substantially by Stokes, I. c. ante,
p. 515.

291-292] ROTATING SPHERE. 525

The latter equation may be written

d?P , 2 dP

;nr + - -j- ~ const->
dr* r dr

or r -;— + 3ce) = const (5),

dr

whence o> = A/1* + B (6).

If the fluid extend to infinity and is at rest there, whilst <y0 is
the angular velocity of the rotating sphere (r = a\ we have

0) = — 0)fl

.(7).

If the external boundary be a fixed concentric sphere of radius

b the solution is

a3 63-r3

ft) = — . 7

. , .

r3 6" — a3

,

(O).

The retarding couple on the sphere may be calculated directly
by means of the formulae of Art. 284, or, perhaps more simply,
by means of the Dissipation Function of Art. 287. We find
without difficulty that the rate of dissipation of energy

-s- dxdvdz
dr I

^}dr

If N denote the couple which must be applied to the sphere to
maintain the rotation, this expression must be equivalent to Na>0,
whence

or, in the case corresponding to (7), where b = oo ,

........................ (11)-

The neglect of the terms of the second order in this problem involves a
more serious limitation of its practical value than might be expected. It is
not difficult to ascertain that the assumption virtually made is that the ratio

* Kirchhofif, Meclumik, c. xxvi.

526 VISCOSITY. [CHAP, xi

o>0a2/i> is small. If we put v = -018 (water), and a = 10, we find that the
equatorial velocity co0a must be small compared with '0018 (c. s.)*.

When the terms of the second order are sensible, no steady motion of
this kind is possible. The sphere then acts like a centrifugal fan, the motion
at a distance from the sphere consisting of a flow outwards from the equator
and inwards towards the poles, superposed on a motion of rotation f.

It appears from Art. 286 that the equations of motion may be written

= ^- + ^%, &c., &c
where

Hence a steady motion which satisfies the conditions of any given problem,
when the terms of the second order are neglected, will hold when these are
retained, provided we introduce the constraining forces

X=2(wr)-v£\ Y=2(u£-w£), Z=2(v£-ur)) ......... (iii)t

The only change is that the pressure p is diminished by ^pg2. These forces
are everywhere perpendicular to the stream-lines and to the vortex-lines, and
their intensity is given by the product 2^co sin ^, where o> is the angular
velocity of the fluid element, and ^ is the angle between the direction of q
and the axis of to.

In the problem investigated in this Art. it is evident d priori that the
constraining forces

X=-A, F=-«fy, Z=0 ............... . ........ (iv).

would make the solution rigorous. It may easily be verified that these
expressions differ from (iii) by terms of the forms -dQ/dx, — dQ/dy, -dQ/dz,
respectively, which will only modify the pressure.

293. The motion of a viscous incompressible fluid, when the
effects of inertia are insensible, can be treated in a very general
manner, in terms of spherical harmonic functions.

It will be convenient, in the first place, to investigate the
general solution of the following system of equations :

VV = 0, VV = 0, W = 0 ............... (1),

* Cf. Lord Rayleigh, " On the Flow of Viscous Liquids, especially in Two
Dimensions," Phil. Mag., Oct. 1893.
t Stokes, 1. c. ante, p. 515.
£ Lord Rayleigh, L c.

292-293] GENERAL PROBLEM OF SLOW MOTION.

527

The functions u, v , w' may be expanded in series of solid har
monics, and it is plain that the terms of algebraical degree n in
these expansions, say un', vn', wn', must separately satisfy (2). The
equations V2wn' = 0, V2vn' = 0, V2wn' = 0 may therefore be put in
the forms

dx

d fdVn d?ln\ _ d

dy V dx dy ) ~ dz \ dz

dz \ dy dz J dx \ dx dy

d fdiin dwn'\ _ d idwn dvn'

dx \ dz dx ] dy \ dy dz

(3).

Hence

_
dy dz ~ dx ' dz

dw^ _ dxn dvn' _ dun' _ dxn
dx ~ dy ' dx dy ~~ dz "

where %n is some function of x, y, z ; and it further appears from
these relations that V2^n = 0, so that %n is a solid harmonic of
degree n.

From (4) we also obtain

- ^ (Win + yvn' + ZWn*) . . . (5),

with two similar equations. Now it follows from (1) and (2) that
V2 (a?un' + y»n + *wn') = 0 .................. (6),

so that we may write

OMn' + yVn -f ZWn = </>?l+1 .................. (7),

where 0n+1 is a solid harmonic of degree n + 1. Hence (5) may be
written

(„ + 1) Un' = *£±! + g dX- _ /Xn ..... (8)
dx dy 1 dz

The factor n + 1 may be dropped without loss of generality ; and
we obtain as the solution of the proposed system of equations :

528 VISCOSITY. [CHAP, xi

dy ^ dz J '

(9),

+ _

dz + y dx * dy)

where the harmonics (f)n, %n are arbitrary*.

294. If we neglect the inertia-terms, the equations of motion
of a viscous liquid reduce, in the absence of extraneous forces, to
the forms

............ (1),

.,, du dv dw /ox

with ;j- + :r 4-:r-:=0 ........................ (2)-

dx dy dz

By differentiation we obtain

0 .............................. (3),

so that p can be expanded in a series of solid harmonics, thus

P = 2pn .............................. (4).

The terms of the solution involving harmonics of different alge
braical degrees will be independent. To obtain the terms in pn
we assume

*f> + £r^± ^,\
da; dx r2™^1

dy dy

dz dz

where r2 = x2 + y2 + z*. The terms multiplied by B are solid
harmonics of degree n + 1, by Arts. 82, 84. Now

dx J dx \ dx y dy dz/ dx dx

-. ,

* Cf. Borchardt, " Untersuchungen uber die Elasticitat fester Korper unter
Beriicksichtigung der Warme," Berl. Monatsber., Jan. 9, 1873; Gesammelte Werke,
Berlin, 1888, p. 245. The investigation in the text is from a paper " On the
Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., t. xiii., p. 51 (1881).

293-294J SOLUTION IN SPHERICAL HARMONICS. 529

Hence the equations (1) are satisfied, provided

Also, substituting in (2), we find

l

whence B=, - ... /rt - rrr^ - ;rr~ ................. CO-

Hence the general solution of the system (1) and (2) is

u = —

* dpn

2(2n + l) dx

= lvf __ rL

H (2 (2n +

, - a Pn , ,

(n + 1) (2n + 1) (2n + 3) dy r™+^

^ \+w'

1) efe (n + 1) (2?i + 1) (2w + 3) dz

......... (8)*,

where w', v7, w' have the forms given in (9) of the preceding Art.

The formulae (8) make

nr2

(9).

Also, if we denote by f, 77, f the components of the angular
velocity of the fluid (Art. 31), we find

dz dy J dx'

o ^ V -^ f„(*/Pn dpn\ ^ / ,T\(*'Xn \ /in\

Zri = — Z ; , . /rt 77 z -j- x -j- I + ^- (?i + I ) ~= — , V...( 1U).

/A (n + 1) (2r? + 1) V «# dz J dy

These make 2 (#f + 3/17 + ^f) = Sn (w + 1) XH (H).

* This investigation is derived, with some modifications, from various sources.
Cf. Thomson and Tait, Natural Philosophy, A.rt. 736 ; Borchardt, 1. c. ; Oberbeck,
"Ueber stationare Fliissigkeitsbewegungen mit Beriicksichtigung der inneren
Reibung," Crelle, t. Ixxxi., p. 62 (1876).

L. 34

530

VISCOSITY.

[CHAP, xi

295. The results of Arts. 293, 294 can be applied to the
solution of a number of problems where the boundary conditions
have relation to spherical surfaces. The most interesting cases
fall under one or other of two classes, viz. we either have

xu -f yv + zw = 0 (1)

everywhere, and therefore pn = 0, (f>n = 0 ; or

af+yi/ + *f=0 (2),

and therefore %n = 0.

1°. Let us investigate the steady motion of a liquid past a fixed spherical
obstacle. If we take the origin at the centre, and the axis of x parallel to the
flow, the boundary conditions are that w = 0, v = 0, w = 0 for r=a (the radius),
and w = u, v = 0, w = 0 for r=o>. It is obvious that the vortex-lines will be
circles about the axis of x, so that the relation (2) will be fulfilled. Again, the
equation (9) of Art. 294, taken in conjunction with the condition to be
satisfied at infinity, shews that as regards the functions pn and 0n we are
limited to surface-harmonics of the first order, and therefore to the cases
•» = 1, n= -2. Also, we must evidently have ^ = 0. Assuming, then,

we find

3B

The condition of no slipping at the surface r=a gives

whence
Hence

These make

:0> *— *2

(V),
(vi).

295]

STEADY MOTION OF A SPHERE.

531

The components of stress across the surface of a sphere of radius r are,
by Art. 283,

If we substitute the values of pxxt pxy, pxt, ..., from Art. 284, we find

d

-j- (xu +yv + zw\

d

d \ d

'-r--i)w+n-r,

In the present case we have

We thus obtain, for the component tractions on the sphere r = a,

If 8S denote an element of the surface, we find

The resultant force on the sphere is therefore parallel to .r, arid equal to

The character of the motion may be most concisely expressed by means of
the stream-function of Art. 93. If we put x=r cos 0, the flux (27r\^) through a
circle with Ox as axis, whose radius subtends an angle 6 at 0 is given by

as is evident at once from (v).

If we impress on everything a velocity - u in the direction of x, we get
the case of a sphere moving steadily through a viscous fluid which is at rest
at infinity. The stream-function is then

«2\

(xiii)*

The diagram on p. 532, shews the stream-lines ^ = const., in this case, for a
series of equidistant values of >//>. The contrast with the case of a Motionless
liquid, depicted on p. 137, is remarkable, but it must be remembered that the

* This problem was first solved by Stokes, in terms of the stream-function,
I.e. ante p. 518.

34—2

532

VISCOSITY.

[CHAP, xi

fundamental assumptions are very different. In the former case inertia was
predominant, and viscosity neglected ; in the present problem these circum
stances are reversed.

If X be the extraneous force acting on the sphere, this must balance the
resistance, whence

(xiv).

It is to be noticed that the formula (xiii) makes the momentum and the

295] RESISTANCE. 533

energy of the fluid both infinite*. The steady motion here investigated
could therefore only be fully established by a constant force X acting on the
sphere through an infinite distance.

The whole of this investigation is based on the assumption that the
inertia-terms udu\dx, ... in the fundamental equations (4) of Art. 286 may
be neglected in comparison with i/v2w, .... It easily follows from (iv) above
that ua must be small compared with v. This condition can always be
realized by making u or a sufficiently small, but in the case of mobile fluids
like water, this restricts us to velocities or dimensions which are, from a
practical point of view, exceedingly minute. Thus even for a sphere of a
millimetre radius moving through water (t/ = '018), the velocity must be
considerably less than -18 cm. per sec.f.

We might easily apply the formula (xiv) to find the * terminal velocity ' of
a sphere falling vertically in a fluid. The force X is then the excess of the
gravity of the sphere over its buoyancy, viz.

where p denotes the density of the fluid, and p0 the mean density of the
sphere. This gives

This will only apply, as already stated, provided u«/i/ is small. For a
particle of sand descending in water, we may put (roughly)

p0 = 2p, y=-018, # = 981,

whence it appears that a must be small compared with -0114 cm. Subject to
this condition, the terminal velocity is u = 12000 a2.

For a globule of water falling through the air, we have
Po = l, p = -00129, ,i = -00017.

This gives a terminal velocity u = 1280000 a2, subject to the condition that a
is small compared with -006 cm.

2°. The problem of a rotating sphere in an infinite mass of liquid is
solved by assuming

av_o ^Y-9 \

«=z -§y*-y-fr>}
"=*%2-* %r

w = y — j — — # — = —
dx dy

.(xvii),

where

* Lord Eayleigh, Phil. Mag., May 1886.
t Lord Eayleigh, 1. c. ante p. 526.

534 VISCOSITY. [CHAP, xi

the axis of z being that of rotation. At the surface r—a we must have

u = — ay, v = ax, w = Q,
if <» be the angular velocity of the sphere. This gives A = <oa3 ; cf. Art. 292.

296. The solutions of the corresponding problems for an
ellipsoid can be obtained in terms of the gravitation-potential of
the solid, regarded as homogeneous and of unit density.

The equation of the surface being

5+1

the gravitation-potential is given, at external points, by Dirichlet's formula*

where A = {(

and the lower limit is the positive root of

This makes

dQ dQ, dQ.

-j— = zirax, -y- = 27raV, —r~
dx ciy ciz

where

d\

We will also write

™ . ...

.............................. (vn) ;

it has been shewn in Art. 110 that this satisfies V2x~^-

If the fluid be streaming past the ellipsoid, regarded as fixed, with the
general velocity u in the direction of #, we assume f

-, ..................... (viii).

dxdy dy' (

-,-
dxdz dz

These satisfy the equation of continuity, in virtue of the relations

* Crelle, t. xxxii. (1846) ; see also Kirchhoff, Mechanik, c. xviii., and Thomson
and Tait, Natural Philosophy (2nd ed.), Art. 494m.
t Oberbeck, I.e. ante p. 529.

295-296] MOTION OF AN ELLIPSOID. 535

and they evidently make w = u, v = 0, w = Q at infinity. Again, they make

so that the equations (1) of Art. 294 are satisfied by

-+ const ............................... (x).

It remains to shew by a proper choice of A, B we can make u, v, w = 0 at
the surface (i). The conditions 0 = 0, w = 0 require

+S<^] =0)
dAj*=o

or 27r^/a2 + £ = 0 ................................. (xi).

With the help of this relation, the condition u = 0 reduces to

2ir4ao-#xo-Hl = 0 .............................. (xii),

where the suffix denotes that the lower limit in the integrals (vi) and (vii) is
to be replaced by zero. Hence

u

(xiii)-

At a great distance r from the origin we have

Q = — A Trabcjr, ^ = 2a6c/r,

whence it appears, on comparison with the equations (iv) of the preceding Art.,
that the disturbance is the same as would be produced by a sphere of radius
a, determined by

(xiv),

"*i£* .............................. (xv)"

The resistance experienced by the ellipsoid will therefore be

(xvi).

In the case of a circular disk moving broadside-on, we have a=0, 6 = c;
whence o0 = 2, ^O=TTOC, so that

a = ^-c=-85c.

O7T

We must not delay longer over problems which, for reasons
already given, have hardly any real application except to fluids of
extremely great viscosity. We can therefore only advert to the
mathematically very elegant investigations which have been
given of the steady rotation of an ellipsoid*, and of the flow

* Edwardes, Quart. Journ. Math., t. xxvi., pp. 70, 157 (1892).

536 VISCOSITY. [CHAP, xi

through a channel bounded by a hyperboloid of revolution (of one
sheet)*.

Some examples of a different kind, relating to two-dimensional
steady motions in a circular cylinder, due to sources and sinks in
various positions on the boundary, have been recently discussed
by Lord Rayleighf.

297. Some general theorems relating to the dissipation of
energy in the steady motion of a liquid under constant extra
neous forces have been given by von Helmholtz and Korteweg.
They involve the assumption that the terms of the second order in
the velocities may be neglected.

1°. Considering the motion in a region bounded by any closed surface 2,
let u, v, w be the component velocities in the steady motion, and u + u', v+v',
w + tv' the values of the same components in any other motion subject only to
the condition that u', v', w' vanish at all points of the boundary 2. By
Art. 287 (3), the dissipation in the altered motion is equal to

where the accent attached to any symbol indicates the value which the
function in question assumes when u, -y, w are replaced by u', v', w'. Now the
formulae (2), (3) of Art. 284 shew that, in the case of an incompressible fluid,

>'VVb +p',gC + 2p'yzf+ ty' „& + 2p'xyk (11),

each side being a symmetric function of a, b, c, /, g, h and a', 6', c', /', ^', /*'.
Hence, and by Art. 287, the expression (i) reduces to

JJJ* dxdydz + JJJ*' dxdy dz + 2 jjj(pxx a' +pvy b' +pz,c'

+ Zpyzf + Vpzxg' + %Pxyh'} dxdydz (iii).

The last integral may be written

du' du' du' .

and by a partial integration, remembering that u', v', iv' vanish at the
boundary, this becomes

or Mp(Xu + YJ+Zvi)dxdyd* ....................... (vi),

* Sampson, 1. c. ante p. 134.

t "On the Flow of Viscous Liquids, especially in Two Dimensions," Phil.
Oct. 1893.

296-297] GENERAL THEOREMS. 537

by Art. 286. If the extraneous forces X, F, Z have a single-valued potential,
this vanishes, in virtue of the equation of continuity, by Art. 42 (4).

