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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| v.- • „ -N V^ V THE THEOKY OF SOUND. THE THEOEY OF SOUND BY JOHN WILLIAM STRUTT, BARON RAYLEIGH, Sc.D., F.R.S. HONORARY FELLOW OF TRINITT COLLEGE, CAMBRIDGE. IN TWO VOLUMES VOLUME li. ^ SECOND EDITION REVISED AND ENLARGED MACMILLAN AND CO., Ltd. " . . / NEW YORK: MACMILLAN & CO. : ..."* -. 1896 , : .: •:•.;■. [All Rights reserved,] TO KEV/ YORK PUDLi: l:3:iary 234(>9SA ASTOR, LCNOX AND TILDEN FOUNDATIONS n 1926 L Fint Edition prinUd 1878. Second EdiHon revUed and enlarged 1896. gaxbrzdob: pbintbo bt j. amd o. r. clat, AT TBB UMITXBSITT FBB88. THE appearance of this second and concluding vohiine has been delayed by pressure of other work that could not well be postponed. Ah in Vol. i. the additions down to § 348 are indicated by square brackets, or by letters following the number of the section. From that point onwards the matter is new with the exception of § 381, which appeared in the first edition as § 348. The additions to Chapter xis. deal with aerial vibrations in ■litTow tubes where the influence of viscosity and heat conduction .r\- important, and with certain phenomena of the second order ii.jwndent upon viscosity. Chapter xx. is devoted to capillary ibrations. and the explanation thereby of many beautiful obser- Ations due to Savart and other physicists. The sensitiveness of dam€s and smoke jets, a very interesting department of acoustics, is considered in Chapter sxi., and an attempt is made to lay the foundations of a theoretical treatment by the solution of problems respecting the stability, or otherwise, of stratified fluid motion. g 371, 372 deal with "bird-calls," investigated by Sondhauss, aud with aeolian tones. In Chapter xxii. a slight sketch is given of ft theory of the vibrations of elastic solids, especially as regards I propagation of plane waves, and the disturbance due to a ; force operative at one point of an infinite solid. The wrtant problems of the vibrations of plates, cylinders aud teres, are perhaps best dealt with in works dtvoted specially to t.h'.' theory of elasticity. The concluding chapter on the facts and theories of audition cwiiid not well have been omitted, but it has entailed labour out of PREFACE. proportion to the results. A large part of our knowledge upon this subject is due to Helmholtz, but most- of the workers who have since published their researches entertain divergent views, in some cases, it would seem, without recognizing how fundamental their objections really ai-e. And on several points the observations recorded by well qualified observers are so discrepant that no satis- factory conclusion can be drawn at the present time. The future may possibly shew that the differences are more nominal than real. In any case I would desire to impress upon the student of this part of our subject the importance of studying Helmholtz's views at lirst hand. lu such a book as the present an imperfect outlic of them is all that can be attempted. Only one thoroughly^ familiar with the TonffinpfiTidwng&i is in a position to appreciate^ many of the observations and criticisms of subsequent writers. Tbbliku Place, With am. Fd»-uar^, 1896 CONTENTS. CHAPTER XI. IgSSe— 254 itatl vibrationi. Equality o( pressure in all directious. Eijuations of motiDn. Equation of oontinuity. Special form for mcompresaible fluid. Uotion in tno dimengioDB. Stremn funalion. Symmetry about an aiis. Velocitj-potential. Logiange's theorem. Btokes' proof. FhyaiMl in- terpretation. Thomgon'B ioTeatigalion. Circulation. Equation of oon- tinuitj in terms of Teloaity-potential, Expression in polar co-ordiuatea. Mution of incompresHible fluid in aimplj' connected B|>acea is determined bj boundary oonditi on B. ExtenBioDtoraultiplyooanectedBpaces. Sphere of irrotntionaUy moving fluid suddenly solidifled irould have no rotation. IiTQiational motion has the least possible energy. Analogy with theories of heat and electricity. Equation of pressure. General equation for sonorous motion. Motion in one dimension. Positive and negative pro- gressiTe waves. Belation between velocity and condensation. Har- monic type. Energy propagated. Half the energy is potential, and half kinetio. Newton's calcnlRtion of velocity of soimd. Laplace'n oor- Kclioo. EipreBsiou of velocity in terms of ratio of speciflo heats. Eiarimenl of Clement and Desormes. Bankine's calcolatiou From !•'■ equivalent. Possible effect of radiation. Stoites' investigation. a& MiSiog of tbe sound. It appears that communication of heat has •ensiblB effect in pcaotice- Velocity dependent upon temperature. ' Variattoo of pitch of organ-pipes. Telocity of sound in water. Exact differential equation for plane waves. Application to waves of theory , ot sleadj motion. Only on one supposition as to [he law connecting T and density can a wave maintain its form without the aBsist- r an impressed force. Explanation of change of type. Poisson's lation. Belation between velocity and condensation in a progressive e of finite amplitude. Diftionlty of ultimate discontinuity. Earn- hlir^ integrals. Kifmann's investigation. Limited initial dlstnrbaoce. '. second order. Repulsion of resonators. Kotatory foroe n a mtpended disc due to vibrations. Btriatious in Kundt's tubes. iBlg*! theory'-] E^iperitnental determinations of tbe velocity of sound. CXJNTENT8. CHAPTER Xn. PAGE §§ 255—266 49 Vibrations in tubes. General form for simple harmonio type. Nodes and loops. Ck>ndition for an open end. In stationary vibrations there must be nodes at intervals of ^X. Beflection of pulses at closed and open ends. Problem in compound vibrations. Vibration in a tube due to external sources. Both ends open. Progressive wave due to disturbance at open end. Motion originating in the tube itsell Forced vibration of piston. Kundt's experiments. Summary of results. Vibrations of the column of air in an organ-pipe. Relation of length of wave to length of pipe. Overtones. Frequency of an organ-pipe depends upon the gas. Com- parison of velocities of sound in various gases. Examination of vibrating column of air by membrane and sand. By Konig*s flames. Curved pipes. Branched pipes. Conditions to be satisfied at the junctions of connected pipes. Variable section. Approximate calcula- tion of pitch for pipes of variable section. Influence of variation of section on progressive waves. Variation of density. CHAPTER XIII. §§267— 272 a 6£ Aerial vibrations in a rectangular chamber. Cubical box. Resonance of rooms. Rectangular tube. Composition of two equal trains of waves. Reflection by a rigid plane wall. [Nodes and loops.] Green's investiga- tion of reflection and refraction of plane waves at a plane sur&ce. Law of sines. Case of air and water. Both media gaseous. Fresnel's ex- pression. Reflection at surface of air and hydrogen. Reflection from warm air. Tyndall's experiments. Total reflection. Reflection from a plate of flnite thickness. [Reflection from a corrugated surface. Case where the second medium is impenetrable.] CHAPTER XIV. §§273—295 9\ Arbitrary initial disturbance in an unlimited atmosphere. Poisson's solu- tion. Verification. Limited initial disturbance. Case of two dimen- sions. Deduction of solution for a disturbance continually renewed. Sources of sound. Harmonic type. Verification of solution. Sources distributed over a surface. Infinite plane wall. Sheet of double souroee. Waves in three dimensions, symmetrical about a point. Har- monic type. A condensed or rarefied wave cannot exist alone. Con- tinuity through pole. Initial ciroumstauoes. Velocity-potential of a given source. Calculation of energy emitted. Speaking trumpet. Theory of conical tubes. Position of nodes. Composition of vibrations from two simple sources of like pitoh, Inttrlteanot of aounds firom etoetEioallj ">>»»*^i"*xi toning tnkM. Polnta of aitenoa. Existence ottea to be inferred froni consideraCioaa of BTmmetrj'. Case of bell. EiperimeDtal raethoda. Mayer's eiperiment, Soudi] shadows. Aperture iD plane acieeu. FresDel's zanee. Oeueral explaoatioQ of ghadovs. Oblique scree n. Condilioim of approiimately ooinplete reQectton. Diverging Waves. Variation of intensity. Foci. ReBection from cnrred finrfaces. Elliptical and parabolic refleotars. Fennat'a prin- ciple. Whimpering gallericB. Observations in Ht Paul's cathedral. Probable eiplanation. Iteaonauoe in buildinKE. Atmospheric refrac- tion of aonnd. Convective equilibrium of temperature. Differential equation to path of cay. Befraclion of sound by wind. Stokea' explanation. Law of refraction. Total reflection from wind overhead. la the caoe of refraction by wind the course of a souod ray is not reversible. Obeervations by Reynolds. TyndaU's observations on fog signals. Law of divergence of aoond. Speaking truijipeL Diffraction of BOand through a small aperture in an infinite screen. [Experiments oD difiraotioc. Circtilar grating. Shadow of circular diec.] Extension of Oreen'a theorem to velocity •potentials, Helmholtz's theorem of reci- ocity. Application to double sources. Variation of total energy within B closed space. CHAPTER XV. |S96— 302 149 Ridaiy naves due to a variation io the medium. Belative importance of I woondacy waves depends upon the wave-lengtb. A region of altered I eotnpreewhility acts like a simple source, a region of altered density like a doable source. Law of inverse fourth powers interred by method of I dimensions. Explanation of harmonic echos. Alteration of character I of ooropound sound. Secondary sooroes dae to eicesaive amplitude. ) Alteration of pitch by relative motion of source and recipient. Eiperi- il illustrations of Doppler's principle. Motion of a simple source. Vibntions in a rectangalar chamber due to internal sources. Simple •ooree situated in an unlimited tube, Energy emitted, Comparison with conical tube. Further discussion of the motion. Calculation of n of the air on a vibrating uircalar plate, whose plane is com- I pleted by a Sied flange. Equation of motion for the plate. Case of xiiDOiiteiice of natural and forced periods. CHAPTER XVI. -322 A of resoaalors. Besonator composed of a piston and air ■ Botontial energy of compression. Periodic time. In a large cl w the oompression is sensibly nniform tbronghout the [ a4>d tfat kinetic energy is sensibly contiued to the ncighboiirhood of the ^ea. Expression of kinetic energy of motion through passage* IS of clectriEBl oouductivily. Calculation of natural pitch. Case P ot san-ral chaniivls, Superior and ioferior limits to conductivity of CONTENTS, ohumeU. Simple apeitnreB. Elliptic aperture, CorapRriaon with oular aperture of equal area. In taan; caites a calculation based on itrea only ia Bnfficlent. Snperior and infcriar limitB to the oondnctiTitj neolia. Correction to leORtb ol pasHage on aoconnt of open end. Con- dactiTit;ot passagen bonuded by nearly eylindriull surlaoee of revolutioA. Comparison of calculated and observed pitah. Multiple reooT Calculation of periode for double reBonator. CominuniCHtioD of energy to external atmosphere. Rate of dissipation. Numericat eiampli Forced Tibralions dne to an eitcmsl source. Eelmholtz's theory c open pipes. Correction to leoglb. Bale of dissipation. InBuence c Hange. EipeiimeDtsI methods of determining the piteb of resonators. Discnseion of motion origicatiiig ntthin an open pipe. Motion doe to external sources. Effect of enlargement at a cloned end. Absorpti soond by resonators. Quincke's tubes. Operation oF a leeouator close to a Bonroe of aouud. Roioforcement o( sound by resonators, td^ resonator. Operation of a resonator close to a double source. Savart's experiment. Two or more resonatora. Question of fornialion of jets during sonorous motion. [Free vibrations initiated. Influence of wind upon pitch of organ-pipes. Maintaining power of wind. Overtones. Mutual influence of neighbouring organ.pipes. Whistling. Maiatenonoe of vibrationa by heat. Trevelyan's rocker. Conununication of heat and Beriol vibrations. Singintt Qtmes. SoDdbaasB' ubseifgtioni. Sounds discovered by Bijke and Bossoha. HelmhoUx's theoi7 of reed "■1 CHAPTER XVII. §§323—333 Applications of Laplace's funotiona Co acoustical problems. Qenerot solution involving! the term of the n"' order. Expression for radial velocity. Di- vergent waves. Origin at a spherical surface. Tlie formation of sonorous waves requires in general a certain area of moving surface ; otherwise the mechanical conditions are satisfied by a local transference of air nithoal appreciable condensaliau or rarefaction. Stokes' discnssion of the effect of Interal motion. Leslie's experiment. Calcutatioo of onmerioal results. The term of zero order is usually deficient when the sound originates in the vibration of a solid body. Beaotion of the BurroundiiiR air on a rigid vibrating sphere. Increase of eBectivo inertia. When the sphere is smaU in comparison with the wave-length, there is but little commu- nication of energy. Vibration of an ellipsoid. Multiple noUTCes. In cases of symmetry Laplace's functions reduce to Legendre's functions. (Table of zonal harmonics.] CalculatioD of the energy emitted from a vibrating spherical surface. Case when the disturbance is limiled to a Btnall part of the spherical surface. Numerical results. Effect of a ■mall sphere situated close to a source of sound. A.nalytiaal trans- formatiODS. Case of continuity through pole. Analytical expressions tor the velocity. potential. Expreasiuu in tcruia of Bcseel's functions of fraationftl order. Particnlar cases. Vibrations of gas conQned within a rigid spherical envelope. Badial vibrations. Diametral vibrations. ~ IS expressed by a Laplace's functiuu of the second order. Trt CONTENTS. of nave-lengths. Belative pitch of varioaa tones. Oenerol motion ex. preuibla b; Bimple vibrations. Case of nniform Initial velooity. Vibra- tiooB of goB inclnileJ between coDoentrio apheriaal Biirfa«eB. Spheric&l ■heet of gw. Investigation of the diatuibanoe produced when plane «■*!■ of wund itnpiiiga upon a epberioal obatacle. Eipanaion of the vtlocitf'potential of plane vavea. Sphere fixed and rigid. Inlennity ct Mcondaty waires, Friniar]> waveg originatini; in a Boiiroe at a finite distanoe. Sfinnietrical expression for Recondary navea. Cbbb or a gMeoiu obBtaole. Equal oompressibilitiea. CHAPTER XVIII. §g 336—343 285 ^^bvblem of a ^^H Fourier'* ^^P pTMwd ii J^K Condition I' to Legem: I of a spherical layer of air. Expansion of velocity- potential in Fourier'* aeries. Diflarantial equation satiafiod by eiich term, Ex- pre«aed in ternu of ^ and of >. Solution for the case of symmetry. BatisGed when the poles are not sources. Heductioa Legendre's tunotions. Conjugate property. Transition from aphe- neal to plane layer. Bessel's tunotion or zero order. Spherical layer bomided by parallels of latitude. Solution for spherical layer bounded by small cirole. Particular cases soluble by Legendre'a func- UoM. General problem for on symmetrical motion. Trsoaitioa to two dimensions. Complete solution for entire sphere in terms of Laplace's (unctions. Expansion of an arbitrary function. Formula of derivation. Cnireeponding formula in Bessel's functions for two dimensions. Indepemtent investigation of plane problem. Trauaverse vibrktions in a oylindricaJ envelope. Case of nniform initial velocity. Sector bounded by radial walla. Application to water waves. Vibra- tioiia, not necessoril; transverse, within a ciroular cylinder with plune end*. Complete solution of differential equation without restriotion ••to Absence of polar source. Formula of derivation. Expressioo of iclodty -potential by deioending aeroi-convergenl series. Case of purely dtvugait wave. Stokes' application to vibrating strings. Importance of loaDding-boarda. Prevention of lateral motion. Velocity-potential of a linear source. Sigtiificanoe of retardation of |X. Problem of plane waves impin^ng upon a cylindrical obstacle. Fixed and rigid cylinder. Uuthematically analogous problem relating to the tronsverae vibntiona of an elastic solid. Application to theory of light. Tyndall's nptnmenb shewing the smallnesa of ibe obstruction to sound offered hy bbrics, whose pores are open. [Reflection from series of eqnidietant uid {iMaUel sheets.] CHAPTER XIX. Friction. Nature of viscosity. Coefficient of risoosity. Independent of the density of the gss. Maxwell's experiments. Comparison of n|uaUona of *isooas notion with those applicable to an elastic solid. that B motion of uniform dilatation or contraction is not Stohes' expression for dissipation function. XIV CONTENTS. PAGE Application to theory of plane wayes. Gradual decay of harmonic waves maintained at the origin. To a first approximation the yelocity of propagation is unaffected by yiscosity. Numerical calculation of coefficient of decay. The effect of viscosity at atmospheric pressure is sensible for very high notes only. A hiss becomes inaudible at a mode- rate distance from its source. In rarefied air the effect of viscosity is much increased. Transverse vibrations due to viscosity. Application to calculate effects of viscosity on vibrations in narrow tubes. Helm- holtz's and Kirchhoff's results. [Kirchhoff's investigation. Plane waves. Symmetry round an axis. Viscosity small.] Observations of Schneebeli and Seebeck. [Exceedingly small tubes. Porous wall. Be- sonanoe of buildings. Dvdrak's observation on circulation due to vibra- tion in Kundt's tubes. Theoretical investigation.] CHAPTER XX. §§ 353—364 343 [Waves moving under gravity and cohesion. Kelvin's formula. Minimum velocity of propagation. Numerical values for water. Capillary tension determined by method of ripples. Values for dean and greasy water. Faraday's crispations. They have a period double that of the support. Lissajous' phenomenon. Standing waves on running water. Scott Bussell's wave pattern. Equilibrium of liquid cylinder. Potential energy of small deformation. Plateau's theorem. Kinetic energy. Frequency equation. Experiments of Bidone and Magnus. Transverse vibrations. Application to determine T for a recently formed surface. Instability. A maximum when X=4'51x2a. Numerical estimates. Application of theory to jet. Savart's laws. Plateau's theory. Experi- ments on vibrations of low frequency. Infiuence of overtones. Bell's experiments. Collisions between drops. Infiuence of electiicity. Obli- que jets. Vibrations of detached drops. Theoretical calculation. Stability due to cohesion may be balanced by instability due to electri- fication. Instability of highly viscous threads, leading to a different law of resolution.] CHAPTER XXI. §§ 365—372 376 [Plane vortex-sheet. Gravity and capillarity. Infinite thickness. Equal and opposite velocities. Tendency of viscosity. General equation for small disturbance of stratified motion. Case of stability. Layers of uniform vorticity. Fixed walls. Stability and instability. Various cases of infinitely extended fluid. Infinities occurring when n + A; 17=0. Sensi- tive flames. Early observations thereon. Is the manner of break-down varioose or sinuous? Nodes and loops. Places of maximum action are loops. Dependence upon azimuth of sound. Prejudicial effect of obetniefcions in the supply pipes. Vaziona explanations. Periodic view of diiinftegnktiiig unoke-jetB. Jets of liquid in liquid. Influence of TiecDitl;. Wana water. Miitura of water and Hloohol. BeU'h eiperi- Bitd-calU. Sondliauas' laws re/jnlatinij (litcU. NuIeg exaliiiDed I bj flsnies. Aeolian toitpa. StroaluirB obnereatioDs. Aeolian harp I Tibnlas ti&nsTerBely to direcliou oF wind. DimensioDal formula.] CHAPTER XXII. 373—381 415 ■ at solid bodieH. Geoeral equations. I'laae waves dilaCational oitional. Btalioaary navee. Initial diBtnrbance limited to a egion. Theory ot PoisaoD and Stokes. Waves from a sinfile Sacondarj waves dispersed from a smaU obstaole. Linear r obstaele. Complete solutioo toi periodic force opera- e at a eiiig;te point of an infinite solid. Uomparisoii with Btokea and Hertx. Beflection of plane waves at perpendioular incidence.] Principle of dynamical similarity. Theory of Hbi|ia and inodL-ls. Applicatioo of prinoiple of similarity to elastic plates. CHAPTER XXIII. i 382—397 Itts and theories of audition. Range of pitch over which the ear is capsblt Of perceivinn sound. EstimBtion of pitch. Preyer'e observatiouB, Amplitude necessary for audibility. Estimate of Toepler and Boltzmann, Antbot's observations by whistle and tuniup-forke. Binaural audition. LooatioD of sounds. Ohm's law of audition, Neceaeary exceptions. Two simple vibrations of neaJ'ly the same pitch, Boeanqnet's observa. tions. Mayer's observation tbat a grave Eound may overwhelm an ■cute sound, but not vici ifrm. EBitct of fatigue. Bow best to bear ijvertones. Helnihollz's theory of audition. Degree of damping of ribrators internal to the ear, Helmholtz's estimate. Mayer's results. How many impolses are required to delimit pitch? Kulilrauscii's resaltH. Beats of overtones. Consonant iuturvals mainly defined thereb,y. Combination -tones. According to Hehnholtz, due to a failure of super- position. In some cases combination -tones exist outside the ear. DiAereuce-tone on harmonium. Helraholtn's theory. Summation- tones- The diffionlty in hearing Ihem perhaps eiplieable by Mayer's observation. Are powerful generators necessary for andibilit; of dilletence-tones? Can beats pass into a diflerence-toneF Periodic changes of suitable pitch are not always recognised as tones. The [ diSerence-toDe involves a vibration of definite amplitude and phase. [ Audible difference-tones from inaudible generators. Consonant intervals I of pure (ones. Helmholtz's views, Detimitatiitn of the Fifth by diffe- J tenti&l tones ot the second order. Order of magnitude of various L differential tones. When the Octave is added, the first differential tone a to delimit the Fifth. Does the ear appreciate phaae-diSereoees? I Eelmholts's observations upon forks. Evidence olmistaned EelTin finds the beats of imperfect harmonics perceptible XVI CONTENTS. when the sounds are fiunt Kdnig's oheenraiions and theories. Beat- tones. The wave-siren. Qnalitj of musical sounds as dependent apon upper partials. Willis' theory of vowel sounds. Artificial imitations. Helmholtz's form of the theory. No real inoonsistenpy. BelatiTe pitch oharaoteristio, versas fixed pitch characteristic. Anerbach's re- sults. Evidence of phonograph. Hermann's condnsions. His analysis of A. Comparison of reeults by various writers. In Lloyd's view double resonance is fundamental. Is the prime tone present? Helm- holtz's imitation of vowels by forks. Hermann's experiment. Whispered vowels.] PAOB Note to § 86* 479 Appendix to Ch. V.* 480 On the vibrations of compound systems when the amplitudes are not infinitely small. Note to § 273« 486 Appendix A. (§ 307)' 487 On the correction for an open end. Index of Authors 492 Index of Subjects 496 1 Appears now for the first time. * Appeared in the First Edition. ERRATA. Vol. I. p. 407, footnote. Add reference to Chree, Camb. PhiL Trans Vol xiv p. 260, 1887. Vol. n. p. 46, for A. K5nig read W. KUnig. Vol. II. p. 236. footnote. Add reference to Gray and Mathews BesstVs Func- tions, Macmill a n , 1895. Since the atmosphere is the almost universal vehicle of tnd, the investigation of the vibrations of a gaseous medium ( always been coneidered the peculiar problem of Physical Acoustics; but in all, except a few specially simple questions, chiefly relating to the propagation of sound in one dimension, the mathematical difficulties are such that progress has been very slow. Even when a theoretical result is obtained, it often happens ihat it cannot be submitted to the test of experiment, io default of accuiate methods of measuring the intensity of vibrations. In ! parts of the subject all that we can do ia to solve those tbiems whose mathematical conditions are sufficiently simple to [kit of solution, and to trust to them and to general principles i to leave us quite in the dark with respect to other questions which we may be interested. Lin the present chapter we shall regard fluids as perfect, that is pay, we shall assume that the mutual action between any two Itions separated by an ideal surface is normal to that surface. Hereafter we shall say something about duid friction; but, in geneml. acoustical phenomena are not materially disturbed by I deviation from perfect fluidity as exists in the case of air I oibar glides, 16 eqoality of pressure in all directions about a given point consequence of perfect fluidity, whether there be or motion, as is proved by considering the equilibrium of a tett^edroD under the operation of the fluid pressures, the 1 EQUATIONS OP FLUID MOTION. [236. impressed forces, and the reactions against acceleration. In the Hmit, when the tetrahedron is taken indefinitely small, the fluid pressures on its sides become paramount, and equilibrium requires that their whole magnitudes be proportional to the areas of the faces over which they act. The pressure at the point a;, y, z will be denoted by p. 237. If pXdV, pYdV, pZdV, denote the impressed forces acting on the element of mass pdV, the equation of equilibrium is dp = p{Xdx+ Ydt/ + Zdz\ where dp denotes the variation of pressure corresponding to chtoges dXf dy, dz in the co-ordinates of the point at which the pressure is estimated. This equation is readily established by considering the equilibrium of a small cylinder with flat ends, the projections of whose axis on those of co-ordinates are respectively dx, dy, dz. To obtain the equations of motion we have, in accord- ance with D'Alembert's Principle, merely to replace X, &c. by X — Du/Dt, &c., where Du/Dt, &a denote the accelerations of the particle of fluid considered. Thus dp dx dp dy dp dz Dtl Dv\ Dt) Dw \ Dt) y 0). In hydrodynamical investigations it is usual to express the veloci- ties of the fluid u, v, w in terms of x, y, z and t They then denote the velocities of the particle, whichever it may be, that at the time t is found at the point x, y, z. After a small interval of time dt, a new particle has reached x, y, z] du/dt . dt expresses the excess of its velocity over that of the first particle, while Du/Dt, dt on the other hand expresses the change in the velocity of the original particle in the same time, or the change of velocity at a point, which is not fixed in space, but moves with the fluid. To this notation we shall adhere. In the change contemplated in ^ d/dt, the position in space (determined by the values of x, y, z) is retained invariable^ while in DIDt it is a certain particle of the Buul ■237.] EQUATION OF CONTINTJITY. 