Under these conditions the dissipation in the altered motion is equal to
\\\$>dxdydz + \\\& dxdydz .......................... (vii),

or 2 (F+F'}. That is, it exceeds the dissipation in the steady motion by the
essentially positive quantity 2F' which represents the dissipation in the
motion u', v', w'.

In other words, provided the terms of the second order in the velocities
may be neglected, the steady motion of a liquid under constant forces having
a single-valued potential is characterized by the property that the dissipation
in any region is less than in any other motion consistent with the same
values of u, v, w at the boundary.

It follows that, with prescribed velocities over the boundary, there is only
one type of steady motion in the region*.

2°. If u, v, w refer to any motion whatever in the given region, we have
2/= JJJ* dxdydz

= 2Stt(pxXa+PyVi>+P*zC + 2pV3f+2pzXsr + 2pxyh)dxdydz ...... (viii),

since the formula (ii) holds when dots take the place of accents.

The treatment of this integral is the same as before. If we suppose that
u, v, w vanish over the bounding surface 2, we find

= -pJJJ(w2 + £2 + M>2) dxdydz + pSH(Xu + Yv+Ziv) dxdydz ...(ix).
The latter integral vanishes when the extraneous forces have a single-
valued potential, so that

F= -P$jj(u2 + v2 + w2)dxdydz ..................... (x).

This is essentially negative, so that F continually diminishes, the process
ceasing only when u=Q, v=0, 10 = 0, that is, when the motion has become

Hence when the velocities over the boundary 2 are maintained constant,
the motion in the interior will tend to become steady. The type of steady
motion ultimately attained is therefore stable, as well as unique f.

It has been shewn by Lord RayleighJ that the above theorem can be
extended so as to apply to any dynamical system devoid of potential energy,

* Helmholtz, " Zur Theorie der stationaren Strome in reibenden Fliissig-
keiten," Verh. d. naturhist.-med. Vereins, Oct. 30, 1868 ; JHss. Abh., t. i., p. 223.

t Korteweg, "On a General Theorem of the Stability of the Motion of a Viscous
Fluid," Phil. May., Aug. 1883.

+ I.e. ante p. 526.

538 VISCOSITY. [CHAP, xi

in which the kinetic energy (T] and the dissipation-function (F) can be
expressed as quadratic functions of the generalized velocities, with constant
coefficients.

If the extraneous forces have not a single-valued potential, or if instead of
given velocities we have given tractions over the boundary, the theorems
require a slight modification. The excess of the dissipation over double the
rate at which work is being done by the extraneous forces (including the
tractions on the boundary) tends to a unique minimum, which is only
attained when the motion is steady*.

Periodic Motion.

298. We next examine the influence of viscosity in various
problems of small oscillations.

We begin with the case of ' laminar ' motion, as this will enable
us to illustrate some points of great importance, without elaborate
mathematics. If we assume that v = 0, w=Q, whilst u is a
function of y only, the equations (4) of Art. 286 require that
p = const., and

du d2u

This has the same form as the equation of linear motion of
heat. In the case of simple-harmonic motion, assuming a time-
factor ei(<Tt+f>, we have

d?u ia-

-j-- — — u ........................... (2),

ajf v

the solution of which is

u = Ae(l+i)f*y + Be~(l+i}M ...................... (3),

provided 0=(<r/2v)* ........................... (4).

Let us first suppose that the fluid lies on the positive side of
the plane xzt and that the motion is due to a prescribed oscillation

w=±rae*'(<rt+e) ........................... (5)

of a rigid surface coincident with this plane. If the fluid extend
to infinity in the direction of y-positive, the first term in (3) is
excluded, and determining B by the boundary-condition (5), we
have

* Cf. Helmholtz, Z.c.

297-298] LAMINAR OSCILLATIONS. 539

or, taking the real part,

s(<rt-0y + e) .................. (7),

corresponding to a prescribed motion

u — a cos (at 4- e) ........................ (8)

at the boundary*.

The formula (7) represents a wave of transversal vibrations
propagated inwards from the boundary with the velocity <r//3, but
with rapidly diminishing amplitude, the falling off within a wave
length being in the ratio e~'27T, or ^.

The linear magnitude

27T//3 Or (4>7TV . 2?r/(7)*

is of great importance in all problems of oscillatory motion which
do not involve changes of density, as indicating the extent to
which the effects of viscosity penetrate into the fluid. In the
case of air (y = '13) its value is 1'28P* centimetres, if P be the
period of oscillation in seconds. For water the corresponding
value is '47P*. We shall have further illustrations, presently,
of the fact that the influence of viscosity extends only to a short
distance from the surface of a body performing small oscillations
with sufficient frequency.

The retarding force on the rigid plane is, per unit area,

— IJL -j- I = fj,/3a {cos (at + e) — sin (at + e)}
L»yJ»=o

= pv* a* a cos (at + e + \ TT) ............... (9).

The force has its maxima at intervals of one-eighth of a period
before the oscillating plane passes through its mean position.

On the forced oscillation above investigated we may superpose any of the
normal modes of free motion of which the system is capable. If we assume
that

u <x A cos my -f B sin my ........................... (i),

and substitute in (1), we find

du

whence we obtain the solution
u = '2,(A

Stokes, I.e. ante p. 518.

540 VISCOSITY. [CHAP, xi

The admissible values of w, and the ratios A : B are as a rule determined
by the boundary conditions. The arbitrary constants which remain are then
to be found in terms of the initial conditions, by Fourier's methods.

In the case of a fluid extending from y= — oo to y = + GO , all real values of
m are admissible. The solution, in terms of the initial conditions, can in
this case be immediately written down by Fourier's Theorem (Art. 227 (15)).
Thus

u = - { dm( /(X)cosw(y-X)e-"m2^X ............... (iv),

"" J 0 J -oo

if t*=/(y) ....................................... (v)

be the arbitrary initial distribution of velocity.

The integration with respect to m can be effected by the known formula

w/todfc-jf-)1*-^ ..................... (vi).

\a/

We thus find u= — i- f° e'^^1** f(\} d\ .. ... (vii).

2<«w_.

As a particular case, let us suppose that f(y}=±U, where the upper or
lower sign is to be taken according as y is positive or negative. This will
represent the case of an initial surface of discontinuity coincident with the
plane y = 0. After the first instant, the velocity at this surface will be zero
on both sides. We find

u= U r
2 (**)*</«

(viii).

By a change of variables, and easy reductions, this can be brought to the
form

where in Glaisher's (revised) notation f

Erf x = e~x"dx .............................. (x).

The function 27r~*Erf x was tabulated by EnckeJ. It appears that u will
equal \U when y/2j/M = '4769. For water, this gives, in seconds and centi
metres,

* Lord Rayleigh, "On the Stability, or Instability, of certain Fluid Motions,"

Proc. Lond. Math. Soc., t. xi., p. 57 (1880).

t See Phil Mag., Dec. 1871, and Encyc. Britann., Art. " Tables."

J BerL Ast. Jahrbtich, 1834. The table has been reprinted by De Morgan,

Encyc. Metrop., Art. "Probabilities," and Lord Kelvin, Math, and Phys. Papers,

t. iii., p. 434.

298-299] DIFFUSION OF ANGULAR ^7ELOC^TY. 541

The corresponding result for air is

t = 8-3 y2.

These results indicate how rapidly a surface of discontinuity, if it could
ever be formed, would be obliterated in a viscous fluid.

The angular velocity («) of the fluid is given by

This represents the diffusion of the angular velocity, which is initially
confined to a vortex-sheet coincident with the plane y = 0, into the fluid on
either side.

299. When the fluid does not extend to infinity, but is
bounded by a fixed rigid plane y = h, then in determining the
motion due to a forced oscillation of the plane y = 0 both terms of
(3) are required, and the boundary conditions give

, sinh(l + i)@(h — y) .. . , /11N

whence u = a — . \ /.. ^ ^ - . et(<rt+g) ............ (11),

sm

as is easily verified. This gives for the retarding force per unit
area on the oscillating plane

_ p = p (i + i) fa coth (l+j)0h. e*^ . . . (12).

The real part of this may be reduced to the form
smh 2/3A cos (o-Z + e -I- ITT) + sin 2/3A, sin (oi

cosh 2£A- cos 2/3h

......... (13).

When @h is moderately large this is equivalent to (9) above ;
whilst for small values of f3h it reduces to

fjLa/h.cos((rt+e) ..................... (14),

as might have been foreseen.

This example contains the theory of the modification introduced by
Maxwell* into Coulomb's method f of investigating the viscosity of liquids by
the rotational oscillation of a circular disk in its own (horizontal) plane. The

* 1. c. ante p. 513.

t Mem. de Vlnst., t. iii. (1800).

542 VISCOSITY. [CHAP, xi

addition of fixed parallel disks at a short distance above and below greatly
increases the effect of viscosity.

The free modes of motion are expressed by (iii), with the conditions that
u = 0 for y = 0 and y = h. This gives .4=0 and mh=STT^ where s is integral.
The corresponding moduli of decay are then given by r = l/vm2.

300. As a further example, let us take the case of a force
X=fcos(at + e) ..................... (1),

acting uniformly on an infinite mass of water of uniform depth h.
The equation (1) of Art. 298 is now replaced by
du d^u

-J1 — V T~o ~r ^ ........................ (^ )•

at dy2

If the origin be taken in the bottom, the boundary-conditions
are u = 0 for y = 0, and dujdy = 0 for y = h ; this latter condition
expressing the absence of tangential force on the free surface.

Replacing (1) by

X=fei«rt+e} ............................ (3),

-, ifL coah(l+i)P(h-y)} .... ,

we find u = -^\l ---- \ ,., \ -\ 01 \ e }

a- ( cosh(l +l)ph J

if £ = (V/2j/)*, as before.

When ph is large, the expression in { } reduces practically to
its first term for all points of the fluid whose height above the
bottom exceeds a moderate multiple of /3~l. Hence, taking the

real part,

f
u = J~ sin (at + e) ........................ (5).

(7

This shews that the bulk of the fluid, with the exception of a
stratum at the bottom, oscillates exactly like a free particle, the
effect of viscosity being insensible. For points near the bottom
the formula (4) becomes

(6),

or, on rejecting the imaginary part,

u = i- sin (<rt + e) - '— e~^ sin (at - fiy + e) (7).

(7 (T

299-300]

PERIODIC TIDAL FORCE.

543

This might have been obtained directly, as the solution of (2)
satisfying the conditions that u = 0 for y = 0, and

u =f/<r . sin (at + e)

for large values of (3y.

The curves J, £, (7, Z>, E, F in the accompanying figure represent
successive forms assumed by the same line of particles at intervals of
one-tenth of a period. To complete the series it would be necessary to add
the images of E, D, C, B with respect to the vertical through 0. The whole
system of curves may be regarded as successive aspects of a properly shaped
spiral revolving uniformly about a vertical axis through 0. The vertical range
of the diagram is one wave-length (2?r//3) of the laminar disturbance.

As a numerical illustration we note that if i/ = '0l78, and 2ir/o- = 12 hours,
we find /3~1 = 15'6 centimetres. This indicates how utterly insensible must
be the direct action of viscosity on oceanic tides. There can be no doubt that
the dissipation of energy by ' tidal friction ' takes place mainly through the
eddying motion produced by the exaggeration of tidal currents in shallow
water. Cf. Art. 310.

544 VISCOSITY. [CHAP. XT

When /3h is small the real part of (4) gives

w = /y(2&-y).cos(<rf + €) ............... (8),

the velocity being in the same phase with the force, and varying
inversely as v.

301. To find the effect of viscosity on free waves on deep
water we may make use of the Dissipation -Function of Art. 287,
in any of the forms there given, the simplest for our purpose
being

since, by Art. 279, the dissipation may, under a certain restriction,
be calculated as if the motion were irrotational.

To put the calculation in a form which shall apply at once to
the case where capillary as well as gravitational forces are taken
into account, we recall that, corresponding to the surface-elevation

77 = a sin k (x — ct) ..................... (2),

we have <f> = aceky cos k (sc — ct) ..................... (3),

since this makes drj/dt = — dfy/dy for y = 0. Hence

and the dissipation is, by (1),

2yitPc2a2 .............................. (5),

per unit area of the surface. The kinetic energy,

has a mean value ^pktfa? per unit area. The total energy, being
double of this, is

2 .............................. (7).

Hence, equating the rate of decay of the energy to the dissipa
tion, we have

a2 .................. (8),

or

(9),

300-302] EFFECT OF VISCOSITY ON WATER-WAVES. 545

whence a = a.Qe~^m ........................ (10).

The ' modulus of decay/ r, is therefore given by r= I/2v&, or,

in terms of the wave-length (X),

T = X2/87T2Z/ (11)*.

In the case of water, this gives

r = -71 2 X2 seconds,

if X be expressed in centimetres. It follows that capillary waves
are very rapidly extinguished by viscosity; whilst for a wave
length of one metre r would be about 2 hours.

The above method rests on the assumption that or is moderately large,
where o-( = kc] denotes the 'speed.' In mobile fluids such as water this
condition is fulfilled for all but excessively minute wave-lengths.

The method referred to fails for another reason when the depth is less than
(say) half the wave-length. Owing to the practically infinite resistance to
slipping at the bottom, the dissipation can no longer be calculated as if the
motion were irrotational.

302. The direct calculation of the effect of viscosity on
water waves can be conducted as follows.

If the axis of y be drawn vertically upwards, and if we assume
that the motion is confined to the two dimensions x, y, we have

du 1 dp

dt p dx

dv _ I dp

dt p dy

du dv

.,,

Wlth

These are satisfied by
dd>

-

~r-> v = -~^- + -j ............ (3),

dy dy dx

=-<» ........................ w,

* Stokes, 1. c. ante p. 518. (Through an oversight in the calculation the value
obtained for T was too small by one-half.)

L- 35

546 VISCOSITY. [CHAP, xi

provided V^ = 0,

(5),

where V,2 = d?lda? + d?jdf .

To determine the ' normal modes ' which are periodic in
respect of as, with a prescribed wave-length 2?r/&, we assume a
time-factor eat and a space-factor eikx. The solutions of (5) are

then

<f> = (Ad* + Be-^) eikx+at

.„.

with

The boundary-conditions will supply equations which are sufficient
to determine the nature of the various modes, and the corre
sponding values of a.

In the case of infinite depth one of these conditions takes the
form that the motion must be finite for y - - oo . Excluding for
the present the cases where m is pure-imaginary, this requires
that B = Q, D = 0, provided m denote that root of (7) which has
its real part positive. Hence

v mCemv) eikx+at,}
ikCtfw) eikx+at j ......

If TJ denote the elevation at the free surface, we must have
drj/dt = v. If the origin of y be taken in the undisturbed level,
this gives

r) = -^(A-iC)eikx+at .................. (9).

If Tj_ denote the surface-tension, the stress-conditions at the
surface are evidently

(10),

to the first order, since the inclination of the surface to the
horizontal is assumed to be infinitely small. Now

dv du

302] CASE OF INFINITE DEPTH. 547

whence, by (4) and (6) we find, at the surface,

Pyy_rd^ = _d^ v<h

p dxz dt dy

+gk + T'fr) A - i (gk + T'ks + Zvkma) C] ... .(12),

(13),

where T' = T1/p> the common factor eikx+at being understood.
Substituting in (10), and eliminating the ratio A : C, we obtain

(cL + 2vk*)* + gk+T'ks = 4>vk*m ............ (14).

If we eliminate m by means of (7), we get a biquadratic in a,
but only those roots are admissible which give a positive value to
the real part of the left-hand member of (14), and so make the
real part of m positive.

If we write, for shortness,

gk + T'k3 = o-2, vk*/<r = 0, a + 2vk2 = x<r ...... (15),

the biquadratic in question takes the form

(a? + iy=160*(a;-0) .................. (16).

It is not difficult to shew that this has always two roots (both
complex) which violate the restriction just stated, and two
admissible roots which may be real or complex according to the
magnitude of the ratio 0. If X be the wave-length, and c (— a/k)
the wave-velocity in the absence of friction, we have

0=vk/c = (2Trv/c)-r-\ .................. (17).

Now, for water, if cm denote the minimum wave-velocity of
Art. 246, we find 2irv/cm = '0048 cm., so that except for very
minute wave-lengths 0 is a small number. Neglecting the square
of 0, we have x=±i, and

OL- — 2vk2 ±ia ........................ (18).