3 flnid on which attention is fixed. The relation between the two kinds of differentiation with respect to time is expressed by D d d d d Dt dt die dy dz ^ muBt be clearly conceived, though in a large class of impor- tnnt problems with which we shall be occupied in the sequel, the ■; ■-^Unction practically disappears. Whenever the motion is very -;riall, the terms udjdx, &c. diminish in relative importance, and ultimately i>//)t = d/d(. 238. We have further to express the condition that there is no creation or annihilation of matter in the interior of the fluid. If a, ^, 7 be the edges of a small rectangular parallelepiped parallel to the axes of co-ordinates, the quantity of matter which passes out of the included space in time dt in excess of that which enters is and this must be equal to the actual loss sustained, or Hence S+-r+'-r^^*=° <>>' the so-called equation of continuity. When p ia constant (with respect to both time and space), the equation assumes the simple form du dv dw ^ -g, dx dy dz In problems connected with sound, the velocities and the varia- lon of density are usually treated as small quantities. Putting .i = p,<l+«), where a. called the condensation, is small, and neg- :> voting the prodacta u dajdx, &c., we find l-£-|+£=» <»>■ III special cases these equations take even simpler forms. In ''w: case of an incompressible fluid whose motion is entirely ,>mlt«t lo the plane of a)y, — J^+r-« w. die dy 1—2 4 STIIEAM-FUNCTION. [238. from which we infer that the expression udy — vdx is a perfect diflferential. Calling it dyjr, we have as the equivalent of (4) where '^ is a function of the co-ordinates which so far is perfectly arbitrary. The function -i/r is called the «^ream-function, since the motion of the fluid is everywhere in the direction of the curves yjr = constant. When the motion is steady, that is, always the same at the same point of space, the curves '^ = constant mark out a system of pipes or channels in which the fluid may be sup- posed to flow. Analytically, the substitution of one function ^ for the two functions u and v is often a step of great consequence. Another case of importance is when there is symmetry round an axis, for example, that of x. Everything is then expressible in terms of x and r, where r = -\/(y' -f- ^), and the motion takes place in planes passing through the axis of symmetry. If the velocities respectively parallel and perpendicular to the axis of symmetry be u and q, the equation of continuity is T-'-^=» <«). which, as before, is equivalent to yjr being the stream-function. 239. In almost all the cases with which we shall have to deal, the hydrodynamical equations undergo a remarkable sim- plification in virtue of a proposition first enunciated by Lagrange. If for any part of a fluid mass udx + vdy + wdz be at one moment a perfect differential (2^, it will remain so for all subsequent time. In particular, if a fluid be originally at rest, and be then set in motion by conservative forces and pressures transmitted from the exterior, the quantities dv dw dw du du dv dz dy* dx dz* dy dx' (which we shall denote by (, fj, (f) can never depart from zero. ■:;39.] laorajjge's theorem. 5 We assume thai p is a function of p. and we shall write for brevity -I'f «• The equations of motion obtained from (1), (2), g 237, are dzi _ „ ^dii ^ du du du dx~ dt dx rfy dz ^ '' with two others of the same form relating to y and z. By bvpotheRis, dX_^dY_ dy dx ' -I > that by differentiating the first of the above equations with ri.'spect to y and the second with respect to x, and subtracting, we eliminate sr and the impressed forces, obtaining equations which may be put into the form D^_da dv /d« , rf"V /•i\ Dt-d^^-^di'' [d^'^ryj^ ^^'' with two others of the same form giving D^jDt, DjjIDt. In the case of an incompressible fluid, we may substitute for dui'dx + dv/dT/ its equivalent —dwidz, and thus obtain Di; du, ^ dv dw . , S-sf + i' + rfjf'*" <*>■ which are the equations used by Helmholtz as the foundation ■ ■{ his theorems respecting vortices. If the motion be continuous, the coefficients of f, jj, f in che above equations are all finite, Let L denote their greatest numerical value, and fl the sum of the numerical values of f , jj, f. By bj'pothesis, ft is initially zero; the question is whether in the course of time it can become finite. The preceding equa- tions shew that it cannot ; for its rate of increase for a given I'^iticle is at any time less than SLil, all the quantities coq- ■ rrnid being positive. Now even if its rate of increase were - great as 3Zfl, li would never become finite, as appears from :ii.' solution of the equation _ ^-i" (^)- 6 lagbange's theorem. [239. A fortiori in the actual case, A cannot depart from zero, and the same must be true of f ,"17, f . It is worth notice that this conclusion would not be disturbed by the presence of frictional forces acting on each particle pro- portional to its velocity, as may be seen by substituting X — ku, Y — tcv, Z-tcw, for X, Y, Z in (2)*. But it is otherwise with the frictional forces which actually exist in fluids, and are de- pendent on the relative velocities of their parts. The first satisfactory demonstration of the important pro- position now under discussion was given by Cauchy; but that sketched above is due to Stokes'. It is not sufficient merely to shew that if, and whenever, f, 17, f vanish, their differential coefficients D(/Dt, &c. vanish also, though this is a point that is often overlooked. When a body falls from rest under the action of gravity, « « «* ; but it does not follow that 8 never becomes finite. To justify that conclusion it would be necessary to prove thai 8 vanishes in the limit, not merely to the first order, but to all orders of the small quantity t; which, of course, cannot be done in the case of a falling body. If, however, the equation had been i x 5, all the differential coefficients of 8 with respect to t would vanish with t, if 8 did so, and then it might be in- feiTed legitimately that 8 could never vary from zero. By a theorem due to Stokes, the moments of momentum about the axes of co-ordinates of any infinitesimal spherical portion of fluid are equal to f, 17, f, multiplied by the moment of inertia of the mass ; and thus these quantities may be regarded as the component rotatory velocities of the fluid at the point to which they refer. If f> Vi ? vanish throughout a space occupied by moving fluid, any small spherical portion of the fluid if suddenly solidified would retain only a motion of translation. A proof of this proposition in a generalised form will be given a little later. Lagrange's theorem thus consists in the assertion that particles of fluid at any time destitute of rotation can never acquire it. ^ By introduoing suoh forces and neglecting the terms dependent on inertia, we should obtain equations applicable to the motion of electricity through uniform conductors. * Cambridge Tram. VoL fzn. p. 807, 1846. B. A. Report on Hydrodynamics, 1847. 240.1 EOTATOEY VELOCITIES. 240. A somewhat differeat mode of investigation haa been ailopled by Thomaon, which affords a highly instructive view of the whole subject'. By the fundamental equations Iho Now Xdx+7d>/ + Zd£ = dR. if the forces Bt Dt --g in which 4,( ida; + vdy + Ddx J Dd^ Dd, Thus. if tr.= i' + 11= + nf, we have 1 j._ dR-§^i..d, + wdy + wds) + idn' ■ D , j_ 1 ..J.. , . j-\ I D . 1 TJt — de. conservative, Dih ""bt ■ {udx -t- vdy + wdt) = d (R + i W ■ ••(1), ..(2). Integrating this equation along any finite arc PsPi, moving with the fluid, we have ^jiudx + vdy + wdz) = {R + ^U'~^),-(R + ^U^- »>).■.. (3), D f j^jiudx + vdy-i od!!) = (4); in which suffixes denote the values of the bracketed function at the points P, and Pj respectively. If the arc be a complete circuit, Dt] words, The line-inteffral of the tangential compotient velocity round any closed curve of a moving fluid remains constant tfiroughaat all lime. LlTbe Une-integral in question is appropriately called the ch-cu- i, and the proposition may be stated :- '!%« circuUttion ill any closed line viooiiiff with the fluid i ■'ing centtant. ' VrjTlci Motion. Edinhiirfh Tratu 8 CIRCULATION. [240. In a state of rest the circulation is of course zero, so that, if a fluid be set in motion by pressures transmitted from the outside or by conservative forces, the circulation along any closed line must ever remain zero, which requires that udx-{-vdy + wdz be a complete differential. But it does not follow conversely that in irrotational motion there can never be circulation, unless it be known that if> is single- valued ; for otherwise Jd<l> need not vanish round a closed circuit. In such a case all that can be said is that there is no circu- lation round any closed curve capable of being contracted to a point without passing out of space occupied by irrotationally moving fluid, or more generally, that the circulation is the same in all mutually reconcilable closed curves. Two curves are said to be reconcilable, when one can be obtained from the other by continuous deformation, without passing out of the irrota- tionally moving fluid. Within an oval space, such as that included by an ellipsoid, all circuits are reconcilable, and therefore if a mass of fluid of that form move irrotationally, there can be no circulation along any closed curve drawn within it. Such spctces are called simply- connected. But in an annular spctce like that bounded by the surface of an anchor ring, a closed curve going round the ring is not continuously reducible to a point, and therefore there may be circulation along it, even although the motion be irrotational throughout the whole volume included. But the circulation is zero for every closed curve which does not pass round the ring, and has the same constant value for all those that do. [In the above theorems "circulation" is defined without reference to masa If the fluid be of uniform density, the moment turn reckoned round a closed circuit is proportional to circulation, but in the case of a compressible fluid a distinction must be drawn. The existence of a velocity-potential does not then imply evanescence of the integral momentum reckoned round a closed circuit.] 241. When udx + vdy + wdz is an exact differential d^, the velocity in any direction is expressed by the corresponding rate of change of ^, which is called the velocity-potential, and du dv dw dx dy d» ( ».] VELOCITV-POTENTIAL. 9 be replaced by d?^ d'J. ^ da?'^ df^ d^- If S denote any closed surface, the rate of flow outwards across the element dS is expressed by dS . dift/dn, where dijyjdn is the rate of vajiation of if> in proceeding outwards along the normal. In the case of constant density, the total loss of Quid in time dt is thus the integration ranging over the whole surface of S. If the space S be full both at the beginning and at the end of the time dt, the loss must vanish ; and thus t 112^'" <"■ ^BDl6 application of this equation to the element dxdydz gives for ^Bbe equation of continuity of an incompressible fluid ■ d^ + df* if-" <^'' ^^k as it is generally written, P V=,^ = (3); when it is desired to work with polar co-ordinates, the trans- formed equation is more readily obtained directly by applying (1) to the corresponding element of volume, than by transforming (2) |l accordance with the analytical rules for effecting changes in the tependoat variables. \ Thus, if we take polaa' co-ordinates in the plane xy, so that j.' = rcoB^, w = rsin 6, ffind „,, d'lf) ld4> ld'<i> d'<f> ,,, ^"^ = 'i + rdh?d^'-ds' '*>' E if we take polar co-ordinates in space, jt = r8in tf cosQ), y = rsin^3inw, z=rcos8, _,, (?<i 2rf0 1 d / . .dd>\ 1 rf"* ,,, '^-^^;d^,^^^^5^(«'°^d^)-^^sin^rf^--<^>- mplor forms are assumed in special cases, such, for example, as mmetry round 2 in (5), 10 PROPBBTY OF IREOTATIONAL MOTION. [241. When the fluid is compressible, and the motion such that the squares of small quantities may be neglected, the equation of con- tinuity is by (8), § 238, | + ^** = (6)' where any form of V'^ may be used that may be most convenient for the problem in hand. 242. The irrotational motion of incompressible fluid within any simply-connected closed space S is completely determined by the normal velocities over the surface of & If /S be a material envelope, it is evident that an arbitrary normal velocity may be im- pressed upon its 8ur£ax;e, which normal velocity must be shared by the fluid immediately in contact, provided that the whole volume inclosed remain unaltered. If the fluid be previously at rest, it can acquire no molecular rotation under the operation of the fluid pressures, which shews that it must be possible to de- termine a function 0, such that V'^ = throughout the space inclosed by 8, while over the surface d^/dn has a prescribed value, limited only by the condition // t^'O (1^ An analytical proof of this important proposition is indicated in Thomson and Tait's Natural Philosophy, §317. There is no difBculty in proving that but one solution of the problem is possible. By Green's theorem, if V*<f> = 0, mt-p^^^'ih?.