The condition pxy — 0 shews that

CJA = - 2ivk*/(oL + 2i/#«) = + 2^2/o- ............ (19),

which is, under the same circumstances, very small. Hence the
motion is approximately irrotational, with a velocity-potential

35—2

548 VISCOSITY. [CHAP, xi

If we put a = + kA/a, the equation (9) of the free surface
becomes, approximately, on taking the real part,

v = ae-2vkHsm(kx±o-t) .................. (21).

The wave-velocity is ajk, or (g/k + Tk)*, as in Art. 246, and
the law of decay is that investigated independently in the last Art.

To examine more closely the character of the motion, as affected by
viscosity, we may calculate the angular velocity (<•>) at any point of the fluid.
This is given by

dv du a

Now, from (7) and (18), we have, approximately,

m = (l±i)& where p = (<r/2v)*.
With the same notation as before, we find

a>=+<rkae-*vk*t+f*y$m{kx±(<rt+py)} ................. (ii).

This diminishes rapidly from the surface downwards, in accordance with
the analogy pointed out in Art. 286. Owing to the oscillatory character of
the motion, the sign of the vortex-motion which is being diffused inwards
from the surface is continually being reversed, so that beyond a stratum of
thickness comparable with 2?r//3 the effect is insensible, just as the fluctuations
of temperature at the earth's surface cease to have any influence at a depth
of a few yards.

In the case of a very viscous fluid, such as treacle or pitch, 6
may be large even when the wave-length is considerable. The
admissible roots of (16) are then both real. One of them is
evidently nearly equal to 20, and continuing the approximation
we find

whence, neglecting capillarity, we have, by (15),

a = -g/2kv ........................ (22).

The remaining real root is 1'09#, nearly, which gives

(23).

The former root is the more important. It represents a slow
creeping of the fluid towards a state of equilibrium with a horizontal
surface ; the rate of recovery depending on the relation between

302-303] MODULUS OF DECAY. 549

the gravity of the fluid (which is proportional to gp) and the
viscosity (/u,), the influence of inertia being insensible. It appears
from (7) and (15) that m = k, nearly, so that the motion is ap
proximately irrotational.

The type of motion corresponding to (23), on the other hand,
depends, as to its persistence, on the relation between the inertia
(p) and the viscosity (/a), the effect of gravity being unimportant.
It dies out very rapidly.

The above investigation gives the most important of the normal modes, of
the prescribed wave-length, of which the system is capable. We know a priori
that there must be an infinity of others. These correspond to pure-imaginary
values of m, and are of a less persistent character. If in place of (6) we
assume

((7cos m'y + D sin m'y}0 '"
with m'2= —W-ajv .................................. (iv),

and carry out the investigation as before, we find

(a2 + 2i^2a +gk + T'&} A - i(gk + T'k*} C- Zivkm'aD = 01

2ik*A + (tf-m'*)C=0) ......... W

Any real value of m' is admissible, these equations determining the ratios
A : C : D ; and the corresponding value of a is

a=-v(k* + m'*) ................................ (vi).

In any one of these modes the plane xy is divided horizontally and
vertically into a series of quasi-rectangular compartments, within each of
which the fluid circulates, gradually coming to rest as the original momentum
is spent against viscosity.

By a proper synthesis of the various normal modes it must be possible to
represent the decay of any arbitrary initial disturbance.

303. The equations (12) and (13) of the preceding Art.
enable us to examine a related question of some interest, viz. the
generation and maintenance of waves against viscosity, by suit
able forces applied to the surface.

If the external forces p'yyi p'xy be given multiples of eikx+at,
where k and a are prescribed, the equations in question deter
mine A and C, and thence, by (9), the value of rj. Thus we find

p'yy (a2 + 2vk*QL + o-2) A - i (o-2 + Zvkma) C

gk(A-iC)

p'xy = a. ^ 2ivk*A + (a + 2^2) 0
#/077 ^ ' J. — iC

550 VISCOSITY. [CHAP, xi

Let us first examine the effect of a purely tangential force.
Assuming p'yy = 0, we find

ia (OL + 2i/£2)2 + a2 - 4i/2A?m
gk ' a + 2z>&2 — 2vkm

For a given wave-length, the elevation will be greatest when
a = ± io-, nearly. To find the force necessary to maintain a train
of waves of given amplitude, travelling in the direction of x-
positive, we put a = — ia. Assuming, for a reason already indi
cated, that vk^ja- and vkm/o- are small, we find

P'xyldP7! = ^vkvjg, or p'xy = 4/^a?; . . . . „ ....... (4).

Hence the force acts forwards on the crests of the waves, and
backwards at the troughs, changing sign at the nodes. A force
having the same distribution, but less intensity in proportion to
the height of the waves than that given by (4), would only retard,
without preventing, the decay of the waves by viscosity. A force
having the opposite sign would accelerate this decay.

The case of purely normal force can be investigated in a
similar manner. If p'xy = 0, we have

p'yy _ (OL + 2i^2)2 + o-2 - 4i/2£3m
gprj~ gk

The reader may easily satisfy himself that when there is no
viscosity this coincides with the result of Art. 226. If we put
a = — ia, we obtain, with the same approximations as before,

Hence the wave-system

77 = a sin (lex — crt) ........................ (7)

will be maintained without increase or decrease by the pressure-
distribution

p' — const. + 4yitfca<7 cos (kx — <rt) ............... (8),

applied to the surface. It appears that the pressure is greatest on
the rear and least on the front slopes of the waves*.

If we call to mind the phases of the particles, revolving in their
circular orbits, at different parts of a wave-profile, it is evident

* This agrees with the result given at the end of Art. 226, where, however, the
dissipative forces were of a different kind.

303] GENERATION OF WAVES BY WIND. 551

that the forces above investigated, whether normal or tangential,
are on the whole urging the surface-particles in the directions in
which they are already moving.

Owing to the irregular, eddying, character of a wind blowing
over a roughened surface, it is not easy to give more than a
general explanation of the manner in which it generates and
maintains waves. It is not difficult to see, however, that the
action of the wind will tend to produce surface forces of the
kinds above investigated. When the air is moving in the direction
in which the wave-form is travelling, but with a greater velocity,
there will evidently be an excess of pressure on the rear-slopes, as
well as a tangential drag on the exposed crests. The aggregate
effect of these forces will be a surface drift, and the residual
tractions, whether normal or tangential, will have on the whole
the distribution above postulated. Hence the tendency will be to
increase the amplitude of the waves to such a point that the
dissipation balances the work done by the surface forces. In like
manner waves travelling faster than the wind, or against the wind,
will have their amplitude continually reduced*.

It has been shewn (Art. 246) that, under the joint influence
of gravity and capillarity, there is a minimum wave-velocity
of 23*2 cm. per sec., or '45 miles per hour. Hence a wind of
smaller velocity than this is incapable of reinforcing waves
accidentally started, which, if of short wave-length, must be
rapidly extinguished by viscosity f. This is in accordance with
the observations of Scott Russell J, from whose paper we make
the following interesting extract :

" Let [a spectator] begin his observations in a perfect calm, when the
surface of the water is smooth and reflects like a mirror the images of
surrounding objects. This appearance will not be affected by even a slight
motion of the air, and a velocity of less than half a mile an hour (8$ in. per sec.)
does not sensibly disturb the smoothness of the reflecting surface. A gentle
zephyr flitting along the surface from point to point, may be observed to
destroy the perfection of the mirror for a moment, and on departing, the
surface remains polished as before ; if the air have a velocity of about a mile
an hour, the surface of the water becomes less capable of distinct reflexion, and

* Cf. Airy, "Tides and Waves," Arts. 265—272; Stokes, Camb. Trans., t. ix.,
p. [62] ; Lord Eayleigh, 1. c. ante p. 526.
t Sir W. Thomson, 1. c. ante p. 446.
t I. c. ante p. 455.

552 VISCOSITY. [CHAP, xi

on observing it in such a condition, it is to be noticed that the diminution of
this reflecting power is owing to the presence of those minute corrugations of
the superficial film which form waves of the third order [capillary waves]....
This first stage of disturbance has this distinguishing circumstance, that the
phenomena on the surface cease almost simultaneously with the intermission
of the disturbing cause so that a spot which is sheltered from the direct action
of the wind remains smooth, the waves of the third order being incapable of
travelling spontaneously to any considerable distance, except when under the
continued action of the original disturbing force. This condition is the
indication of present force, not of that which is past. While it remains it
gives that deep blackness to the water which the sailor is accustomed to
regard as the index of the presence of wind, and often as the forerunner of
more.

" The second condition of wave motion is to be observed when the velocity
of the wind acting on the smooth water has increased to two miles an hour.
Small waves then begin to rise uniformly over the whole surface of the water ;
these are waves of the second order, and cover the water with considerable
regularity. Capillary waves disappear from the ridges of these waves, but are
to be found sheltered in the hollows between them, and on the anterior slopes
of these waves. The regularity of the distribution of these secondary waves
over the surface is remarkable ; they begin with about an inch of amplitude,
and a couple of inches long ; they enlarge as the velocity or duration of the
wave increases ; by and by the coterminal waves unite ; the ridges increase,
and if the wind increase the waves become cusped, and are regular waves of
the second order [gravity waves]*. They continue enlarging their dimensions,
and the depth to which they produce the agitation increasing simultaneously
with their magnitude, the surface becomes extensively covered with waves of
nearly uniform magnitude."

It will be seen that our theoretical investigations give con
siderable insight into the incipient stages of wave-formation. No
sufficient explanation appears however to have been as yet given
of the origin of the regular processions of waves of greater length
which are so conspicuous a result of the continued action of wind
on a large expanse of water.

304. The calming effect of oil on water waves appears to be
due to the variations of tension caused by the extensions and con
tractions of the contaminated surface t- The surface-tension of
pure water is less than the sum of the tensions of the surfaces of
separation of oil and air, and oil and water, respectively, so that a

* Scott Bussell's wave of the first order is the ' solitary wave ' (Art. 234).

t Keynolds, "On the Effect of Oil in destroying Waves on the Surface of
Water," Brit. Ass. Rep., 1880; Aitken, "On the Effect of Oil on a Stormy Sea,
Proc. Roy. Soc. Edin.t t. xii., p. 56 (1883).

303-304] CALMING EFFECT OF OIL ON WAVES. 553

drop of oil thrown on water is gradually drawn out into a thin film.
If the film be sufficiently thin, say not more than two millionths of
a millimetre in thickness, the tension is increased when the thick
ness is reduced by stretching, and conversely. It is evident at
once from the figure on p. 374 that in oscillatory waves any
portion of the surface is alternately contracted and extended,
according as it is above or below the mean level. The consequent
variations in tension produce an alternating tangential drag on the
water, with a consequent increase in the rate of dissipation of
energy.

The preceding formulae enable us to submit this explanation, to a certain
extent, to the test of calculation.

Assuming that the surface tension varies by an amount proportional to
the extension, we may denote it by

where T± is the tension in the undisturbed state, £ is the horizontal displace
ment of a surface particle, and / is a numerical coefficient.

The internal motion of the water is given by the same formulae as in
Art. 302. The surface-conditions are obtained by resolving normally and
tangentially the forces acting on an element of the superficial film. We thus
find, in the case of free waves,

^__
~J

p dx~ da

where T'=Tl/p. In the derivation of the first of these equations a term of
the second order has been neglected.

Since the time-factor is eat, we have ^=u/a, whence, substituting from
Art. 302 (8), (9), (11), we find, as the expression of the surface-conditions (ii),

(a2 + 2^2a +gk+ T'k*} A - i (gk + T'& + 2i//bwa) (7= 0, '

i(2vk*a+fTft?) A + (az + 2vk*a+fT'k*m) (7=0
If we write

the elimination of the ratio A : C between the above equations gives

+ <r2-4,/2£X +fj o-02 ja2+ (l --} o4 = 0 ...... (v).

554 VISCOSITY. [CHAP, xi

This equation, with

* (vi),

determines the values of a and m. Eliminating m, we find

(a + i/F) {/o-02 (a2 + o-2) - 4v2£4a2}2 = M&2 [a2 {(a + 2^2)2 + o-2} -/<r V]2. . .(vii),

or, if we write

a/o-=y, vk*/<r=0 (yiii)>

(y + 0) 1-f^jL (y2 +1) _ 40y\ = 0[~y2{(# + 20)24a}-^/T....(ix).

I CT J L °" J

This equation has an extraneous root y = 0, and other roots are in
admissible as giving, when substituted in (v), negative values to the real part
of m. For all but very minute wave-lengths, 6 is a small number ; and, if we
neglect the squa re of 0, we obtain

This is satisfied by y=±i, approximately; and a closer approximation is
given by

y(y2+l)'2=0 (xi),

leading to

/ ^_\ . 0*
y~ ~\ 2V2/ l 2^2 (XU)'

Hence, neglecting the small change in the 'speed' of the oscillations,
The modulus of decay is therefore

in the notation of Art. 246.

Under the circumstances to which this formula applies the elasticity of the
oil-film has the effect of practically annulling the horizontal motion at the
surface. The dissipation is therefore (within limits) independent of the
precise value of /.

The substitution of (x) for (ix) is permissible when 6 is small compared
with /o-02/cr2, or c small compared with fT'/v. Assuming i/='018, ^'=40,
we have T'/v =2200. Hence the investigation applies to waves whose
velocity is small compared with 2200 centimetres per second. It appears on
examination that this condition is fulfilled for wave-lengths ranging from a
fraction of a millimetre to several metres.

The ratio of the modulus (xiv) to the value (l/2i/&2), obtained on the
hypothesis of constant surface-tension, is 4^/2 (^2/o-)^, which is assumed to
be small. The above numerical data make Xm=l'27, cm = 20. Substituting
in (xiv) we find

r = -30X§x(cffi/c)*.

For X=Xm this gives r='43sec. instead of 1 '41 sec. as on the hypothesis
of constant tension. For larger values of X the change is greater.

304-305] PERIODIC MOTION WITH A SPHERICAL BOUNDARY. 555

When the wave- velocity c is great compared with 2200 c.s., we may
neglect o-y2/^ in comparison with 6. The result is the sams as if we were to
put /=0, so that the modulus of decay has, for sufficiently long waves, the
value l/2f£2 found in Art. 301. The same statement would apply to
sufficiently minute crispations ; but 6 then ceases to be small, and the
approximations break down ab initio. The motion, in fact, tends to become
aperiodic.

305. Problems of periodic motion in two dimensions, with a
circular boundary, can be treated with the help of Bessel's Func
tions*. The theory of the Bessel's Function, whether of the first
or second kind, with a complex argument, involves however some
points of great delicacy, which have been discussed in several
papers by Stokes f. To avoid entering on these, we pass on to
fche case of a spherical boundary ; this includes various problems
of greater interest which can be investigated with much less
difficulty, since the functions involved (the ^rn and M/"n of Art. 267)
admit of being expressed in finite forms.

It is convenient, with a view to treating all such questions on
a uniform plan, to give, first, the general solution of the system of
equations :

(V3 + A2)w' = 0, (V2 + /^' = 0, (V3 + #) w' = 0 (1),

M M M

dx dy dz

in terms of spherical harmonics. This is an extension of the
problem considered in Art. 293. We will consider only, in the
first instance, cases where u', v', wr are finite at the origin.

The solutions fall naturally into two distinct classes. If r
denote the radius vector, the typical solution of the First Class is

y (3),

* Cf. Stokes, 1. c. ante p. 518 ; Steam, Quart. Journ. Math., t. xvii. (1881) ;
and the last paper cited on p. 558.

t "On the Discontinuity of Arbitrary Constants which appear in Divergent
Developments," Camb. Trans., t. x. (1857), and t. xi. (1868).

/ - / 7 \

*-*•<*«•>

d d

556 VISCOSITY. [CHAP, xi

where ^n is a solid harmonic of positive degree n, and tyn is
defined by

d l sin

1.3...(2» + 1)

(4).

It is immediately verified, on reference to Arts. 266, 267, that the
above expressions do in fact satisfy (1) and (2). It is to be
noticed that this solution makes

xu' + yv' + zwf = 0 (5).

The typical solution of the Second Class is

u' = (n + l) ^M (hr) --p - n^n+1 (hr) h2r™+B -r

dx

J fk

(hr]

where <j>n is a solid harmonic of positive degree n. The coefficients
of ^n-i (hr) and ^rw+i (^) in these expressions are solid harmonics
of degrees n — 1 and n + 1 respectively, so that the equations (1)
are satisfied.

To verify that (2) is also satisfied we need the relations

*»'(0 = -Ww.<0 ..................... (1),

which follow easily from (4). The formulae (6) make

xu + yv' + zw = n (n -f 1) (2w + 1) ^rn (hr) $n ...... (9),

the reduction being effected by means of (7) and (8).
If we write

dw' dv' ,du' dw' dvf du'

305] PRELIMINARY ANALYSIS. 557

we find, in the solutions of the First Class,

v- !

2~ -j |V» '*y yn-iv ""/~^7 "T«41V»*/" ^ r2w+1 '

1

2rc + l

(ii);

these make

2 (^ + yV + <zO = - 7i (?i + 1) i^w (&r) %7i (12).

In the solutions of the Second Class, we have

n (fcr) (y ^ - ^) ^»,

-*4>n (13),

2^=-

and therefore

®% + 2/V + *% = 0 (14).

In the derivation of these results use has been made of (7),
and of the easily verified formula

d

_ 2n+1 _

To shew that the aggregate of the solutions of the types (3)
and (6), with all integral values of n, and all possible forms of the
harmonics </>n, %n, constitutes the complete solution of the proposed
system of equations (1) and (2), we remark in the first place that
the equations in question imply

(V8 + Aa)(a«*/ + yw/ + 2W/) = 0 ............... (16),

and (V2 + ^)Of/ + 2/77/ + O = 0 ............... (17).

It is evident from Arts. 266, 267 that the complete solution of
these, subject to the condition of finiteness at the origin, is
contained in the equations (9) and (12), above, if these be
generalized by prefixing the sign 2) of summation with respect to
n. Now when xu -f yv' + zw' and x% ' + yrf + z% ' are given through-

558 * VISCOSITY. [CHAP, xi

out any space, the values of u', v', w' are rendered by (2) completely
determinate. For if there were two sets of values, say u, v, w'
and u", v", w", both satisfying the prescribed conditions, then,
writing

we should have

anii + yvi 4- zw^ = 0,\

du^ dvl dwl_ \

~j -- ' — T -- ' -- T~ — u

dx dy dz }

Regarding u^t vl} wl as the component velocities of a liquid, the
first of these shews that the lines of flow are closed curves lying
on a system of concentric spherical surfaces. Hence the ' circula
tion' (Art. 32) in any such line has a finite value. On the
other hand, the second equation shews, by Art. 33, that the
circulation in any circuit drawn on one of the above spherical
surfaces is zero. These conclusions are irreconcileable unless
ult vl} w1 are all zero.

Hence, in the present problem, whenever the functions (f>n and
%n have been determined by (9) and (12), the values of u', v', w'
follow uniquely as in (3) and (6).

When the region contemplated is bounded internally by a
spherical surface, the condition of finiteness when r = 0 is no longer
imposed, and we have an additional system of solutions in which
the functions ^n(f) are replaced by >PW(?), in accordance with
Art. 267*.

* Advantage is here taken of an improvement introduced by Love, " The Free
and Forced Vibrations of an Elastic Spherical Shell containing a given Mass of
Liquid," Proc. Lond. Math. Soc., t. xix., p. 170 (1888).

The foregoing investigation is taken, with slight changes of notation, from the
following papers :

" On the Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., t. xiii.,
p. 51 (1881) ;

"On the Vibrations of an Elastic Sphere," Proc. Lond. Math. Soc., t. xiii.,
p. 189 (1882) ;

"On the Motion of a Viscous Fluid contained in a Spherical Vessel," Proc.
Lond. Math. Soc., t. xvi., p. 27 (1884).

305-306] SOLUTION IN SPHERICAL HARMONICS.

559

306. The equations of small motion of an incompressible
fluid are, in the absence of extraneous forces,

du 1 dp

-ji = — -7^4-
dt pdx

dv I dp — , nx

-jI = --^-+vV*Vt ..................... (1),

at pay

dw I dp

-JT = — -f +
dt pdz

. , du dv dw /cn

with 3- + j- + ^j~ = ° ..................... V2)-

dx dy dz

If we assume that u, v, w all vary as e^, the equations (1) may
be written

(v^^)

(V2^>

(V+A«)w

where h? = — \/v ........................... (4).

From (2) and (3) we deduce

V2j9 = 0 .............................. (5).

Hence a particular solution of (3) and (2) is

and therefore the general solution is

1 dp , 1 dp , I dp

U = TTJ+U> V = T^^T + V> w = T—-£
*

dz

/t_.
... (7),

where u't v', w' are determined by the conditions of the preceding

Art.

Hence the solutions in spherical harmonics, subject to the
condition of finiteness at the origin, fall into two classes.

560 VISCOSITY.

In the First Class we have

p = const.,

d d

and therefore asu + yv 4- zw — 0

In the Second Class we have

P=Pn,

and

[CHAP, xi

•(8);

.(9).

.(10),
(11),

where f, ?;, f denote the component rotations of the fluid at the
point (as, y, z). The symbols %n, <f>n, pn stand for solid harmonics
of the degrees indicated.

The component tractions on the surface of a sphere of radius r
are given by

- (xu + yv + zw),

= — xp + p

j.

^ (r T^ - 1 J

A6 T

T- +

I

In the solutions of the First Class we find without difficulty

306]

TWO TYPES OF SOLUTION.

561

71 V dx dz

(13),

where

Pn =

(Ar) + (w - 1) ^ (Ar)} ......... (14).

To obtain the corresponding formulae for the solutions of the
Second Class, we remark first that the terms in pn give

-
dr

dx

dx

1 J dx

dx

,

1 ' '

The remaining terms give

(r ^ - l) u' = (n + 1) {Ar^« (Ar) + (n - 2) ^M_1 (Ar)} ^

-_

and

...... (17).

Various reductions have here been effected by means of Art. 305
(7), (8), (15). Hence, and by symmetry, we obtain

rP>, = An |> + Bnr^ ^ ^ + Cn ^

d <l>n
dyr*n+i>

rprz=An
where

..(18),

_ 2(71-1) r2
w~ A2 271 + 1'

Cn = JJL(U + I) {hr^'n^ (hr) + 2 (n - 1) ^v-i
Dn = -

L.

...(19).

36

562 VISCOSITY. [CHAP, xi

307. The general formulas being once established, the applica
tion to special problems is easy.

1°. We may first investigate the decay of the motion of a viscous fluid
contained in a spherical vessel which is at rest.

The boundary conditions are that

u = 0t v = 0, w = 0 .............................. (i),

for r = a, the radius of the vessel. In the modes of the First Class, represented
by (8) above, these conditions are satisfied by

*»(A«) = 0 .................................... (ii).

The roots of this are all real, and the corresponding values of the modulus of
decay (r) are then given by

.............................. (iii).

The modes ?i = 1 are of a rotatory character ; the equation (ii) then
becomes

tan ha = ha .................................... (iv),

the lowest root of which is Aa = 4'493. Hence

In the case of water, we have v = -018 c. s., and

r = 2-75 a2 seconds,
if a be expressed in centimetres.

The modes of the Second Class are given by (10). The surface conditions
may be expressed by saying that the following three functions of #, y, z

' dy

must severally vanish when r = a. Now these functions as they stand satisfy
the equations

V2U = 0, V2V = 0, V2W = 0 (vi),

and since they are finite throughout the sphere, and vanish at the boundary,
they must everywhere vanish, by Art. 40. Hence, forming the equation

dv dv L dw n , ...

-j- + -7- + -r- =0 (vn),

dx dy dz

we find ^n + 1 (Aa) = 0 (viii).

307] MOTION IN A SPHERICAL VESSEL. 568

Again, since .ru+yv + zw = 0 ................................. (ix),

for r = a, we must have

J-^n-f^(n + l)(2w + l)^n(Aa)0n = 0 .................. (x),

where use has been made of Art. 305 (7). This determines the ratio pn : $n.
In the case n = 1 , the equation ( viii) becomes

, .v
(xi),

—
the lowest root of which is ha — 5*764, leading to

» r='0301-.

v

For the method of combining the various solutions so as to represent the
decay of any arbitrary initial motion we must refer to the paper cited last on
p. 558.

2°. We take next the case of a hollow spherical shell containing liquid,
and oscillating by the torsion of a suspending wire*.

The forced oscillations of the liquid will evidently be of the First Class, with
n = 1 . If the axis of z coincide with the vertical diameter of the shell, we find,
putting xi=Ck,

tt-Cf^A^y, v=-Ctyl(hr}x, w = 0 ................. (xii).

If o) denote the angular velocity of the shell, the surface-condition gives

C'Vr1(Aa)=-6) ................................. (xiii).

It appears that at any instant the particles situate on a spherical surface
of radius r concentric with the boundary are rotating together with an
angular velocity

^i(Ar) , . N

T^TT-H o> ................................. (xiv).

If we assume that co = ae(^+e) ................................. (xv),

and put h?=-i<r/v = (I-i)2P* ........................... (xvi),

where, as in Art. 297, /32 = o-/2j/ ................................. (xvii),

the expression (xiv) for the angular velocity may be separated into its real
and imaginary parts with the help of the formula

. ... sin£ cosf

If the viscosity be so small that (3a is considerable, then, keeping only the
most important term, we have, for points near the surface,

* This was first treated, in a different manner, by Helmholtz, 1. c. ante p. 513.

36—2

564 VISCOSITY. [CHAP, xi

and therefore, for the angular velocity (xiv),

the real part of which is

a^e~^(a~r) .cos{o*-j8(r-a) + e} .................. (xxi).

As in the case of laminar motion (Art. 298), this represents a system of waves
travelling inwards from the surface with rapidly diminishing amplitude.

When, on the other hand, the viscosity is very great, /3a is small, and the
formula (xiv) reduces to

o>cos(o-£ + f) ................................. (xxii),

nearly, when the imaginary part is rejected. This shews that the fluid now
moves almost bodily with the sphere.

The stress-components at the surface of the sphere are given by (13). In
the present case the formula reduce to

Pry= -p-

z

(xxiii).

If SS denote an element of the surface, these give a couple
^= - JJtePnr -yprx) <*S= CM (ha) JJ(a* +

9h*a*fa(ha) , . N

-a"-"* ^(ll) " ............... (XX1V)'

by (xiii) and Art. 305 (7).

In the case of small viscosity, where /3a is large, we find, on reference to
Art. 267, putting ha = (l-i) /3a, that

/ fJ \»e*f
2i+n(ha} = (-y(Jl^ - ..................... (xxv),

approximately, where (•=(! — i) j3a. This leads to

^Vr=-|7r/Lia3(l+*)^a« ......................... (xxvi).

If we restore the time-factor, this is equivalent to

............ (xxvii).

The first term has the effect of a slight addition to the inertia of the sphere ;
the second gives a frictional force varying as the velocity.

308. The general formulae of Arts. 305, 306 may be further
applied to discuss the effect of viscosity on the oscillations of a

307-308] OSCILLATIONS OF A VISCOUS SPHEROID. 565

mass of liquid about the spherical form. The principal result of
the investigation can, however, be obtained more simply by the
method of Art. 301.

It was shewn in Arts. 241, 242, that when viscosity is neglected, the
velocity-potential in any fundamental mode is of the form

where Sn is a surface harmonic. This gives for twice the kinetic energy
included within a sphere of radius r, the expression

p ( U(^r*dn = pna(^\2n~'1 j I SJdw . ,42cos2(^ + 6) ...... (ii),

if 8or denote an elementary solid angle, and therefore for the total kinetic
energy

..................... (iii).

The potential energy must therefore be given by the formula

V=$pnaHSn*diff.A*am*(<rt + c) ..................... (iv).

Hence the total energy is

w.A* ........................... (v).

Again, the dissipation in a sphere of radius r, calculated on the assumption
that the motion is irrotational, is, by Art. 287 (vi),

Now

J jqZd&^^j j(j>-^r*d& ..................... (vii),

each side, when multiplied by pdr being double the kinetic energy of the fluid
contained between two spheres of radii r and r + dr. Hence, from (ii),

Substituting in (vi), and putting r = a, we have, for the total dissipation,

2F=2n(n-l)(2n + l)^ f \S,?dw.A2co&(<rt + e} ...... (viii).

The mean dissipation, per unit time, is therefore

2F=n(n-l) (2w + l)^ f fsn2d&. A2 .................. (ix).

If the effect of viscosity be represented by a gradual variation of the
coefficient A, we must have

566 VISCOSITY. [CHAP, xi

whence, substituting from (v) and (ix),

~=-(n-l}(2n+l}^A ........................ (xi).

This shews that A<z e~tlr , where

1 a2

The most remarkable feature of this result is the excessively minute
extent to which the oscillations of a globe of moderate dimensions are affected
by such a degree of viscosity as is ordinarily met with in nature. For a globe
of the size of the earth, and of the same kinematic viscosity as water, we
have, on the c.G. s. system, a = 6*37 x 108, i/='0178, and the value of r for the
gravitational oscillation of longest period (n = 2) is

T = 1*44 x 1011 years.

Even with the value found by Darwin f for the viscosity of pitch near the
freezing temperature, viz. /i = l'3 x 108 x#, we find, taking # = 980, the value

r = 180 hours

for the modulus of decay of the slowest oscillation of a globe of the size of the
earth, having the density of water and the viscosity of pitch. Since this is
still large compared with the period of 1 h. 34 m. found in Art. 241 , it appears
that such a globe would oscillate almost like a perfect fluid.

The investigation by which (xii) was obtained does not involve any special
assumption as to the nature of the forces which produce the tendency to the
spherical form. The result applies, therefore, equally well to the vibrations
of a liquid globule under the surface-tension of the bounding film. The
modulus of decay of the slowest oscillation of a globule of water is, in seconds,

r=ll'2a2,
where the unit of a is the centimetre.

The formula (xii) includes of course the case of waves on a plane surface.
When n is very great we find, putting X = 2n-a/?i,

r = A2/87T2v .. ............................... (xiii),

in agreement with Art. 301.

The same method, applied to the case of a spherical bubble, gives

where v is the viscosity of the surrounding liquid. If this be water we have,
for n = 2, r = 2-8a2.

The above results all postulate that 2yrr is a considerable multiple of the
period. The opposite extreme, where the viscosity is so great that the motion

* Proc. Lond. Math. Soc., t. xiii., pp. 61, 65 (1881).

t "On the Bodily Tides of Viscous and Semi-Elastic Spheroids,...," Phil.
Trans., 1879.

308-309] MODULI OF DECAY. 567

is aperiodic, can be investigated by the method of Arts. 293, 294, the effects of
inertia being disregarded. In the case of a globe returning to the spherical
form under the influence of gravitation, it appears that

n ga
a result first given by Darwin (1. c.). Of. Art. 302 (22).

309. Problems of periodic motion of a liquid in the space
between two concentric spheres require for their treatment
additional solutions of the equations of Art. 306, in which p is of
the form p-n-i , and the functions tyn (hr) which occur in the
complementary functions u't v', w' are to be replaced by ^fn (hr).

The question is simplified, when the radius of the second sphere
is infinite, by the condition that the fluid is at rest at infinity. It
was shewn in Art. 267 that the functions tyn(£)> ^nCO are both
included in the form

d »A&+Ber* n.

In the present applications, we have f = hr, where h is defined
by Art. 306 (4), and we will suppose, for definiteness, that that
value of h is adopted which makes the real part of ih positive.
The condition of zero motion at infinity then requires that A = 0,
and we have to deal only with the function

As particular cases :

(0 = (-

The formulae of reduction for/n(f) are exactly the same as for
^n(f) and^n(f), and the general solution of the equations of
small periodic motion of a viscous liquid, for the space external to
a sphere, are therefore given at once by Art. 306 (8), (10), with
P-n-i written for pn, tmd/n(hr) for tyn(hr).

1°. The case of the rotatory oscillations of a sphere surrounded by an
infinite mass of liquid is included in the solutions of the First Class, with
n=l. As in Art. 307, 2°, we put Xl = Cz, and find

u = Cfl(hr}y, v=-Cfl(fo)z, w = Q .................. (i),

568 VISCOSITY. [CHAP, xi

with the condition Cf-^ (ha) = — o> ................................. (ii),

a being the radius, and o> the angular velocity of the sphere, which we suppose
given by the formula

Putting h = (l — i)$, where /3=(<r/2r)*, we find that the particles on a
concentric sphere of radius r are rotating together with the angular velocity

. .........

/j(Aa) r3 l + iha

where the values of /x (hr\ /: (ha) have been substituted from (3). The real
part of (iv) is

a

-|8(r- a) sin {erf- 0 (*•-«) + «}] ...... (v),

corresponding to an angular velocity

(vi)

of the sphere.

The couple on the sphere is found in the same way as in Art. 307 to be

03

l+iha
Putting ha = (\ — i) /3a, and separating the real and imaginary parts we find

This is equivalent to

,r
~

dt ~
The interpretation is similar to that of Art. 307 (xxvii) *.