^ (*)■ the integration on the left-hand side ranging over the volume, and on the right over the surface of 8. Now if ^ and ^ + A^ be two functions, satisfjring Laplace's equation, and giving pre- scribed surface-values of d(f>ldn, their difference A^ is a function also satisfying Laplace's equation, and making dl^^jdn vanish over the surface of 8, Under these circumstances the double integral in (2) vanishes, and we infer that at every point of S d^(^/da, d^(f>ldy, dA^/dz must be equal to zero. In other words A^ most be constant, and the two motions identical As a par- ikr OMe» there can be no motion of the irrotational kind I the volome S, independently of a motion of the surface. 3 restriction to simply-connected spaces is rendered necessarj' ' the failure of Green's theorem, which, as was first pointed fut by Helmholtz, is otherwise possible. When the space S is multiply- connected, the irrotational lotion ia still determinate, if besides the normal velocity at reiy point of S there be given the values of the constant orculatioDS in all the possible irreconcilable circuits. For a complete discussion of this question we must refer to Thomson's original memoir, and content ourselves here with the case of a doubly -connected space, which will suffice for illustration. Let ABCD be an endless tube within which fluid moves irrotational 1 J. For this motion there must exist a velocity-poten- tial, whose differential coefficients, expressing, as they do, the com- ponent velocities, are necessarily single- valued, but which need not itself be single- valued. The simplest way of attacking the difficulty pre- sented by the ambiguity of <t>< is to conceive a barrier AB taken across the ring, so as to close the passage. The space ABCDBAEF is then simply continuous, and Green's theo- rem applies to it without modifica- tion, if allowance be made for a possible finite difference in the value of ^ on the two sides of the barrier. This difference, if it I enat, is necessarily the same at all points of AB, and in the )dynamicai application espresses the circulation round the Fig. 5i. I In applying the equation m^pf.)^--i.h'> (^ ) have to calculate the double integral over the two faces of B baiTier as well as over the original surface of the ring. Now has the same value on the two sides, ^ydS (over two faces oi AB) 'jldn" 12 MULTIPLY-CONNECTED SPACES. [242. if K denote the constant difference of ^. Thus, if k vanish, or there be no circulation round the ring, we infer, just as for a simply-connected space, that <f> is completely determined by the surface-values of d<f>/dn. If there be circulation, if> is still determined, if the amount of the circulation be given. For, if <l> and ^-f-A^ be two functions satisfjring Laplace's equation and giving the same amount of circulation and the same normal velocities at 8^ their difference. Aif> also satisfies Laplace's equa- tion and the condition that there shall be neither circulation nor normal velocities over 8. But, as we have just seen, under these circumstances A^ vanishes at every point. Although in a doubly-connected spax^e irrotational motion is possible independently of sur&ce normal velocities, yet such a motion cannot be generated by conservative forces nor by motions imposed (at any previous time) on the bounding surfieice, for we have proved that if the fluid be originally at rest, there can never be circulation along any closed curve. Hence, for multiply-connected as well as simply-connected spaces, if a fluid be set in motion by arbitrary deformation of the boundary, the whole mass comes to rest so soon as the motion of the boundary ceases. If in a fluid moving without circulation all the fluid outside a reentrant tube-like surface of uniform section become instan- taneously solid, then also at the same moment all the fluid within the tube comes to rest. This mechanical interpretation, however unpractical, will help the student to understand more clearly what is meant by a fluid having no circulation, and it leads to an extension of Stokes* theorem with respect to mole- cular rotation. For, if all the fluid (moving subject to a velocity-potential) outside a spherical cavity of any radius be- come suddenly solid, the fluid inside the cavity can retain no motion. Or, as we may also state it, any spherical portion of an irrotationally moving [incompressible] fluid becoming suddenly solid would possess only a motion of translation, without rotation^. A similar proposition will apply to a cylinder disc, or cylinder with flat ends, in the case of fluid moving irrotationally in two iimensions only. ^ Thomson on Vortex Motion, loc, eiU !■! ANALOGY WITH HEIAT AND ELECTRICITY. 13 i ^H^ The motion of an incompressible fluid which has been once ^^B rest partakes of the remarkable property (§ 7d) common to that ^^3r all s^slema which are set in motion with prescribed velocities, DAmely, that the energy is the least possible. If any other motion be proposed satisfying the equation of continuity and ^e boiindaiy conditions, its energy is necessarily greater than ^^ut of the motion which would be generated from rest'. B 243. The fact that the irrotational motion of incompressible fluid depends upon a velocity-potential satisfying Laplace's equation, is the foundation of a far-reaching analogy between the motion of such a fluid, and that of electricity or heat in a uniform conductor, which it is often of great service to bear iu mind. The same may be said of the connection between all the branches of Physics which depend mathematically on a potential, for it often happens that the analogous theorems are far from equally obvious. For example, the analytical that, if V'4> = 0, j dn ^dS=0 over a closed surface, is most readily suggested by the fluid interpretation, but once obtained may be interpreted for electric -or magnetic forces. W Agiun, in the theory of the conduction of heat or electricity, •ift is obvious that there can be no steady motion in the interior of S, without transmission across some part of the bounding soiiace, but this, when interpreted for incompressible fluids, gives I important and rather recondite law. 241. When a velocity-potential exists, the equation to deter- e the pressure may be put into a simpler form. We have from ). § 240. D aice by integration Dt ..(1), [Tlic reader who wislies to pursue the eloAy of g«Deral hjdrodynamice is the Iieatitee of Lamb and BMset.] 14 EQUATION OF PBESSURE. [244. Now r?= ^ + ^' + ^ + w;*; JJt at 80 that /f-*-t-j^ (2). which is the form ordinarily given. If p be constant, / — is replaced, of course, by - . The relation between p and ^ in the case of impulsive motion from rest may be deduced from (2) by integration. We see that -\pdt = — <f> ultimately. The same conclusion may be arrived at by a direct application of mechanical principles to the circumstances of impulsive motion. If j9 =2 Kpy equation (2) takes the form *logp = iJ-^-iI7« (3). If the motion be such that the component velocities are always the same at the same point of space, it is called steady, and ^ becomes independent of the time. The equation of pressure is then / ^^R-hU* (4). r or in the case when there are no impressed forces, / ^ = C-\U* (5). In most acoustical applications of (2), the velocities and condensa- tion are small, and then we may neglect the term \ U*, and sub- stitute ^ for I — , if Sp denote the small variable part of p ; thus Bp „ d4> jr-^'dt <^>' which with | + ^'^ = (7) are the equations by means of which the small vibrations of an elastic fluid are to be investigated. ■4.] PLANE WAVES. JI <f = dp/dp, so that Bp = a^p^, (6) ..(8). and we get on elimination of e. dp dt -(9). 245. The simplest kind of wave-motion is that in which the excursions of every particle are parallel to a fixed line, and are the eame in all planes perpendicular to that line. Let us therefore (aaiuning that R = Q) suppose that is a function of x (and t) nlj. Onr equation (9) § 244 becomes dF~°' d^ ■ -(1). s as that already considered in the chapter on Strings. ffe there found that the general solution is j>=f{x-at)-\-F{x-¥at) (2), anting the propagation of independent waves in the pasitive 1 negative directions with the common velocity a. Within such limits as allow the application of the approximate equation (1), the velocity of sound is entirely independent of the form of the wave, being, for example, the same for simple waves ^ = Aco6— (x — at), whatever the wave-length may be. The condition satisfied by the rdtive wave, and therefore by the initial disturbance if a posi- i wave alone be generated, is ddi dd> by (8) § 2M »-»«-0 (3). Similarly, for a negative wave u + aa = (4). ktever the initial disturbance may be (and u and a are both atrary), it can alwaj's be divided into two parts, satisfying stively (3) and (4), which are propagated undisturbed. In -0, ^■pute ^^Britra 16 PLANE PROGRESSIVE WAVES. [245. each component wave the direction of propagation is the same b& that of the motion of the condensed parts of the fluid. The rate at which energy is transmitted across unit of area of a plane parallel to the front of a progressive wave may be re- garded as the mechanical measure of the intensity of the radiatioa In the case of a simple wave, for which <f> = A cos — (x — at) (5), the velocity ^ of the particle at x (equal to d^jdx) is given by i ^Asm^ix-aJt) (6), and the displacement f is given by i=--<ioa^{x-<U) (7). The pressure p =pt + Bp, where by (6) § 244 Bp = -^p,aABm^(a,-at) ...(8). Hence, if W denote the work transmitted across unit area of the plane x in time t, dW dt = (po + ^)f = iPoa(-r") -4' + periodic terms. If the integration with respect to time extend over any number of complete periods, or practically whenever its range is sufficiently long, the periodic terms may be omitted, and we may take W:t=hPoa(^jA' (9); or by (3) and (6), if ^ now denote the maximum value of the velocity and 8 the maximum value of the condensation, Tr=W«^ = iPoaV^ (10). Thus the work consumed in generating waves of harmonic type is the same as would be required to give the maximum velocity | to the whole mass of air through which the waves extend*. 1 The earliest Btatement of the principle embodied in equation (10) that I ha^e met with is in a paper by Sir W. Thomson, **0n the possible density of tha Inminiferons medium, and on the meohanieal yalae of a onbio mile of ami-lis^" Phil. Mag. a. 1^. M. 1S66. where T(=X/a) is the periodic time. In a ffiven medium the tnechauical measure of the intensity is proportional to the square of the amplitude directly, and to the square of the periodic time inversely. The reader, however, must be on his guard against supposing that the mechanical measure of intensity of undulations of different wave lengths is a proper measure of the loudness of the corresponding sounds, as perceived by the ear. In any plane progressive wave, whether the type be harmonic or not, the whole energj' is equally divided between the potential and kinetic fomis. Perhaps the simplest road to this result is to consider the formation of positive and negative waves from an initial disturbance, whose energj' is wholly potential'. The total energies of the two derived progi-essive waves are evidently equal, and make up together the energy of the original disturbance. Moreover, in each progressive wave the condensation (or rare- faction) is one-half of that which existed at the corresponding point initially, bo that the potential energj- of each progressive wave is one-quaHer of that of the original disturbance. Since, as wfi have just seen, the whoh energy is one-half of the same quantity, it follows that in a progressive wave of any type one- half of the energy is potential and one-half is kinetic. The same conclusion may also be drawn from the general expreasiona for the potential and kinetic energies and the relations between velocity and condensation expressed in (3) and (4). The potential energj' of the element of volume dV is the work that would be gained during the expansion of the corresponding quantity of gas from its actual to its normal volume, the expansion being opposed throughout by the normal pressure p^. At any B of the expansion, when the condensation is «', the effective rare Sp is by § at-t a'p„a', which pressure has to be multiplied ' the corresponding increment of volume dV.da'. The whole work gained during the expansion from dV to dF(l+B) is therefore a'^padV.j^s'ds' or ^a^p^dV-s''. The genera! expressions for the potential and kinetic energies are accordingly ' Bounqnet, Phil. Mag. xlv. p. 173. 1ST3. * PMl Hag. (6) 1. p. aaO. 1B76. 18 Newton's investigation. [245. potential energy ^ ^a^po 1 1 1 si^ dV (12), kinetic energy = hpoJlJ^^dV (18), and these are equal in the case of plane progressive waves for which u= ± flW. If the plane progressive waves be of harmonic type, u and 8 at any moment of time are circular functions of one of the space co-ordinates (x\ and therefore the mean value of their squares is one-half of the maximum value. Hence the total energy of the waves is equal to the kinetic energy of the whole mass of air concerned, moving with the maximum velocity to be found in the waves, or to the potential energy of the same mass of air when condensed to the maximum density of the wavea [It may be worthy of notice that when terms of the second order are retained, a purely periodic value of u does not correspond to a purely periodic motion. The quantity of fluid which passes unit of area at point x in time dt is pudt, or po(l + 8)udt If u be periodic, Judt = 0, but Jsudt may be finite. Thus in a positive progressive wave Jsudt = aj^dt, and there is a transference of fluid in the direction of wave propagation.] 