2°. In the case of a ball pendulum oscillating in an infinite mass of fluid,
which we treat as incompressible, we take the origin at the mean position of
the centre, and the axis of x in the direction of the oscillation.

The conditions to be satisfied at the surface are then

w=u, 0 = 0, w — 0 .............................. (x),

for r = a (the radius), where u denotes the velocity of the sphere. It is evident
that we are concerned only with a solution of the Second Class. Again, the
formula) (10) of Art. 306, when modified as aforesaid, make

(hr)<l>n ...... (xi) ;

~_n
* Another solution of this problem is given by Kirchhofif, Mechanik, o. xxvi.

309] OSCILLATIONS OF A SPHERE. 569

and by comparison with (x), it appears that this must involve surface
harmonics of the first order only. We therefore put n = l, and assume

<$>i = Bx ........................... (xii).

Hence

w -*

A d x „- /r N 70 . d x

w-

The conditions (x) are therefore satisfied if

2f0(ha) B = n .................. (xiv).

The character of the motion, which is evidently symmetrical about the
axis of X) can be most concisely expressed by means of the stream-function
(Art. 93). From (xi) or (xiii) we find

or, substituting from (3),

If we put ,£=rcos0, this leads, in the notation, and on the convention as
to sign, of Art. 93 to

Writing u = a«i(<rt+f) ................................. (xviii),

and therefore h = (l — i) /3, where /3 = (cr/2i/)*, we find, on rejecting the imagina
part of (xvii),

cos <r^ ~ ^r " a) + e }

At a sufficient distance from the sphere, the part of the disturbance which
is expressed by the terms in the first line of this expression is predominant.
This part is irrotational, and differs only in amplitude and phase from the
motion produced by a sphere oscillating in a Motionless liquid (Arts. 91, 95).
The terms in the second line are of the type we have already met with in the
case of laminar motion (Art. 298).

To calculate the resultant force (X] on the sphere, we have recourse to
Art. 306 (18). Substituting from (xii), and rejecting all but the constant

570 VISCOSITY. [CHAP, xi

terms in prx, since the surface-harmonics of other than zero order will
disappear when integrated over the sphere, we find

z + ClBa .................. (xx),

where B_^-\a\ <7X = 2/iAa/0' (ha) ..................... (xxi),

by Art. 306 (19). Hence, by (xii) and (3),

{2/0/ (ha} ' Wa*h (ha}}

This is equivalent to

X- - 1 ^ *+ -3,P«V + u ...... (xxiii).

The first term gives the correction to the inertia of the sphere. This
amounts to the fraction

^ + 4^

of the mass of fluid displaced, instead of ^ as in the case of a frictionless
liquid (Art. 91). The second term gives a frictional force varying as the
velocity*.

310. We may next briefly notice the effect of viscosity on
waves of expansion in gases, although, for a reason to be given, the
results cannot be regarded as more than illustrative.

In the case of plane waves f in a laterally unlimited medium,
we have, if we take the axis of as in the direction of propagation,
and neglect terms of the second order in the velocity,

du _ I dp d?u m

Tt~~^Tx + *vdx* .................. !

by Art. 286 (2), (3). If s denote the condensation, the equation
of continuity is, as in Art. 255,

ds _ du

dt~ dx ......

* This problem was first solved, in a different manner, by Stokes, 1. c. ante
p. 518. For other methods of treatment see 0. E. Meyer, "Ueber die pendelnde
Bewegung einer Kugel unter dem Einflusse der inneren Keibung des umgebenden
mediums," Crelle, t. Ixxiii. (1871) ; Kirchhoff, Mechanik, c. xxvi. The variable
motion of a sphere in a liquid has been discussed by Basset, Phil. Trans., 1888;
Hydrodynamics, c. xxii.

t Discussed by Stokes, I. c. ante p. 518.

309-310] DAMPING OF AIR-WAVES. 571

and the physical equation is, if the transfer of heat be neglected,

p=p^^p,s (3),

where c is the velocity of sound in the absence of viscosity.
Eliminating p and s, we have

d*u A*u . d*u

To apply this to the case of forced waves, we may suppose that
at the plane x = 0 a given vibration

u = aei<Tt (5)

is kept up. Assuming as the solution of (4)

u = aePrt+f>x (6),

we find /3s (c2 + ^Vcr) = - cr2 (7),

whence 8=4- . ^ ... ,

c V 6 c2

If we neglect the square of i/cr/c2, and take the lower sign, this
gives

c 3 c3

Substituting in (6), and taking the real part, we get, for the waves
propagated in the direction of ^-positive

u — ae~x!l cos a

H

where l = $&/vo* ........................ (11).

The amplitude of the waves diminishes exponentially as they
proceed, the diminution being more rapid the greater the value of
<7. The wave-velocity is, to the first order of i/<7/c2, unaffected by
the friction.

The linear magnitude I measures the distance in which the
amplitude falls to l/e of its original value. If X denote the wave
length (27rc/<r), we have

it is assumed in the above calculation that this is a small ratio.

572 VISCOSITY. [CHAP, xi

In the case of air- waves we have c = 3'32 x 104, j/ = '132, C.G.S., whence

i/o-/c2 = 27ri//Xc = 2-50X-1xlO-5, £=9'56X2xl03,
if X be expressed in centimetres.

To find the decay of free waves of any prescribed wave-length
(2-7T/&), we assume

and, substituting in (4), we obtain

Gt2 + ±vk2a = -k-c2 ..................... (14).

If we neglect the square of vk/c, this gives

a = -*vk2±ikc ..................... (15).

Hence, in real form,

u = ae-t/Tcosk(a}±ct) .................. (16),

where T = 3/2z/&2 ........................ (17)*.

The estimates of the rate of damping of aerial vibrations,
which are given by calculations such as the preceding, though
doubtless of the right order of magnitude, must be actually under
the mark, since the thermal processes of conduction and radiation
will produce effects of the same kind, of comparable amount, and
ought therefore, for consistency, to be included in our calculations.
This was first pointed out distinctly by Kirchhoff, who has
investigated, in particular, the theoretical velocity of sound-waves
in a narrow tubef. This problem is important for its bearing on
the well-known experimental method of Kundt. Lord Rayleigh
has applied the same principles to explain the action of porous
bodies in absorption of sound J.

311. It remains to call attention to the chief outstanding
difficulty of our subject.

It has already been pointed out that the neglect of the terms
of the second order (adu/dx, &c.) seriously limits the application
of many of the preceding results to fluids possessed of ordinary

* For a calculation, on the same assumptions, of the effect of viscosity in
damping the vibrations of air contained within spherical and cylindrical envelopes
reference may be made to the paper ' ' On the Motion of a Viscous Fluid contained
in a Spherical Vessel," cited on p. 558.

t " Ueber den Einfluss der Warmeleitung in einem Gase auf die Schallbewegung,"
Pogfl. Ann., t. cxxxiv. (1868) ; Ges. Abh., p. 540.

J " On Porous Bodies in relation to Sound," Phil. Mag.t Sept. 1883.

310-311] TURBULENT FLOW OF A LIQUID. 573

degrees of mobility. Unless the velocities be very small, the
actual motion, in such cases, so far as it admits of being observed,
is found to be very different from that represented by our formulae.
For example, when a solid of ' easy ' shape moves through a liquid,
an irregular, eddying, motion is produced in a layer of the fluid
next to the solid, and a widening trail of eddies is left behind,
whilst the motion at a distance laterally is comparatively smooth
and uniform.

The mathematical disability above pointed out does not apply
to cases of rectilinear flow, such as have been discussed in Arts.
288, 289 ; but even here observation shews that the types of
motion there investigated, though always theoretically possible,
become under certain conditions unstable. The case of flow
through a pipe of circular section has been made the subject of
a very careful experimental study by Reynolds*, by means of
filaments of coloured fluid introduced into the stream. So long
as the mean velocity (w0) over the cross-section falls below a
certain limit depending on the radius of the pipe and the nature
of the fluid, the flow is smooth, and in accordance with Poiseuille's
laws ; but when this limit is exceeded the motion becomes wildly
irregular, and the tube appears to be filled with interlacing and
constantly varying streams, crossing and recrossing the pipe. It
was inferred by Reynolds, from considerations of dimensions, that
the aforesaid limit must be determined by the ratio of wQa to v,
where a is the radius, and v the (kinematic) viscosity. This was
verified by experiment, the critical ratio being found to be,
roughly,

Thus for a pipe one centimetre in radius the critical velocity for
water (v — '018) would be 18 cm. per sec.

Simultaneously with the change in the character of the motion,
when the critical ratio is passed, there is a change in the relation
between the pressure-gradient (dp/dz) and the mean velocity w0.
So long as w^a/v falls below the above limit, dp/dz varies as wQt as

* "An Experimental Investigation of the Circumstances which determine
whether the Motion of Water shall be Direct or Sinuous, and of the Law of Resist
ance in Parallel Channels," Phil. Trans., 1883.

t The dependence on v was tested by varying the temperature.

574 VISCOSITY. [CHAP, xi

in Poiseuille's experiments, but when the irregular mode of flow
has set in, dpjdz varies more nearly as WQ-.

The practical formula adopted by writers on Hydraulics, for
pipes whose diameter exceeds a certain limit, is

R = tfpwf ........................... (2),

where R is the tangential resistance per unit area, WQ is the mean
velocity relative to the wetted surface, and f is a numerical
constant depending on the nature of the surface. As a rough
average value for the case of water moving over a clean iron
surface, we may take f— '005*. A more complete expression for
R, taking into account the influence of the diameter, has been
given by Darcy, as the result of his very extensive observations on
the flow of water through conduits t-

The resistance, in the case of turbulent flow, is found to be
sensibly independent of the temperature, and therefore of the
viscosity of the fluid. This is what we should anticipate from
considerations of ' dimensions,' if it be assumed that R oc w*\.

If we accept the formula (2) as the expression of observed
facts, a conclusion of some interest may be at once drawn.
Taking the axis of z in the general direction of the flow, if w
denote the mean velocity (with respect to the time) at any point
of space, we have, at the surface,

dw

"X •

if w0 denote the general velocity of the stream, and S?? an element
of the normal. If we take a linear magnitude I such

w0/l = dw/dn,

then I measures the distance between two planes moving with a
relative velocity w0 in the regular ' laminar ' flow which would give
the same tangential stress. We find

(3).

* See Kankine, Applied Mechanics, Art. 638; Unwin, Encyc. Britann., Art.
" Hydromechanics."

t Recherches experimentales relatives au mouvement de Veau dans les tuyaux,
Paris, 1855. The formula is quoted by Bankine and Unwin.

J Lord Bayleigh, " On the Question of the Stability of the Flow of Fluids,"
Phil. Mag., July 1892.

311] SKIN-RESISTANCE. 575

For example, putting v = '018, ?#0 = 300 [c. s.], /='005, we obtain
I = '024 cm. * The smallness of this result suggests that in the
turbulent flow of a fluid through a pipe of not too small diameter
the value of w is nearly uniform over the section, falling rapidly
to zero within a very minute distance of the walls *K

Applied to pipes of sufficient width, the formula (2) gives

— Tra2 -/- = 2-7TCLR = 7rfpaw02,

CiZ

-l-d/=fw-l ........................ (4).

p dz a

The form of the relation which was found to hold by Reynolds, in
his experiments, was

1 dp vz-mw0m

~ ~

where wi = l723.

The increased resistance, for velocities above a certain limit,
represented by the formula (2) or (4), is no doubt due to the action
of the eddies in continually bringing fresh fluid, moving with a
considerable relative velocity, close up to the boundary, and so
increasing the distortion -rate (dw/dn) greatly beyond that which
would obtain in regular 'laminar' motion}.

The frictional or ' skin-resistance '§ experienced by a solid of
' easy ' shape moving through a liquid is to be accounted for on the
same principles. The circumstances are however more complicated
than in the case of a pipe. The friction appears to vary roughly
as the square of the velocity ; but it is different in different parts
of the wetted area, for a reason given by W. Froude, to whom the
most exact observations || on the subject are due.

* Cf. Sir W. Thomson, Phil. Mag., Sept. 1887.

t This was in fact found experimentally by Darcy, I. c. The author is indebted
for this reference to Prof. Keynolds.

J Stokes, Math, and Phys. Papers, t. i., p. 99.

§ So called by writers on naval architecture, to distinguish it from the ' wave-
resistance' referred to in Arts. 221, 228.

|| "Experiments on the Surface-friction experienced by a Plane moving through
Water," Brit. Ass. Rep., 1872, p. 118.

" The portion of the surface that goes first in the line of motion, in experiencing
resistance from the water, must in turn communicate to the water motion, in the
direction in which it is itself travelling. Consequently the portion of the water
which succeeds the first will be rubbing, not against stationary water, but against
water partially moving in its own direction, and cannot therefore experience as
much resistance from it."

576 VISCOSITY. [CHAP, xi

312. The theoretical explanation of the instability of linear
flow, under the conditions stated, and of the manner in which
eddies are maintained against viscosity, is at present obscure. We
may refer, however, to one or two attempts which have been .made
to elucidate the question.

Lord Rayleigh, in several papers*, has set himself to examine
the stability of various arrangements of vortices, such as might be
produced by viscosity. The fact that, in the disturbed motion,
viscosity is ignored does not seriously affect the physical value
of the results except perhaps in cases where these would imply
slipping at a rigid boundary.

As the method is simple, we may briefly notice the two-dimensional form
of the problem.

Let us suppose that in a slight disturbance of the steady laminar motion

u= £7, #=0, w = 0,
where U is a function of y only, we have

u=U+u', v = v', io = 0 (i).

The equation of continuity is

The dynamical equations reduce, by Art. 143, to the condition of constant
angular velocity DQDt = 0, or

3

..... ...(iv).

dy ) * dy

Hence, neglecting terms of the second order in u', v',

Contemplating now a disturbance which is periodic in respect to #, we
assume that u', v' vary as eikx+i<Tt, Hence, from (ii) and (v),

and .•(r^JH7)fihp'__fr.O ..................... (vii).

* " On the Stability or Instability of certain Fluid Motions," Proc. Lond. Math.
Soc., t. xi., p. 57 (1880), and t. xix., p. 67 (1887) ; " On the Question of the Stability
of the Flow of Fluids," Phil. Mag., July 1892.

312] DISTURBED LAMINAR MOTION. 577

Eliminating %', we find

(v+kU)(**-W\ - &'=0 (viii),

W / dy*

which is the fundamental equation.

If, for any value of y, dUjdy is discontinuous, the equation (viii) must be
replaced by

where A denotes the difference of the values of the respective quantities on
the two sides of the plane of discontinuity. This is obtained from (viii) by
integration with respect to y, the discontinuity being regarded as the limit of
an infinitely rapid variation. The formula (ix) may also be obtained as the
condition of continuity of pressure, or as the condition that there should be no
tangential slipping at the (displaced) boundary.

At a fixed boundary, we must have v' = Q.

1°. Suppose that a layer of fluid of uniform vorticity bounded by the
planes y= ± h, is interposed between two masses of fluid moving irrotationally,
the velocity being everywhere continuous. This is a variation of a problem
discussed in Art. 225.

Assuming, then, U=u for y>ht U=uy/h for h>y> -h, and U= — u for
y< -h, we notice that d2 £//<%2 -— 0, everywhere, so that (viii) reduces to

$-"-' ....................... • ............ «•

The appropriate solutions of this are :

v' = Ae-kv, for y>h-, \

v' = Be-*y + C<*v, for h>y>-k; I ..................... (xi).

v'=Dekv, for y<-h

The continuity of v' requires

With the help of these relations, the condition (ix) gives
2 (a- + /hi) Cfe** - 1 (Be-** + C<*h) = 0,
2 (o--/hl) Bekh+

Eliminating the ratio B : (7, we obtain

(xiii).

(xiv).
87

578 VISCOSITY. [CHAP, xi

For small values of kh this makes <r2 = - k*ii2, as in the case of absolute
discontinuity (Art. 225). For large values of &/*, on the other hand, <r= ±&u,
indicating stability. Hence the question as to the stability for disturbances
of wave-length X depends on the ratio X/2A. The values of the function in
{ } on the right-hand of (xiv) have been tabulated by Lord Eayleigh. It
appears that there is instability if X/2A>5, about ; and that the instability is
a maximum for X/2A = 8.

2°. In the papers referred to, Lord Rayleigh has further investigated
various cases of flow between parallel walls, with the view of throwing light
on the conditions of stability of linear motion in a pipe. The main result is
that if d2 U/dy* does not change sign, in other words, if the curve with y as
abscissa and U as ordinate is of one curvature throughout, the motion is
stable. Since, however, the disturbed motion involves slipping at the walls,
it remains doubtful how far the conclusions apply to the question at present
under consideration, in which the condition of no slipping appears to be
fundamental.

3°. The substitution of (x) for (viii), when d2U/dy2 = 0, is equivalent to
assuming that the rotation £ is the same as in the undisturbed motion ; since
on this hypothesis we have

du' dv' ., , , '

dfr-dE-1*" ................................. (XV)'

which, with (vi), leads to the equation in question.

It is to be observed, however, that when d2U/dyz=0) the equation (viii)
may be satisfied, for a particular value of y, by <r-f ££7 = 0. For example, we
may suppose that at the plane y = 0 a thin layer of (infinitely small) additional
vorticity is introduced. We then have, on the hypothesis that the fluid is
unlimited,

v/ = A**kv ................................. (xvi),

the upper or the lower sign being taken according as y is positive or negative.
The condition (ix) is then satisfied by

0 ................................. (xvii),

if A/-o.. ............. (xviii),

where U0 denotes the value of U for y=Q. Since the superposition of a
uniform velocity in the direction of x does not alter the problem, we may
suppose U0 — 0, and therefore <r = 0. The disturbed motion is steady ; in
other words, the original state of flow is (to the first order of small quantities)
neutral for a disturbance of this kind*.

Lord Kelvin has attacked directly the very difficult problem of
determining the stability of laminar motion when viscosity is taken

* Cf. Sir W. Thomson, " On a Disturbing Infinity in Lord Kayleigh's solution
for Waves in a plane Vortex Stratum," Brit. Ass. Rep., 1880.

312] REFERENCES. 579

into account*. He concludes that the linear flow of a fluid through
a pipe, or of a stream over a plane bed, is stable for infinitely
small disturbances, but that for disturbances of more than a certain
amplitude the motion becomes unstable, the limits of stability
being narrower the smaller the viscosity. A portion of the
investigation has been criticised by Lord Rayleighf ; and there
can be no question that the whole matter calls for further
elucidation^.

* "Rectilinear Motion of Viscous Fluid between two Parallel Planes," Phil.
Mag., Aug. 1887 ; " Broad River flowing down an Inclined Plane Bed," Phil. Mag.,
Sept. 1887.

t I. c. ante p. 574.

t The most recent contribution to the subject is a paper by Reynolds, " On the
Dynamical Theory of Incompressible Viscous Fluids and the Determination of
the Criterion," the full text of which has not yet been published. Proc. Roy. Soc.,
May 24, 1894.

37—2

CHAPTER XII.

EQUILIBRIUM OF ROTATING MASSES OF LIQUID.

313. THIS subject had its origin in the investigations on the
theory of the Earth's Figure which so powerfully engaged the
attention of mathematicians near the end of the last and the
beginning of the present century.

Considerations of space forbid our attempting more than a
rapid sketch, with references to the original memoirs, of the case
where the fluid is of uniform density, and the external boundary is
ellipsoidal. With this is incorporated a slight account of some
cognate investigations by Dirichlet and others, which claim notice
not only on grounds of physical interest, but also by reason of the
elegance of the analytical methods employed.

We write down, in the first place, some formulae relating to
the attraction of ellipsoids.

If the density p be expressed in ' astronomical ' measure, the
gravitation-potential (at internal points) of a uniform mass en
closed by the surface

IS

where A = {(a2 + \)(&2 + X)(c2 + \)}1 (3).

This may be written

n = 7rp(ci^ + {30y* + y^-Xo) (4),

where

•(5),

* For references see p. 534. The sign of ft has been changed from the usual
reckoning.

313] ATTRACTION OF ELLIPSOIDS. 581

fao ^T^

and x0 = a&cj ^- ........................ (6).

The potential energy of the mass is given by

..................... (7),

where the integrations extend over the volume. Substituting
from (4) we find

V= iro

(8).

This expression is negative because the zero of reckoning
corresponds to a state of infinite diffusion of the mass. If we
adopt as zero of potential energy that of the mass when collected
into a sphere of radius a, = (abc)3, we must prefix to the right-
hand of (8) the term

........................... (9).

If the ellipsoid be of revolution, the integrals reduce. If it
be of the planetary form we may put

2c ..................... (10),

and obtain* «„ = &, = (?' + 1) footr1 ?- ?2,

......... (12),

provided the zero of V correspond to the spherical form.
For an ovary ellipsoid we put

a = 6 = (g2"1^c ..................... (13),

* Most simply by writing c2 + X = (a2 - c2) tt2.

582 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII
which leads to

(I*).

(15).

The case of an infinitely long elliptic cylinder may also be
noticed. Putting c = oo in (5), we find

26 2a n .

««%-+&• ft-j+V 7o = 0 ............ (16).

The energy per unit length of the cylinder is

• Fl-.A^Wfcg^J ............... (17),

if a2 = tt6.

314. If the ellipsoid rotate in relative equilibrium about the
axis of zt with angular velocity ?i, the component accelerations
of the particle (#, T/, #) are — ?&2#, — ?iay, 0, so that the dynamical
equations reduce to

1 dp d£l 1 dp d£l 1 dp d£l

-nzx = --- £• — -*—, —n*y = --- f—~T, 0 = --- f-—~r

p ax ax p ay ay p az dz

............... (1).

Hence ^ = \ n? (a* + f) - O + const ................ (2).

The surfaces of equal pressure are therefore given by

In order that one of these may coincide with the external
surface

we must have

(5).

In the case of an ellipsoid of revolution (a = b), these con
ditions reduce to one :

313-814] MACLAURIN'S ELLIPSOID. 583

Since a?l(a? + \) is greater or less than c2/(c2 4- X), according as
a is greater or less than c, it follows from the forms of «0, 70 given
in Art. 313 (5) that the above condition can be fulfilled by a
suitable value of n for any assigned planetary ellipsoid, but not
for the ovary form. This important result is due to Maclaurin*.

If we substitute from Art. 313 (11), the condition (6) takes the
form

-(3f*+l)?cot->f-8f ............... (7).

The quantity f is connected with the excentricity e of the
meridian section by the relations

The equation (7) was discussed, under slightly different forms,
by Simpson, d'Alembert-)-, and (more fully) by Laplace J. As
f decreases from oo to 0, and e therefore increases from 0 to 1, the
right-hand side increases continually from zero to a certain maxi
mum (-2247), corresponding to e = '9299, a/c = 2*7198, and then
decreases asymptotically to zero. Hence for any assigned value of
n, such that n*/27rp < '2247, there are two ellipsoids of revolution
satisfying the conditions of relative equilibrium, the excentricity
being in one case less and in the other greater than '9299.
If n9/2irp > "2247, no ellipsoidal form is possible.

When £ is great, the right-hand side of (7) reduces to T4- 1~2 approximately.
Hence in the case of a planetary ellipsoid differing infinitely little from a
sphere we have, for the ettipticity,

< = (a-c)/a = K-2 = if^ .............................. (i).

If g denote the value of gravity at the surface of a sphere of radius a, of the
same density, we have </ = ^7rpa, whence

Putting ?*%/#= say 5 we nnd that a homogeneous liquid globe of the same
size and mass as the earth, rotating in the same period, would have an
ellipticity of TJ|T.

* I. c. ante pp. 322, 367.

+ See Todhunter, Hist, of the Theories of Attraction, etc., cc. x., xvi.

£ Mecanique Celeste, Livre 3mc, chap. iii.

584 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

The fastest rotation which admits of an ellipsoidal form of revolution,
for such a mass, has a period of 2 h. 25 m.

If m be the total mass, h its angular momentum, we have

whence we find

f

m

This gives the angular momentum of a given volume of given
fluid in terms of f, and thence in terms of the excentricity e.
It appears from the discussion of an equivalent formula by
Laplace, or from the table given below, that the right-hand
side increases continually as f decreases from oo to 0. Hence
for a given volume of given fluid there is one, and only one,
form of Maclaurin's ellipsoid having any prescribed angular mo
mentum.

The following table, giving numerical details of a series of Maclaurin's
ellipsoids, is derived from Thomson and Tait*, with some modifications intro
duced for the purpose of a more ready comparison with the corresponding
results for Jacobi's ellipsoids, obtained by Darwin (see Art. 315). The unit of
angular momentum is m^ a^.

e

a/a

c/a

7i2/27r/>

Angular
momentum

0

1-0000

1-0000

0

0

•1

1-0016

•9967

•0027

•0255

•2

1-0068

•9865

•0107

•0514

•3

1-0159

•9691

•0243

•0787

•4

1-0295

•9435

•0436

•1085

•5

1-0491

•9086

•0690

•1417

•6

1-0772

•8618

•1007

•1804

•7

1-1188

•7990

•1387

•2283

•8

1-1856

•7114

•1816

•2934

•9

1-3189

•5749

•2203

•4000

•91

1-341

•5560

•2225

•4156

•92

1-367

•5355

•2241

•4330

•93

1-396

•5131

•2247

•4525

•94

1-431

•4883

•2239

•4748

•95

1-474

•4603

•2213

•5008

•96

1-529

•4280

•2160

•5319

•97

1-602

•3895

•2063

•5692

•98

1-713

•3409

•1890

•6249

•99

1-921

•2710

•1551

•7121

1-00

GO

0

0

CC

* Natural Philosophy, Art. 772.

314-315] JACOBI'S ELLIPSOID. 585

315. To ascertain whether an ellipsoid with three unequal
axes is a possible form of relative equilibrium, we return to the
conditions (5). These are equivalent to

(a0-/30)a2&2 + 7oC2(a2-&2) = 0 ............ (10),

and

If we substitute from Art. 313, the condition (10) may be
written

2 ^-o an

" -

The first factor, equated to zero, gives Maclaurin's ellipsoids,
discussed in the preceding Art. The second factor gives

0 (13),

which may be regarded as an equation determining c in terms of
a, b. When c2 = 0, every element of the integral is positive, and
when c2 = a262/(a2 + b'2) every element is negative. Hence there is
some value of c, less than the smaller of the two semiaxes a, 6,
for which the integral vanishes.

The corresponding value of n is given by (11), which takes the
form

so that n is real. It will be observed that as before the ratio
M2/27r/D depends only on the shape of the ellipsoid, and not on its
absolute size.

The possibility of an ellipsoidal form with three unequal axes
was first asserted by Jacobi in 1834*. The equations (13) and
(14) were carefully discussed by C. O. Meyerf, who shewed that
when a, b are given there is only one value of c satisfying (13),
and that, further, rffairp has its greatest value ('1871), when
a = 6 = l'7!61c. The Jacobian ellipsoid then coincides with one
of Maclaurin's forms.

* " Ueber die Figur des Gleichgewichts," Pogg. Ann., t. xxxiii. (1834) ; see also
Liouville, " Sur la figure d'une masse fluide hoinogeue, eu dquilibre, et doue'e d'un
mouvement de rotation," Journ. de VEcole Polytechn., t. xiv., p. 290 (1834).

t "De aequilibrii formis ellipsoidicis," Crelle, t. xxiv. (1842).

586 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII
If in the second factor of (12) we put a = b, and write

we find
whence

cot-1 £=

s_n du

u>] 14
13£+3£3

(v)*.

It may readily be verified that this has only one finite root, viz. £='7171,
which makes e = *8127.

As n2/27rp diminishes from the above limit, the ratio of one
equatorial axis of Jacobi's ellipsoid to the polar axis increases,
whilst that of the other diminishes, the asymptotic form being
an infinitely long circular cylinder (a = oo , b = c).

Thomson and Tait, Art. 778'. The / of these writers is equal to our f ~1.

315]

NUMERICAL EXAMPLES.

587

The following table of numerical data for a series of Jacobi's ellipsoids
has been computed by Darwin. The subject is further illustrated by the
annexed figures. The first of these gives the meridian section of the ellipsoid
of revolution which is the starting point of the series. The remainder,
adopted from Darwin's paper*, give the principal sections of two other forms.

Axes

n*
torp

Angular
momentum

a/a

»/»

c/a

1-197

1-197

•698

•1871

•304

1-216

1-179

•698

•187

•304

1-279

1-123

•696

•186

•306

1-383

1-045

•692

•181

•313

1-601

•924

•677

•166

•341

1-899

•811

•649

•141

•392

2-346

•702

•607

•107

•481

3-136

•586

•545

•067

•644

5-04

•45

•44

•026

1-016

GO

0

0

0

• co

607

2.346

* " On Jacobi's Figure of Equilibrium for a Kotating Mass of Fluid," Proc. Hoy.
Soc., Nov. 25, 1886.

588 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

There is a similar solution for the case of an elliptic cylinder rotating
about its axis*. The result, which may be easily verified, is

2np (a + b)2'"

316. The problem of relative equilibrium, of which Maclaurin's
and Jacobi's ellipsoids are particular cases, has in recent times
engaged the attention of many able writers, to whose investi
gations we can here only refer. These are devoted either
to the determination, in detail, of special forms, such as the
annulus*^, and that of two detached masses at a greater or less
distance apart J, or, as in the case of Poincare's celebrated paper §,
to the more general study of the problem, and in particular to the
inquiry, what forms of relative equilibrium, if any, can be obtained
by infinitesimal modification of known forms such as those of
Maclaurin and Jacobi.

The leading idea of Poincare's research may be stated as
follows. With a given mass of liquid, and a given angular
velocity n of rotation, there may be one or more forms of relative
equilibrium, determined by the property that the value of V—T0
is stationary, the symbols V, T0 having the same meanings as in
Art. 195. By varying n we get one or more 'linear series' of
equilibrium forms. Now consider the coefficients of stability of
the system (Art. 196). These may, for the present purpose, be
chosen in an infinite number of ways, the only essential being
that V — TO should reduce to a sum of squares ; but, whatever
mode of reduction be adopted, the number of positive as well as of
negative coefficients is, by a theorem due to Sylvester, invariable.
Poincare proves that if, as we follow any linear series, one of the
coefficients of stability changes sign, the form in question is as it

* Matthiessen, "Neue Untersuchungen iiber frei rotirende Fliissigkeiten,"
Schriften der Univ. zu Kiel, t. vi. (1859). This paper contains a very complete list
of previous writings on the subject.

t First treated by Laplace, " Memoire sur la theorie de 1'anneau de Saturne,"
Mem. de VAcad. des Sciences, 1787 [1789]; Mecanique Celeste, Livre 3mc, c. vi.
For later investigations, with or without a central attracting body, see Matthiessen,
1. c. ; Mine. Sophie Kowalewsky, Astron. Nachrichten, t. cxi. , p. 37 (1885) ; Poincare,
I. c. infra', Basset, Amer. Journ. Math., t. xi. (1888); Dyson, I. c. ante p. 166.

t Darwin, "On Figures of Equilibrium of Rotating Masses of Fluid," Phil.
Trans., 1887; a full account of this paper is given by Basset, Hydrodynamics,
c. xvi.

§ "Sur 1'equilibre d'une masse fluide animde d'un mouvement de rotation,"
Acta Math., t. vii. (1885).

315-317] OTHER FORMS OF EQUILIBRIUM. 589

were the crossing-point with another linear series. For this
reason it is called a 'form of bifurcation.' A great part of
Poincare's investigation consists in ascertaining what members of
Maclaurin's and Jacobi's series are forms of bifurcation.

Poincare also discusses very fully the question of stability, to
which we shall briefly revert in conclusion.

317. The motion of a liquid mass under its own gravitation,
with a varying ellipsoidal surface, was first studied by Dirichlet*.
Adopting the Lagrangian method of Art. 13, he proposes as the
subject of investigation the whole class of motions in which the
displacements are linear functions of the velocities. This has
been carried further, on the same lines, by Dedekindf and
RiemannJ. More recently, it has been shewn by Greenhill§ and
others that the problem can be treated with some advantage by
the Eulerian method.

We will take first the case where the ellipsoid does not change
the directions of its axes, and the internal motion is irrotational.
This is interesting as an example of finite oscillation of a liquid
mass about the spherical form.

The expression for the velocity-potential has been given in
Art. 107 ; viz. we have

with the condition of constant volume

%M =
a b c

The pressure is then given by

* " Untersuchungen iiber ein Problem der Hydrodynamik," Gott. Abh., t. viii.
(1860) ; Crelle, t. Iviii. The paper was posthumous, and was edited and amplified
by Dedekind.

t Crelle, t. Iviii.

J "Beitrag zuden Untersuchungen iiberdieBewegung ernes fliissigen gleicharti-
gen Ellipsoides," Gott. Alh., t. ix. (1861); Math. Werke, p. 168.