246. The first theoretical investigation of the velocity of sound was made by Newton, who assumed that the relation be- tween pressure and density was that formulated in Boyle's law. If we assume p = tep, we see that the velocity of sound is expressed by »i/K, or \/p -7- *Jp, in which the dimensions of p (= force -r area) are [ilf ] [Z]"* [^~*> a»d those of p (= mass -i- volume) are [ilf ] [Z]^. Newton expressed the result in terms of the * height of the hanuh geneom atmosphere^ defined by the equation gp^^p (1). where p and p refer to the pressure and the density at the earth's surface. The velocity of sound is thus \/(jrA), or the velocity which would be acquired by a body falling fi-eely under the action of gra\dty through half the height of the homogeneous atmosphere. To obtain a numerical result we require to know a pair of simultaneous values oip and p. 16.] Laplace's correction. [It is fouod by esperimeiit' that at 0° Cent, under the prasBore-l Ine (at Paris) to 760 mm. of mercury at 0" the density of dry air ■0012033 gms. per cubic centimetre. If we aasiime as the laity of mercury at 0" 13'59-53% and 5=9«0039, we have ' C.O.a measure p = 760 X 13-.5953 x 980-939, p = 0012933, whence a = >Jiplp) = 27994-0 ; J that the velocity of sound at 0° would be 279945 metres per jwcond, falling short of the result of direct observation by about a iuth part.] Newton's investigation established that the velocity of sound ibould be independent of the amplitude of the vibration, and also Bf the pitch, but the discrepancy between his calculated value ^ablished in 1687) and the experimental value was not explained mtil Laplace pointed out that the use of Boyle's law involved lie aesumption that in the conden^tions and i-arefactions ac- eompAnying sound the temperature remains constant, in contra- diction to the known fact that, when air is suddenly compressed, B tempei-ature rises. The laws of Boyle and Charles supply only le relation between the thiee quantities, pressure, volume, Mad temperature, of a gas, viz. pv = R0 (2). ■where the temperature d is measured from the zero of the gas tbermometer ; and therefore without some auxiliary assumption it I impossible to specify the connection between p and v (or p). lAptace considered that the condensations and rarefactions con- seroed in the propagation of sound take place with such rapidity itliat the heat and cold produced have not time to pass away, and ibat therefore the relation between volume and pressure is sensibly ) same as if the air were confined in an absolutely non-con- hctiDg vessel. Under these circumstances the change of pressure responding to a given oondeusation or Kirefaction is greater n on the hj'pothesis of constant terapemture, and the velocity tfaottnd is accordingly increased. I On Ui« Oeufiities of the Principal Gbhcb, Proc. ICoy. Soc. vol. i .Ti. Tal. nn. p. 321, 1881. U7. i 20 laplace'b COBRECnOX. [246. In equation (2) let v denote the volume and p the pressure of the unit of tnsas, and let be expressed in centigrade degrees reckoned from the absolute zero'. The condition of the gas (if uniform) is defined by any two of the three quantities p, v, $, and the third may be expressed in terms of them. The relation between the simultaneous variations of the three quantities is ^.* + ^ (3). In order to effect the change specified by dp and dv, it is in general necessary to communicate heat to the gas. Calling the necessm-y quantity of heat dQ, we may write ^«=©^-©^^ » Suppose now (a) that dp = 0. Equations (3) and (4) give where Jz (p const.) expresses the specific heat of the gas under a constant pressure. This being denoted by Xp, we have -(SI ('^ Again, suppose (b) that dv^O. We find in a similar manner that, if K, denote the specific heat under a constant volume, -©I (»> In order to obtain the relation between dp and dv when there is no communication of heat, we have only to put dQ « 0. Thus ©*-©*-». or, OD substituting for the differential coefficients of Q their values in terms of «,, «,,, «j.7 + *-y-o (^>- Since « = !//>, diiv= —dpjp; so that a- = ^»S«t_J, (8), dp pr. p^ ^ >■ ■ OiitlMordii)«i7Miiti8nd»i4»letli«*lMolDWMroi<abont -tlV. s.] EXPERIMENT OF CLEMENT AND DES0RM1«. 21 , as unuil. the ratio of the specific heats be denoted by 7, laplace'B value of the velocity of sound is therefore greater than Newton's in the ratio of Vy '■ 1. By integration of (8), we obtain for the relation between p and p, on the supposition of no communication of heat, i tiff <»'^ ^Bnt'ere p,. p, are two simultaneous values. Under the same ^TSreumstances the relation between pressure and temperature is bv (3) '(£f ..(10). The magnitude of 7 cannot be determined with accuracy by direct experiment, but an approximate value may be obtained by a method of which the following is the principle. Air is compressed into a reservoir capable of being put into communication with ■he external atmosphere by opening a wide valve. At first the t«mpemture of the compressed air is raised, but after a time the superSuous heat passes away and the whole moss assumes the temperature of the atmosphere ©. Let the pressure (measured by a manometer) be p. The valve is now opened for as short a time as ia sufficient to pcnuit the equilibrium of pressure to be completely established, that is, until the internal pressure has become equal to that of the atmosphere P. If the experiment be properly arranged, this operation is so quick that the air in the vessel has not sufficient time to receive heat from the sides, and therefore expands nearly according to the law expressed in (9). Its temperature 6 at the moment the operation is complete ^M therefore determined by ■ HW-' <"> ^^pe enclosed air is next allowed to absorb heat until it has ^^Wptined the atmospheric temperature 6, and its pressure (p') is ^^■en observed. During the last change the volume is constant, ^^fad therefore the relation between pressure and temperature ..(12); ■ It U here uiiii>)«d tbat 7 t conitBDt. Thii eqiuUoa app«ari to have b«eB iSPECIFIC HEATS. so that by elimination of 6I&, P -m whence [24JI ..{13). log;,- l og P logp-logp By experiments of this nature Clement and Desormes de- termined 7= 1'3492 ; but the method is obviously not susceptible of any great accuracy. The value of 7 required to reconcile the calculated and observed velocities of sound is 1-408, of the Bubstautial correctnes.'< of which there can be little doubt. We are not, however, dependent on the phenomena of sound for our knowledge of the magnitude of 7. The value of Kj, — the specific heat at constant pres-sure— has been determined experimentally by Regnault; and although on account of in- herent difficulties the experimental method' may fail to yield a aatisfector}' result for «„, the information sought for may be obtained indirectly by means of a relation between the two specific heats, brought to light by the modem science of Thermo- djTi amies. If from the equations 1«. dv dp we elimuiate dp, there results ic.de ..(15). hift Let us suppose that dQ = 0, or that there is no coQimunicatiot) of heat. It is known that the heat developed during the com- pression of an approximately perfect gas. such as air, is almofiti exactly the thermal equivalent of the work done in compi it. This important principle was assumed by Mayer in celebrated memoir on the dynamical theory of heat, though on grounds which can hardly bo considered adequate. However that may be, the principle itself is very nearly true, as has siQCfe been proved by the experiments of Joule and Thomson. If we measure heat in dynamical units, Mayer's principle Ktd6 = pdv oil the understanding that there i ' [S«e, Jjowever, Joly, Phil. Trmis. to\, ci.-vk.j.u. k, \%5V.\ > communication of heat. Comparing tliis with (15), we se x,-K,= R (16), ,.(17). The value of pv in gravitation measure (gramme, centimetre) 1 1033 ~ 001293, at 0° Cent, ao that „_ 1033 -001293 X2T2-85' By Begnauit's experiments the specific heat of air is '2379 I that of water ; and in order to raise a gramme of water one legree Cent., 42350 gramme-centimetres of work must be done Hence with the same units as for R, ytp = -2379x42350. Calculating from these data, we find 7 = 1*410, agreeing almost exactly with the value deduced from the velocity of sound. This investigation is due to Rankine, who employed in it 1850 to Scutate the specific heat of air, taking Joule's equivalent 1 the observed velocity of sound as data. In this way he nticipated the result of Reguault's experiments, which were lot publLthed until 1853. 247. Laplace's theory has often been the subject of niis- npprehension among students, and a stumblingblock to those tnarkable persons, called by De Morgan ' paradoxers.' But there I be no reasonable doubt that, antecedently to all calculation, 9ie hypothesis of no communication of heat is greatly to be referred to the equally special hypothesis of constant temperature. "here would be a real difficulty if the velocity of sound were lot decidedly in excess of Xewton's value, and the wonder is gather that the cause of the excess remained so long undiscovered. The only question which can possibly be considered open, il whether a small part of the heat and cold developed may not Bcape by conduction or radiation before producing its full effect. Everything must depend on the rapidity of the alternations. Below a certain limit of slowness, the heat iu excess, or defect, .would have time to adjust itself, and the temperature would seiisibh- coustant. In this case the relation between 24 STOKES INVESTIGATION pressure and density woiild be that which leads to Newton's value of the velocity of sound. On the other hand, above a certain limit of quickness, the gas would behave as if confined in a non-conducting vessel, as supposed in Laplace's theory. Now although the circumstances of the actual problem are better represented by the latter than by the former supposition, there may still (it may be said) be a sensible deviation from the law of pressure and density involved in Laplace's theory, entailing a somewhat slower velocity of propagation of sound. This question has been carefully discussed by Stokes in a paper published in 1851 \ of which the following is an outline. The mechanical equations for the small motion of air are l-4> (')■ with the equation of continuity ds ^du dv dw .^. dt^d^^Ty^Tz^^ (2>- The temperature is supposed to be uniform except in so fieu* as it is disturbed by the vibrations themselves, so that if denote the excess of temperature, p^Kp{l-\'S-\'a0) (3). The effect of a small sudden condensation « is to produce an elevation of temperature, which may be denoted by fis. Let dQ be the quantity of heat entering the element of volume in time dt, measured by the rise of temperature that it would produce, if there were no condensation. Then (the distinction between DjDt and djdt being neglected) de_ ds dQ dt dt dt dQ/dt being a function of and its differential coefficients with respect to space, dependent on the special character of the dissipation. Two extreme cases may be mentioned; the first when the tendency to equalisation of temperature is due to conduction, the second when the operating cause is radiation, and the transparency of the medium such that radiant heat is ^ PhiL Mag. (4) i. 806. 247.] <»* lEEBCT <? ^ATWA^mra^ -9 not senflibly mbnuth ud -viiinii m rfimmnp of In the fionner caae 4fi*ii x V^, and il laie iKCses:, ^ndsa- 2^ selected lij Sfeakes ior amLhrxdoL ixzi^gza^Bskai^ ^ydrxt^^i, Newton's law off fifiatic p kang mwiiiiiiii^ k & sn&aeia s^onni* matkn to the tmdL We hsx^e liian A=^s"** -- ''' In the case of plme wat^k, 10 wiiidi we sfaali cnrL-fine our attention, rand vTuikh, vMk ilji, «, ^are fsmcsaons of x (and/) only. EHiminadng^ and ■ herween di, <f land (Si, wefmd ^- /*t . ^ firom which and (5 ) we get (d <ft ' <f *t ^, \dl^^.d^'^''?dL^^/d^ '^^ if 7 be written (in the aune sense as be£oie> htl-rv^. If the vifaratioDS be hannonk, we maj suppose that « raiies as ^, and the equation becomes Let the coefficient of « in (7) be pat into the form /A-e"^, where '^ ""ic* ' }« + 7*n* ^ '^ and 2Vr-tan-:^.tan-^ = tan-<'?^:^i>^'? OX Ek[uation (7) is then satisfied by terms of the form ^i>iooe ^r—t tin ^)x but iiL being positive, and -^ less than \ir) if we wish for the expression of the wave travelling in the positive direction, we must take the lower sign. Discarding the imaginary part, we find as the appropriate solution « — -4e"''**"**cos(n*-;ACos'^a;) (10). 