§ "On the Rotation of a liquid Ellipsoid about its Mean Axis," Proc. Camb.
Phil. Soc.t t. iii. (1879); "On the general Motion of a liquid Ellipsoid under the
Gravitation of its own parts," Proc. Camb. Phil. Soc., t. iv. (1880).

590 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

by Art. 21 (4); and substituting the value of O from Art. 313
we find

-n = - \ (-** + 1 f- + ~

p \U/ U v

.................. (4).

The conditions that the pressure may be uniform over the
external surface

/yi2 rt/2 /y^

J + t + J-1 ........................ <5>>

are therefore

v

g + 2*p«,) a' = fa +2,rp/30) b* = g + 2^-py.) c* . . . . (6).

These equations, with (2), determine the variations of a, b, c.
If we multiply the three terms of (2) by the three equal magni
tudes in (6), we obtain

ail + 1)b 4- cc + 2?r/) (a0ad + fijbb + %cc) = 0 ......... (7).

If we substitute the values of «0, /30, 70 from Art. 313, this has the
integral

r°° /1\
a2 + 62 + c2 - 4>7rpabc I -r- = const ............. (8).

7 o ^

It has been already proved that the potential energy is

/*°° /7"X
F= const. - ft TrytfbW I -^ ............... (9),

and it easily follows from (1) that the kinetic energy is

+62 + c2) .................. (10).

Hence (8) is recognized as the equation of energy

T+F=const ....................... (11).

When the ellipsoid is of revolution (a = b), the equation (8),
with a2c = a3, is sufficient to determine the motion. We find

(l + |0 c2 + F= const (12).

The character of the motion depends on the total energy. If
this be less than the potential energy in the state of infinite

317] VARYING ELLIPSOIDAL SURFACE. 591

diffusion, the ellipsoid will oscillate regularly between the prolate
and oblate forms, with a period depending on the amplitude ;
whilst if the energy exceed this limit it will not oscillate, but will
tend to one or other of two extreme forms, viz. an infinite line of
matter coinciding with the axis of z, or an infinite film coincident
with the plane xy*.

If, in the case of an ellipsoid of revolution, we superpose on the
irrotational motion given by (1) a uniform rotation o> about the axis of zt the
component angular velocities (relative to fixed axes) are

d d c ...

i(,-=. — x — w?/, v = — y-\-G)X) w:=- z • \*J'

a a c

The Eulerian equations then reduce to

d . d 2 _ 1 dp dQ, ^

d d 2 1 dp do,

-y + d>x + 2-G>x-o>2y= f- — 5-,

a* a pdy dy'

c I dp do.

-z = f r i

c p dz dz '

The first two equations give, by cross-differentiation,

t+2'a = ° <•**>>

or o>a2 = G>0a02 (iv),

which is simply the expression of von Helmholtz' theorem that the ' strength '
of a vortex is constant (Art. 142). In virtue of (iii), the equations (ii) have
the integral

/n /ri. \ ^"

,(V).

Introducing the value of Q from Art. 313 (4), we find that the pressure will
be constant over the surface

provided + 2irpa0-a>2 «

In virtue of the relation (iii), and of the condition of constancy of volume

2- + -

a c

* Dirichlet, /. c. When the amplitude of oscillation is small, the period
must coincide with that obtained by putting n — 2 in the formula (10) of Art. 241.
This has been verified by Hicks, Proc. Camb. Phil. Soc., t. iv., p. 309 (1883).

592 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

this may be put in the form

c + 2 (a)2ad + o>oJa2) + 47rpa0«d -f 2irpyQcc = 0 ............ (ix),

whence 2a2 + c-2 + 2o)2«2-47rpa2c - - - r = const ..... (x).

This, again, may be identified as the equation of energy.
In terms of c as dependent variable, (x) may be written

(xi).

If the initial circumstances be favourable, the surface will oscillate regularly
between two extreme forms. Since, for a prolate ellipsoid, V increases with
e, it is evident that, whatever the initial conditions, there is a limit to the
elongation in the direction of the axis which the rotating ellipsoid can attain.
On the other hand, we may have an indefinite spreading out in the equatorial
plane *.

318. For the further study of the motion of a fluid mass
bounded by a varying ellipsoidal surface we must refer to the
paper by Riemann already cited, and to the investigations of
Brioschi'f', Lipschitzj, Greenhill§ and Basset ||. We shall here
only pursue the case where the ellipsoidal boundary is invariable
in form, but rotates about a principal axis (z)*\\.

If Uy v, w denote the velocities relative to axes #, y rotating in their own
plane with constant angular velocity n, the equations of motion are, by
Art. 199,

Du _ 9 I dp do.

-=r- - znv - nLx— f- - -T- ,

Dt p dx dx

Dv 2 1 dp do,

Dt pdy dy'

Dw 1 dp do.

Dt p dz dz

If the fluid have an angular velocity o> about lines parallel to zt the actual
velocities parallel to the instantaneous positions of the axes will be

* Dirichlet, 7. c. f Crelle, t. lix. (1861).

t Crelle, t. Ixxviii. (1874). § I.e. ante p. 589.

|| "On the Motion of a Liquid Ellipsoid under the Influence of its own
Attraction," Proc. Lond. Math. Soc., t. xvii., p. 255 (1886) ; Hydrodynamics, c. xv.

II Greenhill, "On the Rotation of a Liquid Ellipsoid about its Mean Axis,"
Proc. Carnb. Phil. Soc., t. iii. (1879).

317-318] SPECIAL CASES. 593

since the conditions are evidently satisfied by the superposition of the irrota-
tional motion which would be produced by the revolution of a rigid ellipsoidal
envelope with angular velocity n — eo on the uniform rotation o> (cf. Art. 107).
Hence

Substituting in (i), and integrating, we find

-fl + const
Hence the conditions for a free surface are

a262 262

~ " ~

= -7rpy0c2 (v).

This includes a number of interesting cases.

1°. If we put ?i=o>, we get the conditions of Jacobi's ellipsoid (Art. 315).

2°. If we put ft = 0, so that the external boundary is stationary in space,
we get

f 2<*262

Woo-

These are equivalent to

and

It is evident, on comparison with Art. 315, that c must be the least axis
of the ellipsoid, and that the value (viii) of a>2/2irp is positive.

The paths of the particles are determined by

262

-

whence x = ka cos (<rt + e), y = ^6sin(o-^ + e), ^=0 ............... (x),

-" ................................. «

and &, e are arbitrary constants.

These results are due to Dedekind*.

* I. c. ante p. 589. See also Love, "On Dedekind's Theorem,...," Phil. Mag.,
Jan. 1888.

L. 38

594 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

3°. Let o>=0, so that the motion is irrotaticmal. The conditions (v)
reduce to

These may be replaced by

and

The equation (xiii) determines c in terms of a, 6. Let us suppose that
a>6. Then the left-hand side is easily seen to be positive for c = a, and nega
tive for c = b. Hence there is some real value of c, between a and 6, for which
the condition is satisfied ; and the value of nt given by (xiv) is then real, for
the same reason as in Art. 315.

4°. In the case of an elliptic cylinder rotating about its axis, the condition

(v) takes the form

4a262

If we put n = a>, we get the case of Art. 315 (i).

If ft = 0, so that the external boundary is stationary, we have

If o) = 0, i.e. the motion is irrotational, we have

319. The small disturbances of a rotating ellipsoidal mass
have been discussed by various writers.

The simplest types of disturbance which we can consider are
those in which the surface remains ellipsoidal, with the axis of
revolution as a principal axis. In the case of Maclaurin's ellipsoid,
there are two distinct types of this character ; in one of these the
surface remains an ellipsoid of revolution, whilst in the other
the equatorial axes become unequal, one increasing and the other
decreasing, whilst the polar axis is unchanged. It was shewn by
Biemann'l' that the latter type is unstable when the eccentricity
(e) of the meridian section is greater than "9529. The periods of

* Greenhill, 1. c. ante p. 589.

t 1. c. ante p. 589. See also Basset, Hydrodynamics, Art. 367. Biemann has
further shewn that Jacobi's ellipsoid is always stable for ellipsoidal disturbances.

318-320] ORDINARY AND SECULAR STABILITY. 595

oscillation in the two types (when e < '9529) have been calculated
by Love*.

The theory of the stability and the small oscillations of
Maclaurin's ellipsoid, when the disturbance is unrestricted, has
been very fully worked out by Bryan f, by a method due to
Poincare'. It appears that when e < '9529 the equilibrium is
thoroughly stable. For sufficiently great values of e there is
of course instability for other types, in addition to the one above
referred to.

320. In the investigations here cited dissipative forces are
ignored, and the results leave undetermined the more important
question of 'secular' stability. This is discussed, with great
command of mathematical resources, by Poincare.

If we consider, for a moment, the case of a fluid covering a
rigid nucleus, and subject to dissipative forces affecting all relative
motions, there are two forms of the problem. It was shewn in
Art. 197 that if the nucleus be constrained to rotate with constant
angular velocity (n) about a fixed axis, or (what comes to the same
thing) if it be of preponderant inertia, the condition of secular
stability is that the equilibrium value of V — T0 should be station
ary, V denoting the potential energy, and T0 the kinetic energy of
the system when rotating as a whole, with the prescribed angular
velocity, in any given configuration. If, on the other hand, the
nucleus be free, the case comes under the general theory of
'gyrostatic' systems, the ignored coordinates being the six co
ordinates which determine the position of the nucleus in space.
The condition then is (Art. 235) that the equilibrium value of
V+K should be a minimum, where K is the kinetic energy of
the system moving, as rigid, in any given configuration, with the

* " On the Oscillations of a Rotating Liquid Spheroid, and the Genesis of the
Moon," Phil. Mag., March, 1889.

f " The Waves on a Rotating Liquid Spheroid of Finite Ellipticity," Phil. Trans.,
1889 ; " On the Stability of a Rotating Spheroid of Perfect Liquid," Proc. Roy. Soc.,
March 27, 1890. The case of a rotating elliptic cylinder has been discussed by
Love, Quart. Journ. Math., t. xxiii. (1888).

The stability of a rotating liquid annulus, of relatively small cross-section,
has been examined by Dyson, I. c. ante p. 166. The equilibrium is shewn to
be unstable for disturbances of a "beaded" character (in which there is a periodic
variation of the cross-section as we travel along the ring) whose wave-length exceeds
a certain limit.

596 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII

component momenta corresponding to the ignored coordinates
unaltered. The two criteria become equivalent when the disturb
ance considered does not alter the moment of inertia of the
system with respect to the axis of rotation.

The second form of the problem is from the present point of
view the more important. It includes such cases as Maclaurin's
and Jacobi's ellipsoids, provided we suppose the nucleus to be
infinitely small. As a simple application of the criterion we may
examine the secular stability of Maclaurin's ellipsoid for the
types of ellipsoidal disturbance described in Art. 319*.

Let n be the angular velocity in the state of equilibrium, and h the
angular momentum. If I denote the moment of inertia of the disturbed
system, the angular velocity, if this were to rotate, as rigid, would be h//.
Hence

and the condition of secular stability is that this expression should be a
minimum. We will suppose for definiteness that the zero of reckoning of V
corresponds to the state of infinite diffusion. Then in any other configuration
V will be negative.

In our previous notation we have

c being the axis of rotation. Since abc=a?, we may write

where /(a, 6) is a symmetric function of the two independent variables «, b.
If we consider the surface whose ordinate is / (a, b), where a, b are regarded
as rectangular coordinates of a point in a horizontal plane, the configurations
of relative equilibrium will correspond to points whose altitude is a maximum,
or a minimum, or a ' minimax,' whilst for secular stability the altitude must
be a minimum.

For a=oo, or 6 = 00, we have f (a, 6) = 0. For a=0, we have F=0, and
/(a, 6) oc 1/62, and similarly for b=Q. For a = 0, &=0, simultaneously, we have
/ (a, b) = oo . It is known that, whatever the value of h, there is always one
and only one possible form of Maclaurin's ellipsoid. Hence as we follow the
section of the above-mentioned surface by the plane of symmetry (a = 6), the
ordinate varies from QO to 0, having one and only one stationary value in the

* Poincare, I. c. For a more analytical investigation see Basset, " On the
Stability of Maclaurin's Liquid Spheroid," Proc. Camb. Phil. Soc., t. viii., p. 23
(1892).

320] STABILITY OF MACLAURIN's ELLIPSOID. 597

interval. It is easily seen from considerations of continuity that this value
must be always negative, and a minimum*. Hence the altitude at this point
of the surface is either a minimum, or a minimax. Moreover, since there
is a limit to the negative value of F, viz. when the ellipsoid becomes a
sphere, there is always at least one finite point of minimum (and negative)
altitude on the surface.

Now it appears, on reference to the tables on pp. 584, 586, that when
h<'304m^a^, there is one and only one ellipsoidal form of equilibrium,
viz. one of revolution. The preceding considerations shew that this corre
sponds to a point of minimum altitude, and is therefore secularly stable (for
symmetrical ellipsoidal disturbances).

When h > "304 m^a^, there are three points of stationary altitude, viz. one
in the plane of symmetry, corresponding to a Maclaurin's ellipsoid, and two
others symmetrically situated on opposite sides of this plane, corresponding to
the Jacobian form. It is evident from topographical considerations that the
altitude must be a minimum at the two last-named points, and a minimax at
the former. Any other arrangement would involve the existence of additional
points of stationary altitude.

The result of the investigation is that Maclaurin's ellipsoid is
secularly stable or unstable, for ellipsoidal disturbances, according
as e is less or greater than '8127, the eccentricity of the ellipsoid
of revolution which is the starting point of Jacobi's series f.

The further discussion of the stability of Maclaurin's ellipsoid,
though full of interest, would carry us too far. It appears that the
equilibrium is secularly stable for deformations of any type so long
as e falls below the above-mentioned limit. This is established by
shewing that there is no form of bifurcation (Art. 316) for any
Maclaurin's ellipsoid of smaller eccentricity.

Poincare has also examined the stability of Jacobi's ellipsoids.
He finds that these are secularly stable provided the ratio a : b
(where a is the greater of the two equatorial axes) does not
exceed a certain limit.

The secular stability of a rotating elliptic cylinder has been in
vestigated directly from the equations of motion of a viscous fluid
by

* It follows that Maclaurin's ellipsoid is always stable for a deformation such
that the surface remains an ellipsoid of revolution. Thomson and Tait, Natural
Philosophy (2nd ed.), Art. 778".

t This result was stated, without proof, by Thomson and Tait, 1. c.

t Proc. Camb. Phil. Soc., t. vi. (1888).

LIST OF AUTHORS CITED.

The numbers refer to the pages.

Airy, Sir G. B., 277, 278, 286, 292,
297, 300, 301, 356, 357, 360, 376,
377, 499, 503, 551

Aitken, J., 552

d'Alembert, J. le B., 583

Ampere, A. M., 232

Argand, A., 74

Basset, A. B., 166, 197, 221, 251, 570,

588, 592, 594, 596
Beltrami, E., 93, 99, 156
Bernoulli, D., 23, 26, 367
Bertrand, J., 201
Bessel, F. W., 305
Bjerknes, C. A., 156, 165, 207
Bobyleff, D., 112, 519
Boltzmann, L., 114, 220
Borchardt, C. W., 528, 529
Borda, J. C., 27
Boussinesq, J., 418, 420, 523
Brioschi, F., 592

Bryan, G. H., 126, 195, 595, 597
Burnside, W., 380, 388
Burton, C. V., 217
Byerly, W. E., 305

Cauchy, A., 18, 225, 380
Cayley, A., 257
Christoffel, E. B., 90
Clebsch, A., 165
Colladon and Sturm, 466
Coulomb, C. A., 541
Craig, T., 181

Darcy, H., 574, 575
Darwin, G. H., 301, 350, 353, 354,
356, 357, 367, 368, 369, 566, 586, 588

Dedekind, E., 589, 593

Delaunay, C., 502

De Morgan, A., 399

Dirichlet, P. L., 130, 589, 591, 592

Dyson, F. W., 166, 261, 588, 595

Earnshaw, S., 473
Edwardes, D., 535
Encke, J. F., 540
Euler, L., 3, 483

Fawcett, Miss, 191, 197

Ferrel, W., 360, 502

Ferrers, N. M., 117, 127, 145, 146,

155
Forsyth, A. E., 74, 103, 121, 146,

304, 305, 405, 473, 519
Fourier, J. B., 483
Frost, P., 43

Froude, E. E., 403, 404, 575
Froude, W., 24, 27, 403, 404

Gauss, C. F., 43, 74, 113, 121
Gerstner, F. J. von, 412, 414, 416
Glaisher, J. W. L., 123, 399, 487,

540

Graham, T., 29
Green, G., 50, 67, 131, 163, 292, 375,

376
Greenhill, A. G., 89, 90, 97, 98, 184,

189, 248, 388, 390, 426, 434, 435,

522, 589, 592, 594
Grobli, W., 261
Guthrie, F., 207

Hadley, G., 322

Hamilton, Sir W. E., 198, 201

LIST OF AUTHORS CITED.