26 THE AMPLITUDE IS MORE [247. The first thing to be noticed is that the sound cannot be propagated to a distance unless sin ^ be insensible. The velocity of propagation ( V) is F=n/i""^ sec ^ -(11). which, when sin y^ is insensible, reduces to r^nfi-' (12). Now from (9) we see that y^ cannot be insensible, unless q/n is either very great, or very small. On the first supposition from (11), or directly from (7), we have approximately, V=^k (Newton); and on the second, V=iy/(Ky), (Laplace), as ought evidently to be the case, when the meaning of g in (5) is con- sidered. What we now learn is that, if q and n were comparable, the eflfect would be not merely a deviation of V fi:t)m either of the limiting values, but a rapid stifling of the sound, which we know does not take place in nature. Of this theoretical result we may convince ourselves, as Stokes explains, without the use of analysis. Imagine a mass of air to be confined within a closed cylinder, in which a piston is worked with a reciprocating motion. If the period of the motion be very long, the temperature of the air remains nearly constant, the heat developed by compression having time to escape by conduction or radiation. Under these circumstcmces the pressure is a function of volume, and whatever work has to be expended in producing a given compression is refunded when the piston passes through the same position in the revei'se direction; no work is consumed in the long run. Next suppose that the motion is so rapid that there is no time for the heat and cold developed by the condensations and rarefactions to escape. The pressure is still a function of volume, and no work is dissipated. The only difference is that now the variations of pressure are more considerable than before in comparison with the variations of volume. We see how it is that both on Newton's and on Laplace's hypothesis the waves travel without dissipation, though \vith different velocities. But in intermediate cases, when the motion of the piston is neither so slow that the temperature remains constant nor so quick that the heat has no time to adjust itself, the result is different. The work expend^ in producing a small oondensa- ISFLUEKCED THAN THE VELOCITY. ttoii is DO longer completely refunded during the corresponding rarefaction on account of the diminished tenipemture, part of the heat developed by the compiession having in the meantime escaped. In fact the passage of heat by conduction or radiation from a warmer to a finitely colder body always involves dissipa- tion, a principle which occupies a fundamental position in the -cience of Thermodynamics. lu order therefore to maintain the iiiotion of the piston, energy must be supplied from without, and if there be only a limited store to be drawn from, the motion rit ultimately subside. Another jwiut to be noticed is that, if q and » were com- parable, V would depend upon ». viz. on the pitch of the sound, a state of things which from experiment we have no reason to suspect. On the contraiy the evidence of observation goes to prove that there is no such connection. From (10) we see that the falling off in the intensity, esti- K" i per wave-length, is a maximum with tan -Jr, or i/c ; and ) -^ is a maximum when q : n = 'Jy. In this case fi. = nic~i y~i, 2^ = tan"'-/* — tan~'7~' (13), ce. if we take 7 = 1-.%, 2^/^ = 8 47'. .ftlculating from these data, we find that for each wave- length of arlvanc«, the amplitude of the vibration would be diminished in the ratio 6172. LTo take a numerical example, let "rfn *^^ * secoud, X = wave-length = 44 inches [112 cm.]. • In 20 yards [1828 em.] the intensity would be diminished in the ratio of about 7 millions to one, ■ Corresponding to this, 9 = 219S (U). If the value of q were actually that just written, sounds of the pitch in question would be very rapidly stifled. We there- fore infer that ? is in £act either much greater or else much less. But even ao large a value as 2000 is utterly inattmif^sible, as we may convince ourselves by considering the significance of EFFECT OF CONDUCT] Suppose that by a rigid envelope transparent to radiant heat. the volume of a small mass of gas were maintained constant, then the equation to determine its thermal condition at any time is = Ae~'^.. ..(15), whence where A denotes the initial excess of temperature, proving that after a time \jq the excess of tempei-ature would fall to leas than half its original value. To suppose that this could happen in &. two thousandth of a second of time would be in contradiction to the most superficial observatii We are therefore justified in assuming that q is very email in comparison with it, and our equations then become ap- proximately «*'/' ' = ^e-"-'"''»''=f' ms — ^Vt-x) ..(16). The effects of a small radiation of heat are to be sought for rather in a damping of the vibration than in an altered velocity of propagation. Stokes calculates that if 7 = 1'414, V= 1100, the ratio (N : 1) in which the intensity is diminished in passing over a distance a^ is given by logi, N = '0001 156 qx in foot-second measure. Although we are not able to make precise measurements of the intensity of sound, yet the fact that audible vibrations can be propagated for many miles excludes any such value of 5 as could appreciably affect the velocity of transmission. Neither is it possible to attribute to the air such a conducting power as could materially disturb the application of Laplace' theory. In order to trace the effects of conduction, we have only to replace q in (■')) by —q'd-liln?. Assuming as a particu) solution we Snd in- inicy = t^ + rjn^nC-Kiir 247.] VELOCITY DEPENDENT UPON" TEMPERATURE, whence, if 9' be relatively small. 1 ?■« .■ 2«7 , Thus the solution in real quantities is ..(17). t-A.Ea^(-'i^ -2,-^)0 •"■("'■ ..(18), leaving the velocity of propagation to this order of approximatioa still equal to i/(icy). From (18) it appears that the firat effect of conduction, as of radiation, is on the amplitude rather than on the velocity of propagation. In truth the conducting power of gasea is so feeble, and in the case of audible sounds at any rate the time during which conduction can take place is so short, that disturbance from this cause is not to be looked for. In the preceding discussions the waves are supposed to be propagated in an open space. When the air is confined within a tube, whose diameter is small in comparison with the wave- length, the conditions of the problem ai'e altered, at least in the case of conduction. What we have to say on this head will. however, come more conveniently in another place. 248. From the expression *J(py!p), we see that in the same gas the velocity of sound is independent of the density, because if the temperature be constant, p varies as p(p = RpS). On the other hand the velocity of sound is proportional to the square root of the absolute temperature, so that if a^ be its value at O' Cent. = 0^^1 + 27 ..(1), 273 ►ere the temperature is measured in the ordinary manner from B freezing point of water. i The most conspicuous effect of the dependence of the velocity (sound on temperature is the variability of the pitch of organ We shall see in the following chapters that the period [the note of a flue organ-pipe is the time occupied by a pulse in running over a distance which is a definite multiple of the length of the pipe, and therefore varies inversely as the velocity of pcopiiation. The uicoDvenJence arising from this alteration 30 VELOCITY OF SOUND IN WATER. [248. of pitch b aggravated by the fact that the reed pipes are not similarly affected ; so that a change of temperature puts an organ out of tune with itself. Prof. Mayer* has proposed to make the connection between temperature and wave-length the foundation of a pyrometric method, but I am not aware whether the experiment has ever been carried out. The correctness of (1) as regards air at the temperatures of 0' and 100° has been verified experimentally by Kundt. See § 260. In different gases at given temperature and pressure a is inversely proportional to the square roots of the densities, at least if 7 be constant*. For the non-condensable gases 7 does not sensibly vary from its value for air. [Thus in the case of hydrogen the velocity is greater than for air in the ratio V(l-2933) : V(-08993), or 3-792 : 1.] The velocity of sound is not entirely independent of the degree of dryness of the air, since at a given pressure moist air is somewhat lighter than dry air. It is calculated that at 50" F. [10' C], air saturated with moisture would propagate sound between 2 and 3 feet per second faster than if it were perfectly dry. [1 foot = 30-5 cm.] The formula a* = dpjdp may be applied to calculate the velocity of sound in liquids, or, if that be known, to infer conversely the coefficient of compressibility. In the case of water it is found by experiment that the compression per atmosphere is '0000457. Thus, if dp = 1033 x 981 in absolute C.G.S. units, dp = -0000457, since p=l. Hence a = 1489 metres per second, which does not differ much from the observed value (1435). 249. In the preceding sections the theory of plane waves has been derived from the general equations of motion. We 1 On an Acoustic Pyrometer. PhiU Mag, xlv. p. IS, 1S78. - According to the kinetic theory of gaseB, the \'elocity of sound ii detmnined solely by, and is proportional to, the mean velocity of the molecules. Fieitoii, Phil Mag. (5) in. p. 441, 1877. [See also Waterston (1846), Phil, Tram. ^ CLxzxin. A, p. 1, 1892.] uow proceed to an indepeucieiit investigation in which the motion - expressed in tenns of the actnal position of the layera of air ;;istead of by means of the velocity-potential, whose aid is no ionger necessary inasmuch as in one dimension there can be no iiuestion of molecular rotation. If y. y + dtfjdx.dx, define the actual positions at time t of neighbouring layera of air whose equilibrium positions aie defined by j: and x + dx, the density p of the included slice is given by P : p.. = \ whence by (9) g 246. p : p..- dx ' ..(1). ••(2), the expansions and condensations being supposed to take place according to the adiabatic law. The mass of unit of area of the slice is p,dx, and the corresponding moving force is (3). -dpldx.dx, gi •ing for the equal on of motion Bl tweeu (2) and (3) p is to be eliminated. \dx} dt' p, ds^ Thus, .(4). Equation (4) ia an exact equation defining the actual abscissa f ID terms of the equiiibrinm abscissa x and the time. If the tiou be assumed to be small, we may replace {dyldn:)''-*'^, which BOUTS as the coefficient of the small quantity d/'yjdC, by its aimate value unity ; and (4) then becomes dp p, lie"" 8 ordinary approximate equation. If the expan&ioa be isothermal, as ..(5), Newton's theory, the idons corresponding to (4) and (5) are obtained by merely ling 7=1. Whatever may be the relation betwe^^pudo, depending on 32 WAVES OF PERMANENT TYPE. [249. the constitution of the medium, the equation of motion is by (1) and (3) \dx} dt'^dp dai' ^^^' from which p, occurring in dp/dp, is to be eliminated by means of the relation between p and dy/dx expressed in (1). 250. In the preceding investigations of aerial waves we have supposed that the air is at rest except in so far as it is disturbed by the vibrations of sound, but we are of course at liberty to attribute to the w^hole mass of air concerned any common motion. If we suppose that the air is moving in the direction contrary to that of the waves and with the same actual velocity, the wave form, if permanent, is stationary in space, and the motion is steady. In the present section we will consider the problem under this aspect, as it is important to obtain all possible clearness in our views on the mechanics of wave propaga- tion. If Wo I Pof Po denote respectively the velocity, pressure, and density of the fluid in its undisturbed state, and if u, p, phe the corresponding quantities at a point in the wave, we have for the equation of continuity pn-=poUo (1), and by (5) § 244 for the equation of energy f'^ = J«.'-i«' (2). Eliminating w, we get rf-t»--('-^) <"• determining the law of pressure under which alone it is possible for a stationary wave to maintain itself in fluid moving with velocity Uq. From (3) t-i ■<*>■ or » = constant — - ^-^ ...(5). p V ^ Since the relation between the pressure and the density of actual gases is not that expressed in (5), we conclude that a self- maintaining stationary aerial wave is an impofleibilityy whatever WAVES OF PERMANENT TYPE. mnv be the velocity u,, of the general current, or in other words that a ^vave cannot be propagated relatively to the undisturbed parts ■ if the gas without undergoing an alteration of tj-pe. Nevertheless. liT^n the changes of density concemed are small, (5) may be itisfied approximately; and we see from (4) that the velocity of '.ream necessary to keep the wave stationary is given by vWJ- ..(6), which is the same as the velocity of the wave estimated relatively to the fluid. This method of regarding the subject shews, perhaps more clearly than any other, the nature of the relation between velocity Liid cnndensatiou § 245 (3), (4). In a stationary wave-form a loss 'if velocity accompanies an augmented density according to the principle of energy, and therefore the fluid composing the con- lionsed parts of a wave moves forward more slowly than the Midiaiurbed portions. Relatively to the fluid therefore the ii.rtion of the condensed parts is in the same direction as that in .. hich the waves are propagated. When the relation between pressure and density is other than that expressed in (5), a stationary wave can be maintained only by the aid of an impressed force. By (1) and (2) § 237 we have, on the supposition that the motion is steady, A' = dn , 1 dp " J- + - J •■ ax pax ■ m. while the relation between « and p is given by (1 ). If we suppose that p = a}p, (7) becomes X = (rf ..(8), i-?wing that an impressed force is necessary at every place \Phere is variable and unequal to a. 251. The reason of the change of type which endues when a wave it left to itself is not difficult to nndei-stand. From the .ordinary theory we know that an inflnitely small disturbance is ;nled with a certain velocity a, which velocity is relative t Uw parts of the medium undisturbed hy the wave. Let us J of a wave so long that the variations of 34 SUPERPOSITION OF PAHTICLE VELOCTTY. [251. velocity and density are insensible for a considerable distance along it, and at a place where the velocity (u) is finite let lu imagine a small secondary wave to be superposed. The velocity with which the secondary wave is propagated through the medium is a, but on account of the local motion of th'e mediam itself the whole velocity of advance is a -f u, and depends upon the part of the long wave at which the small wave is placed. What has been said of a secondary wave applies also to the parts of the long wave itself, and thus we see that after a time t the place, where a certain velocity u is to be found, is in advance of its original position by a distance equal, not to at, but to (a + u) f : or, as we may express it, it is propagated with a velocity a+u. In symbolical notation u=f[x — (a + u)t], where/ is an artatrary function, an equation tirst obtained by Foisson*. From the argument just employed it might appear at first sight that alteration of type was a necessary incident in the progress of a wave, independently of any particular supposition as to the relation between pressure and density, and yet it was proved in § 250 that in the ca^e of one particular law of pressure there would be no alteration of tj-pe. We have, however, tacitly assumed in the present section that a is constant, which is tanta- mount to a restriction to Boyle's law. Under any other law of pressure ij{dpldp) is a function of p, and therefore, as we shall see presently, of u. In the case of the law expressed in (5) § 250, the relation between u and p for a progressive wave is such that t/(dpjdp) + u is constant, as much advance being lost by slower propagation due to augmented density as is gained by superpon- tion of the velocity v. So far as the constitution of the medium itself is concerned there is nothing to prevent our ascribing arbitrarj- values to both n and p, but in a progressive wave a relation between these two quantities must be satisfied. We know already (§ 245) that this in the case when the disturbance is small, and the followii^ argument will not only shew that such a relation is to be expected in cases where the square of the motion must be retained, but will even define the form of the relation. Whatever may be the law of pressure, the velocity of propagv tion of small disturbances is by § 245 equal to •/(dpjdp), ood in 1 Ufimobe mr U Tbiorie dn Son. Journal de I'ttoU folyttetelgw, fc tn- p. 819. 1B06. Jal.J -v^ ~<l\ if this of wkntr I ^ a B^Mtn* «n«. ItKckw in order «» pc«eM «ke iaaMii that the aanreri* Ik fH^H will be geaamuA a> arr ftm things ia the inEMerfiMa aag^ tbp st«t« of An^ a> a £^h detennined by tfae auenoB I9P applying tfass etztenoa we an enodeimtioDS, not ■laof nt ely, but iclatii<e)y to thaw pnTuUiiy ta ibe twigfabmniiig pnti cf Ae »«dhm. s» that the fbna of (I) proper &T the ] -"V®? wlieoce -M-^) inch U the reUtioo betveen u and p neceaswy (or a {x«iti\v irugreesive vave. Enyation (2) ms obtained «Dal,\*tio9ilt,v by EATTishavr'. In the case of Boyle's lav, 'J{dp!dp) is constant, and thi> ivln- ■i'ln betwe«ii velocity and density, given first, I beUev«. by M> Imboltz*. it " = 010?^^ (*). if f»t be the deuaity coirespoDdiog to k = 0. In this case Poisson'e integral allows us to fiirm a dt.'liiiiu> idea f the change of type accompanjHng tho eorlior Wngos nf the i':>tgr«3«« of the wave, aud it finally leads us to n dirticiilty which liM not sa yet been aurmonnted'. If we draw a curve Id ri'iin-soiit Traia. I8SS, p. 146. ItU der Phytik, IV. p, lOti. 1SS2. On a diflioiillr in the Tbeory ol & : Mas. Nov. In4f*, 3— a 36 ULTIBIATE DISCONTINUITY. the distributiou of velocity, taking x for abscissa and u fst ordicate, we may find the corresponding curve after the lapee of time t by the following construction. Through any point on the original curve draw a. straight line in the positive direction parallel to X, and of length equal to (a + u)t, or, as we are concerned with the shape of the curve only, equal to u t. The locus of the ends of these lines is the velocity curve ait«r a time t. But this law of derivatioa cannot hold good indefinitely. The crests of the velocity curve gain continually on the troughs and must at last overtake them. After this the curve would indicate two values of u for one value of x, ceasing to represent anything that could actually take place. In fact we are not at liberty to push the application of the integral beyond the point at which the velocity becomes discontinuous, or the velocity curve has a vertical tangent. In order to find when this happens let us take two ni'ighbouring points on any part of the curve which slopes down- wards in the positive direction, and inquire after what time this part of the curve becomes vertical. If the difference of abscissee be dx, the hinder point will overtake the forward point in the time dx-^{—du). Thus the motion, as determined by Poisson's equation, becomes discontinuous after a time equal to the reci- procal, taken positively, of the greatest negative value of dnjdx. For example, let us suppose that u-Ucos~{x-{a-\-u)t]. where JJ ia the greatest initial velocity. When ( = 0, the greatest negative value of dujdx is — tirUjX ; so that discontinuity will commence at the time t = \j2-tTU. When discontinuity sets in, a state of things exists to which the usual differential equations are inapplicable ; and the subse- <|uent progress of the motion has not been determined. It is pi-obable, as suggested by Stokes, that some sort of reflection would ensue. In regard to this matter we must be careful to keep purely mathematical questions distinct from physical ones, ti practice we have to do with spherical waves, whose divergency may of itself be sufficient to hold in check the tendency to discontinuity. In actual gases too it is certain that before dis- continuity could enter, the law of pressure would begin to change its form, and the influence of viscosity could no longer be neglected. But these considerations have nothing to do with the mathetoatinl ^•] EARSSHAW 9 IXVTSTIGATION. problem of determining what would happen to waves of finite Amplitude in a medium, free &x)in viscosity, whose pressuiv is luider all circumstances exactly proportional to its density; and this problem has not been solved. It is worthy of remark that, although we may of course conceive a wave of finite disturbance to exist at any moment, there is a limit to the duration of its previous independent existence By drawing lines in the negative instead of in the positive direction we may trace the hi!»tory of the velocity cun'e , and wu see that HS we push our inquiry further and further into past time the forward slopes become easier and the backward slopes stee]Kr. At a time, equal to the greatest positive value of dai/du, antecedent to that at which the curve is first contemplated, the velocity would be discontinuous. 262. The complete integration of the exact equationN (■!) and (G) g 2-*9 in the case of a progressive wave was first effected by '. Finding reason for thinking that in a sound wavtj I equation dt \dxj .,(1) t always be satisfied, he observed that the result of differeii- Kog (I) with respect to (. viz. rfP ( Wi da? t2). I by means of the arbitrary function F he made to ooincidp with any dj-namieal equation in which the ratio of d'jf/dP and d'yidai' is expressed in terms of dy/dx. The form of the function f being thus determined, the Bolutiyn may be completftd by the J process applicable to such ca«e«*. ■'Writing for brevity a in place of djf/dx. we hav« I the integral ie to be toaoA by feiitninatin^ c Wiwimih iIh. iations y- equal to fu'f. and 4 beiag m* arbhtary ftmntt/M EAENSHAW 8 INVESTIGATION. If p — a*/>, the exact equation (6 § 249) ii [dnJ df (&■•■■■ ■•(*). r-M-^ (5), by comparison of which with (2) we see that or on integration F(a) = C±a\oga (6), as might also have been inferred from (4 J § 2J>1. The constant G vanishes, if F{<i}, viz. u, vanish when a«l, or p = p,; otherwise it represents a velocity of the medium as a whole, having nothing to do with the wave as such. For a positive progressive wave the lower signs in the ambiguities are to be used. Thus in place of (3), we have y~ax-alogat + <f>(a)\ - 0~ax-at +af (a)| ^ •'' and ti = — aloga = alog- (8). If we subtract the second of equations (7) from the first, we get y — erf + (rt log « = (a) — « ^' (a), from which by (8) we see that y - (a + u) £ is an arbitrary function of a, or of u. Conversely therefore u is an arbitrary function of y — {a + u)t, and we may write "=/[y-(«+")(l (9)- Equation (9) is Poisson's integral, considered in the preceding section, where the symbol x has the same meaning as here attaches to y. 263. The problem of plane waves of finite amplitude attracted also the attention of Riemann, whose memoir was communicated to the Royal Society of Gottingen on the 28th of November, 1859'. Riemann's investigation is founded on the general hydrodynamioal equations investigated in ^ 237, 238, and is not restricted to any particular law of pressure. In order, however, not unduly to > Dtber die For^iduuimg ebener Loltwellen von endlioher Sohmnimngcwirite. CKittiiigen, Abhandlmigat, t. vm. 1860. See ftlw ui eiMUent kbetnet in Oh ForUekritU &tr PhgM, xr. p. 128. [BefomuM mKf be nude almt to ■ pi^w ^ C. T. Bortou, FhO. Mag- xxrr. p. SIT, ia»S.] dt'^"'dj; " cir (2). If we roiihiply (2) by ±a, and afterwards add it to (1), we d\Qgp d\ogp_ du ■-53.] RIEMANN'a EQUATIONS. 39 f xWnd the discuasion of this part of our subject, already perhaps treated at gi-eat«r length than its acoustical importance would warrant, we shall here confine ourselves to the case of Boyle's law of pressure. Applying equations (1), (2) of § 237 and (1) of § 238 to the circumstances of the present problem, we get [ hn r -*-di These equations are more general than Poisson'a and Earuahaw's in that they are not limited to the case of a single positive, or I'gative. progressive wave. From (5) we learn that whatever ■i:iy be the value of P corresponding to the point x and the time ' the same value of P corresponds to the point x-¥(u4-a)dt at the time t + dt; and in the same way from (G) we see that Q remains unchanged when x and ( acquire the increments (ti - a) dt and dl respectively. If P and Q be given at a certain instant of me as functions of x, and the representative curves be drawn, we ^y deduce the corresponding value of it by (4), and thus, as in !51, construct the curves representing the values of P and Q r the small interval of time dt, from which the new values t tt and p in their turn become known, and the process can be The element of the fluid, to which the valuea of P and Q at r moment belong, is itself moving with the velocity u, so that Sie velocities of P and Q relatively to the element are numerically the Bamc, and equal to a, that of P being in the positive direction I and that of Q in the negative direction. di~ "'+°'<;^' dt- <" "'d,, ™ where P = a[ogp + ii, Q = alogp-i« (4). ,Xhu« liP-j^((ij!-<ll+o)(((| (5), dQ-^[d>:-(u-a)dl] (6). 