599

Hankel, H., 37

Hanlon, G. 0., 27

Hansen, P. A., 305

Hayward, R. B., 171, 201

Heine, E., 117, 122, 124, 145, 305, 486

Helmholtz, H. von, 24, 59, 84, 102,
103, 214, 222, 223, 224, 226, 228,
231, 235, 300, 390, 409, 421, 423,
503, 513, 537, 538, 563

Herman, R. A., 143

Hicks, W. M., 97, 143, 166, 261, 591

Hill, M. J. M., 253, 264

Hugoniot, 28

Jacobi, C. G. J., 157, 201, 585

Kelland, P., 273, 432

Kelvin, Lord, 8, 35, 37, 42, 51, 60, 61,
145, 157, 169, 170, 176, 191, 196,
207, 211, 216, 218, 220, 221, 222,
223, 224, 227, 231, 250, 260, 261,
331, 334, 335, 348, 350, 360, 361,
362, 368, 380, 383, 391, 403, 407,
409, 436, 438, 445, 446, 455, 502,
551, 575, 578, 579

Kirchhoff, G., 42, 44, 59, 90, 102,
106, 109, 110, 114, 115, 167, 168,
176, 177, 184, 191, 204, 220, 226, 249,
251, 426, 428, 490, 525, 534, 568, 570

Koch, S., 513

Korteweg, D. J., 537

Kowalewski, Mme Sophie, 588

Lagrange, J. L., 3, 9, 18, 71, 76, 201,

227, 263, 274
Lame, G., 157, 159
Laplace, P. S., 120, 319, 343, 346,

354, 356, 360, 361, 362, 363, 364,

368, 466, 495, 583, 584, 588
Larmor, J., 182, 210, 221, 262
Legendre, A. M., 123
Lewis, T. C., 262
Liouville, J., 481, 585
Lipschitz, R., 592
Lodge, A., 336
Lommel, E., 305
Love, A. E. H., 253, 261, 388, 558,

593, 595

McCowan, J., 277, 301, 421
Macdonald, H. M., 434, 435

Maclaurin, C., 322, 583

Matthiessen, L., 588

Maxwell, J. C., 27, 36, 37, 43, 118, 156,

173, 220, 231, 234, 241, 257, 316,

442, 448, 513, 516, 541
Mehler, F. G., 321
Meissel, E., 305
Meyer, C. 0., 585
Meyer, 0. E., 570
Michell, J. H., 102, 411
Murphy, R., 321

Nanson, E. J., 226, 227
Navier, C. L. M. H., 515
Neumann, C., 73, 143, 211
Newton, Sir L, 466, 524
Niven, W. D., 155

Oberbeck, A., 529, 534
Ostrogradsky, M. A., 424

Piotrowski, G. von, 522

Poincare, H., 327, 588, 595, 596

Poiseuille, J. L. M., 521

Poisson, S. D., 310, 380, 385, 475,

481, 482, 487, 490, 515
Purser, F., 221

Rankine, W. J. M., 30, 71, 91, 138, 412,
416, 475, 574

Rayleigh, Lord, 43, 89, 106, 109, 110,
111, 125, 266, 271, 279, 293, 303,
305, 309, 310, 311, 314, 320, 321,
382, 383, 384, 388, 391, 392, 393,
402, 410, 412, 418, 427, 428, 442,
445, 448, 455, 456, 458, 460, 461,
467, 471, 476, 483, 488, 489, 4'JO,
494, 496, 504, 526, 533, 536, 537,
540, 551, 572, 574, 576, 579

Reusch, E., 261

Eeynolds, 0., 28, 261, 382, 513, 552,
573, 579

Riemann, B., 59, 95, 471, 589, 594

Routh, E. J., 214, 215, 253, 266, 318,
326, 496, 507

Russell, J. Scott, 418, 455, 456, 551

St Venant, B. de, 28
Sampson, R. A., 134, 536
Schlomilch, 0., 399
Schwarz, H. A., 90

600

LIST 0V AUTHORS CITED.

Schwerd, 483

Simpson, T., 583

Steam, H. T., 555

Stefan, J., 135

Stokes, Sir G. G., 18, 33, 37, 97, 130,
131, 132, 134, 138, 142, 143, 226,
227, 228, 232, 264, 277, 307, 382,
386, 388, 409, 411, 412, 415, 421,
430, 432, 467, 475, 476, 480, 489,
491, 514, 515, 518, 524, 526, 531,
539, 545, 551, 555, 570, 575

Sylvester, J., 119, 588

Tait, P. G., 261

Tarleton, K. A., 247

Thomson, J., 27

Thomson, J. J., 90, 239, 242, 243, 261

Thomson, Sir W., see Lord Kelvin

Thomson and Tait, 37, 50, 51, 97,
117, 118, 119, 121, 167, 173, 178,
184, 197, 201, 205, 214, 216, 266,
321, 322, 324, 327, 365, 367, 496,
503, 529, 534, 584, 586, 597

Todhunter, I., 117, 146, 305

Topler, A., 114

Turner, H. H., 367

Unwin, W. C., 574
Vince, S., Ill

Webb, E. R, 388, 461
Weber, H., 16, 17
Whetham, W. C. D., 522

Young, T., 270, 286, 289

INDEX.

The numbers refer to the pages.

Air- waves, 464

effect of viscosity on, 570
Aperture,

flow through rectilinear, 82
circular, 152
elliptic, 160
Atmospheric oscillations, 491, 494

Basin, tidal oscillations in a circular,

304, 312

tides in a rotating, 335, 341
Bessel's functions, 305, 306, 310

connexion with spherical harmonics,

321
Borda's mouthpiece, 27, 103

Canal, 'long' waves in uniform, 271
Canal of variable section, 291, 294
Canal-theory of the tides, 286
Canal of triangular section, standing

waves in, 426, 429, 432
Canals, general theory of waves in, 429
Capillarity, 442
Capillary waves, 443
Circular sheet of water, waves on, 304,

312, 426

'Circulation' denned, 35
Circulation-theorem, 38, 57
Coaxal circles, 80
Complex variable, 74
Confocal conies, 82

quadrics, 158
Conjugate functions, 73
Curved stratum of fluid, motion of,

114, 253

Curvilinear coordinates, 156, 158
Cylinder,

motion of a circular, 85

with cyclosis, 88

Cylinder, elliptic, translation, 92
rotation, 95, 98
rotating, in viscous fluid, 523
Cylindrical obstacle, flow past a, 87

Discontinuity,

instability of surfaces of, 390, 391
impossible in a viscous fluid, 541
Discontinuous motions, 100
Disk, motion of a circular, 152
Dissipation-function, 504, 518
Dissipation of energy by viscosity, 517, 536
Dissipative forces, general theory of,
496, 503

Eddies, 573
Efflux of gases, 28
Efflux of liquids, 26,

through capillary tubes,

520, 522

'Elasticity of volume,' 461
Electro-magnetic analogy (vortex-mo
tion), 231

Ellipsoid, Jacobi's, 585
Maclaurin's, 583
translation of an, 147, 149, 152, 154,

162

in viscous fluid, 534
rotation of an, 149, 154, 164
Ellipsoidal Harmonics, 145, 155
Ellipsoidal mass of liquid, rotation,

of, 582, 585

Ellipsoidal shell, motion of fluid in, 155
Elliptic aperture, flow through, 160
Energy, dissipation of, 517, 536
equation of, 10
of air-waves, 469

of irrotationally moving liquid, 50,
52, 62, 73, 134

39

602

INDEX.

Energy, of 'long' waves, 278

of solid moving through liquid,

130, 172

of surface waves, 378
of vortex-systems, 239, 242
superficial, 442

Equation of continuity, 5, 6, 15

Equations of motion,

of Motionless fluid, 3, 14
of a solid in a liquid, 171, 176, 192
of a viscous fluid, 514, 516
relation to moving axes, 322

Equilibrium of rotating masses of
liquid, 580, 588

Expansion, waves of, 464

Fish-line problem, 455

Flapping of sails and flags, 392

'Flow' defined, 35, 39

Flow of a viscous fluid through a

crevice, 519
Flow of a viscous fluid through a pipe,

520, 522

'Flux' defined, 41
Forced oscillations, 269, 329
Fourier's theorem, 380, 395

Generalized coordinates, 197

Gerstner's waves, 412, 416

Globe, oscillations of a liquid, 436

tides on a rotating, 343
Globule, vibrations of a, 461
Green's theorem, 50,

Lord Kelvin's extension of, 60
Group-velocity (of waves), 381, 401,

445, 448
Gyrostatic system, 211, 503

Harmonics, spherical, 117, 316

conjugate property
of, 127

ellipsoidal, 145, 155

zonal, 121, 122

sectorial, 126

tesseral, 125

Helicoid, motion of a, 191
Hydrokinetic symmetry, 181
Hypergeometric series, 121

Ignoration of coordinates, 214
Image of a double-source, 138, 262

'Impulse' defined, 169
theory of, 169, 173, 194
in vortex-motion, 237, 248

Impulsive motion, 12

Inertia of a solid, effect of fluid in
modifying, 130, 172, 192

Instability of linear flow in a pipe, 573

' Irrotational motion ' defined, 38

Jets, theory of, in two dimensions,

103, 105
capillary phenomena of, 457, 459

'Kinematic' coefficient of viscosity, 513
Kinetic energy of a solid in a liquid, 173
Kinetic stability, 327

Lagrange's equations, 201, 207
Lagrange's (velocity-potential) theorem,

18, 38
Lamina, impact of a stream on, 94,

107, 109, 112
'Laminar' motion defined, 34

in viscous fluid, 538,

541, 545

Laplace's tidal problem, 345
Limiting velocity, 24
Lines of motion, see stream-lines.

Minimum energy, 51, 63
' Modulus of decay,' 497

of water-waves, 545

of sound-waves, 572
Multiple-connectivity, 53, 57

'Normal modes' of oscillation, 268

Oil, effect of a film of, on water-
waves, 552

Orbits of particles (in wave-motion),
373, 376

'Ordinary' stability, 327

Orthogonal coordinates, 156

Oscillations, see Small oscillations, and
Waves.

Oscillatingplane,inviscousfluid,538,541

Pendulum, oscillating in air, 490

oscillating in viscous fluid, 568
Periodic motion of a viscous fluid,
538, 555, 559

INDEX.

603

' Periphractic ' regions, 43, 70

Pipe, flow of viscous fluid in, 520, 522

Poiseuille's experiments, 521

Pressure-equation, 21

Pressure, resultant, 177

Pressures on solids in moving fluid, 218

Progressive waves, 374

Eeflection of waves, 280

Eesistance of a lamina, 107, 109, 112

(viscous) to moving sphere, 533
Retardation and acceleration of tides

(frictional), 501, 502
Eevolution, motion of a solid of, in

frictionless fluid, 184, 189, 196
Eipples and waves, 447
Eotating liquid, 29

Eotating sheet of water, tides on, 322
' Botationar motion, 222
Eotation, electromagnetic, 32
Eotation of a liquid mass under its own
attraction, 580

Sector, rotation of a circular, 97

' Secular stability,' 327, 595

Ship-waves, 403

'Simple-source' of sound, 484

'Simply-connected' regions, 40

Skin-resistance, 575

Slipping, resistance to, at the surface

of a solid, 514, 521
Small oscillations, general theory, 266

relative to rotating solid, 324
Smoke-rings, 261

Solid, motion of, through a liquid, 167
' Solitary ' wave, 418
Sound, velocity of, 466
Sound-waves, plane, 464, 468
energy of, 469
general equation of, 480
of finite amplitude, 470, 471, 473, 475
spherical, 477, 479
'Sources' and 'sinks,' 63, 118, 128,

129, 135

Source, Simple-, of sound, 484
'Speed' defined, 268
Sphere, motion of, in infinite mass of

liquid, 130
in liquid bounded by
concentric spheri
cal envelope, 132

Sphere, motion of, in cyclic region, 143,

217
in viscous fluid, 524,

530, 533
Spheres, motion of two, in a liquid,

139, 205, 206

Spherical harmonics, see Harmonics.
Spherical vortex, 264
Spherical mass of liquid, gravitational

oscillations, 436
Spherical sheet of water, waves and

tides on, 314, 440
of air, vibrations of, 489
Stability of a cylindrical vortex, 250
of a jet, 457, 459
of rotating masses of liquid, 326,

594
of a water-surface, with wind, 389,

392, 449
of steady motion of a solid in a

liquid, 178, 188, 189
of the ocean, 362
'ordinary' and ' secular,' 327
Standing waves, 372, 424
Stationary waves on the surface of a

current, 421

' Steady motion,' defined, 22
general conditions for, 262
of a solid in a liquid, 178
of a solid of revolution, 190, 196
of a viscous fluid, 519, 528, 536
Stokes' theorem, 37
Stream-function, Lagrange's, 70

Stokes', 133, 255
Stream-lines, 20

of a circular disk, 153
of a circular vortex, 258
of a cylinder, 86

with circulation, 88
of a liquid flowing past an oblique

lamina, 94

of an elliptic cylinder, 93, 99
of a sphere, 137
Stream-lines, of a sphere in viscous

fluid, 532

Stream-lines, of a vortex-pair, 80, 246
Stresses in a viscous fluid, 512
Superficial energy, 442
Surface-conditions, 8, 9, 371, 514
Surface - distributions of sources and
sinks, 65, 66, 235

604

INDEX.

Surface disturbance of a stream, 393,

395, 401, 404, 450, 451, 455
Surfaces of discontinuity, 100

instability of, 389, 391
Surface-tension, 442
Surface-waves, 370
Symmetry, hydrokinetic, 181

Tangential stress, 1, 509

Tension, surface-, 442

Terminal velocity of sphere in viscous
fluid, 533

Tidal waves, 266
friction, 499

Tide-generating forces, 364

Tides, canal theory of, 286, 287, 289, 290
change of phase by friction, 501
equilibrium theory of, 365
of second order, 300
on open sheets of water, 301, 303,

304, 312, 314, 320
on rotating sheet of water, 331, 341
on rotating globe, 343, 348, 355, 356
retardation of spring-, 502

Torsional oscillations of a sphere, effect
of viscosity on, 563, 567

Trochoidal waves, 411, 414

Tube, flow of viscous liquid through
a, 520

Turbulent flow of a liquid, 573

Velocity-potential, 18, 40, 56

due to a vortex, 233
Vena contracta, 27, 106
Vibrations of a cylindrical jet, 457, 459
of a spherical globule, 461
of gas in spherical envelope, 483,

487, 488
Viscosity, 508

coefficients of, 512, 513
effect of, on sound-waves, 570

on vibrations of a liquid

globe, 564

on water-waves, 544, 545
Viscous fluid, problems of steady mo
tion, 519

problems of oscillatory motion, 538,
559

Viscous fluid, motion of, when inertia

can be neglected, 526, 528
Vortex, elliptic, 251

spherical, 264
Vortex-filament, 223
Vortex-line, 222
Vortex-pair, 246
Vortex-ring, 257
Vortex-sheet, 234
Vortices, in curved stratum of fluid, 253

persistence of in frictionless fluid,
224

rectilinear, 243

circular, 254

Wave-resistance, 383

Waves, effect of oil on, 552

Waves due to gravity and cohesion,

combined, 445
Waves due to inequalities in the bed

of a stream, 407
Waves, capillary, 443
Waves in uniform canal, general theory

of, 429

(triangular section), 426, 429, 432
Waves, effect of viscosity on, 544, 545
Waves, 'long,' 271, 277, 282

in canal of variable section, 291, 294
of finite amplitude, 297
of expansion, 464

Waves of permanent type, 409, 418, 421
Waves on open sheets of water, 301,

304, 311, 312

on a spherical sheet, 314, 319
on the common surface of two fluids,

385, 446

on the common surface of two cur
rents, 388, 391, 448
Waves, Gerstner's, 412, 416
Waves produced by surface-disturbance
of a stream, 393, 395, 401, 404,
450, 451, 455
Waves, Ship-, 403
Wave, 'solitary,' 418
Wind, effect of, on stability of a water-
surface, 389, 449
operation of, in generating waves, 551

CAMBRIDGE: PRINTED BY j. & c. r. CLAY, AT THE UNIVERSITY PRESS.

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