40 LIMITED INITIAL DISTrRBANCE. [253. We are now in a position to ti-ace the consequences of au initial disturbance which is confined to a finite portion of the medium, e.g. between oc = a and x = ff, outside which the medium is at rest and at its normal density, so that the values of P and Q are a logpo- Each value of F propagates itself in turn to the ele- ments of fluid which lie in front of it, and each value of Q to those that lie behind it. The hinder limit of the region in which P is variable, viz. the place where P first attains the constant value a log Pq, comes into contact first with the variable values of Q, and moves accordingly with a variable* velocity. At a definite time, requiring for its determination a solution of the differential equa- tions, the hinder (left hand) limit of the region through which P varies, meets the hinder (right hand) limit of the region through which Q varies, after which the two regions separate themselves, and include between them a portion of fluid in its equilibrium condition, as appears from the fact that the values of P and Q are both alogpo. In the positive wave Q has the constant value a log />o, so that t^ = a log (p/po), as in (4) § 251 ; in the negative wave P has the same constant value, giving as the relation between w and p, M » — a log (p/po)- Since in each progressive wave, when isolated, a law prevails connecting the quantities u and p, we see that in the positive wave dii vanishes with dP, and in the negative wave du vanishes with dQ. Thus from (5) we leara that in a positive progressive wave du vanishes, if the increments of x and t be such as to satisfy the equation etc — (w + a) df = 0, from which Poisson's integral immediately follows. It would lead us too far to follow out the analytical develop- ment of Riemann's method, for which the reader must be referred to the original memoir ; but it would be improper to pass over in silence an error on the subject of discontinuous motion into which Riemann and other writers have fallen. It has been held that a state of motion ia possible in which the fluid is divided into two parts by a surface of discontinuity pi-opagating itself with constant velocity, all the fluid on one side of the surface of discontinuity being in one uniform condition as to density and velocity, and on the other side in a second uniform condition in the same respects. Now, if this motion were possible, a motion of the same kind in which the surfiBM^ of discontinuity is at rest would also be 1 At this point an error seems to have crept into Biemann's work, which is eotreoted in the abstract of the FortachritU der Phytik, « POISSONS INTEGRAL. 41 move possible, as we maj see by supposiug a velocity equal and ipoaite to that with which the surface of discontinuity at first moves, to be impressed upon the whole mass of fluid Id order to the relations that must subsist between the velocity and ity on the one side (m,, p,) and the velocity and density on the eiile (tij, ps), we notice in the first place that by the principle ff conservation of matter p,Uj= p^n-i. Again, if we consider the 'iKimentum of a slice bounded by parallel planes and including the -urface of discontinuity, we see that the momentum leaving the •iice in the unit of time is for each unit of area {/>iHa = (>i»ii)"i, >hile the momentum entering it is pjM,'. The difference of mo- iiit^iitum must be balanced by the pressures acting at the boundanes 'i the slice, so that whence p. "i ("i - "i) =Pi -P, = a? (p. - p,), -VC:)' =V©' ■ (')■ The motioD thus determined is, however, not possible ; it satisfies indeed the conditions of mass and momentum, but it violates the condition of energy (§ 244) expressed by the equation i M-j' - i I'l' = n' logp,-(i= logp, (8). This argument has been already given in another form in § 250, which would alone justify us in rejecting the assumed motion, since it ftppeare that no steady motion is possible except under the law of density there determined. From equation (8) of that section we oan find what impressed foi*ces would be necessary to maintain the notion defined by (7). It appears that the force A', though con- fined to the place of discontinuity, is made up of two parts of f^fposite signs, since by (7) u passes through the value ii. The vhole moving force, viz. JX/> dx, vanishes, and this explains how it is chat the condition relating to momentum is satisfied by (7), though the force X be ignored altogether. 2S3a. Among the phenomena of the second order which ttdmic of a ready explanation, a prominent place must be lo the repulsion of resonators discovered independently CvoF&k' and Mayer'. These obser\'ers found that an air resonator ' any kind (Ch. XVI.) when exposed to a powerful source p. i%. 18TC ; Witd. Ann. t p. aas, 1873. , p, 336, 1876. gned ■ r by I nator H )urce I 42 REPULSION OF RESONATORS. [253 OU of sound experiences a force directed inwards from the mouth, somewhat after the manner of a rocket. A combination of four light resonators, mounted anemometer fashion upon a steel point, may be caused to revolve continuously. If there be no impressed forces, equation (2) § 244 gives j'i-t-i'^- (>^ Distinguishing the values of the quantities at two points of space by suflSxes, we may write This equation holds good at every instant. Integrating it over a long range of time we obtain as applicable to every case of fluid motion in which the flow between the two points does not continually increase Jtsr,dt-'Jtsrodt=yUo'dt''^JUi'dt (3). The first point (\vnth suflix 0) is now to be chosen at such a distance that the variation of pressure and the velocity are there insensible. Accordingly j'ur^dt^-^jUHt (4). This equation is true wherever the second point be taken. If it be in the interior of a resonator, or at a comer where three fixed walls meet, Ui = 0, and therefore /(«^i-«^o)d«=0 (5), or the mean value of ^ in the interior is the same as at a distance outside. "^y (^) § ^*^6' ^f ^^6 expansions and contractions be adiabatic, pxpy\ and w = j!}'y-i'/y. Thus v/i(r"->i*-« <«^ If in (6) we suppose that the difference between pi and p^ is comparatively small, we may expand the function there contained by the binomial theorem. The approximate result may be expressed /^-"=rj(^)'* ■ •■••('). shewing that the mean value of (ih — jPo) is positive, or in other I irds that the mean pressure in the resonator is in excess of the .iiuiospheric presaure'. The resonator therefore tends to move a^ if impelled by a force acting normally over the area of its aperture and directed inwards. The experiment may be made (after Dvoriik) with a Helmholtz resonator by connecting the nipple with a horizontal and not too narrow glass tube in which moves a piston of ether. When a fork of suitable pitch, e.g. 256 or 512, is vigorously excited and presented to the mouth of the resonator, the movement of the ether shews an augmentation of pressure, while the similar presentation of the non-vibrating fork is without effect. If to the first order of small quantities (p-^,;)ip^=Pco3 lit (8). its mean value correct to the second order is P'j4^, in which for air and the principal gases y= Vi. If the expansions and coiitraotions be supposed to take place isothermally, the corresponding result is arrived at by putting ¥ »(7). 6. Iq § 253 ([ the effect to be eicplaiued is intimately 'connected with the compressibility of the Huid which occupies the interior of the resonator. In the class of phenomena now to be coDfiidored the compressibility of the fluid is of secondary import- ance, and the leading features of the explanation may be given upon the supposition that the fluid retains a constant density throughout. ^^L If p be constant, (4) § 253 tt may be written ^B jip,-p„)dt=-i[pju,'dt (1), ^Boewing that the mean pressure at a place where there is motion is less than in the undisturbed parts of the fluid — a theorem due to Kelvin', and applied by him to the explanation of the attractions observed by Guthrie and other experimenters. Thus a vibrating tuning-fork, presented to a delicately suspended rec tangle of paper, appears to exercise an atti-action, the mean J of f7* being greater on the face exposed to the fork than 1 Che back. • Phil. 31'ig.\Q\. n. p. 270. 1878. ' Froe. Rfn. Son. vol. six. p. 271, 1887. 44 A ROTATORY FORCE OPERATIVE [253 fc. In the above experiment the action depends upon the prox- imity of the source of disturbance. When the flow of fluid, whether steady or alternating, is uniform over a large region, the effect upon an obstacle introduced therein is a question of shape, [n the case of a sphere there is manifestly no tendenc}'' to turn ; and since the flow is symmetrical on the up-stream and down- stream sides, the mean pressures given by (1) balance one another. Accordingly a sphere experiences neither force nor couple. It is otherwise when the form of the b<Kly is elongated or flattened. That a flat obstacle tends to turn its flat side to the stream* raav be inferred from the general character of the lines of flow round' it. The pressures at the various points of the surface BC (Fig. 54 a) depend ujwn the velocities of the fluid there obtaining. The full pressure due to the complete stoppage of the stream is to be found at two points, where the current divides. It is pretty evident that upon the up- stream side this lies (P) on AB, and upon the down-stream side upon AC at the con-esponding point Q, The resultant of the pressures thus tends to turn .^fi so as to face the stream. When the obstacle is in the form of an ellipsoid, the mathe- matical calculation of the forces can be effected; but it must suffice here to refer to the particular case of a thin circular disc, whose normal makes an angle with the direction of the un- disturbed stream. It may be proved^ that the moment M of the couple tending to diminish has the value given by J/=Jp«»TPsin25 (2), a being the radius of the disc and W the velocity of the stream. If the stream be alternating instead of steady, we have merely to employ the mean value of W\ as appeai-s from (1). The observation that a delicately suspended disc sets itself across the direction of alternating currents of air originated in the attempt to explain certain anomalies in the behaviour of a magnetometer mirrorl In illustration, " a small disc of paper, about the size of a sixpence, was hung by a flne silk fibre across I Thomson and Tait's Natural Philotophy, g 886, 1867. > W. KSnig, Wied. Ann. t. xuu. p. 51, 1891. * Proe. Roy. Soc, vol. zxxn. p. 110, 1881. rPOS A SrSPBXDED DISC. the mouth of a resonator of pitch 12S. When a sound of this pitch is excited in the neighbourhood, there is a powerful rush of air into and out of the resonator, and the disc sets itself promptly across the passage. A fork of pitch 12S may be held near the resonator, but it is better to use a second resonator at a little distance in order to avoid any possible disturbance due to the ut>ighbourhood of the vibrating prongs. The experiment, though rather less striking, was also successful with forks and resonators ..f pitch 256." Upon this principle an instrument may be constructed for iieaauring the intensities of aerial vibrations of selected pitch". \ tube, measuring three quarters of a wave length, is open at one ud and at the other is closed air-tight by a plate of glass. At ..iL- quarter of a wave length's distance from the closed end in hung by a silk fibre a light min'or with attached magnet, such as is tised for reflecting galvanometers. In its undisturbed condition the plane of the mirror makes an angle of 45' with the AXIS of the tube. At the side is provided a glass window, through which light, entering along the axis and reHected by the mirror, is able to escape from the tube and to form a suitable image upon a. divided scale, The tube as a whole acts as a resonator, and the alternating currents at the loop (§ 255) deflect the mirror through an angle which is read in the usual manner, In an instniment constnicted by Boys' the sensitiveness is exalted to an extraordinary degree. This is effected partly by the use of a very light mirror with suspension of quartz fibre^ and partly by the adoption of double resouance. The large resonator is a heavy brass tube of about 10 cm. diameter, closed at one end, and of such length as to resound to e'. The mirror is hung in a short lateral tube forming a commmiication between the large resonator and a small glass bulb of suitable capacity. J T he external vibrations may be regarded as magnified first by the ^|hrge resonator and then agai'n by the small one, so that the ^Bliiror ia affected by powerful alternating currents of air. The ^BslectioD of pitch is so definite that there is hardly any response ^Hto sounds which are a semi-tone too high or too low. ^Pr Perhaps the most striking of all the effects of alternating Meiial currents is the rib-like structure assumed by cork filings in ' FhiL Hag. vol. Dv. p. 196, 1883. I^JhlMr*, vol. ILU. p. 604. 1890. 46 EXPLANATION OF THE [253 6. Fig. 54 5. Fig. 54 c. Kundt's experiment § 260. Close observation, while the vibrations are in progress, shews that the filings are disposed in thin laminae transverse to the tube and extending upwards to a certain distance from the bottom. The effect is a maximum at the loops, and disappears in the neighbourhood of the nodes. When the vibra- tions stop, the laminae necessarily fall, and in so doing lose much of their sharpness, but they remain visible as transverse streaks. The explanation of this peculiar behaviour has been given by A, Konig^ We have seen that a single spherical obstacle experiences no force from an alternating current. But this condition of things is disturbed by the presence of a neighbour. Consider for simplicity the case of two spheres at a moderate distance apart, and so situated that the line of centres is either parallel to the stream. Fig. 54 6, or perpendicular to it, Fig. 54 c. It is easy to recognise that the velocity between the spheres will be less in the first case and greater in the second than on the averted hemi- spheres. Since the pressure increases as the velocity diminishes, it follows that in the first position the spheres will repel one another, and that in the second position they will attract one another. The result of these forces between neighbours is plainly a tendency to aggregate in laminae. The case may be contrasted with that of iron filings in a magnetic field, whose direction is parallel to that of the aerial current. There is then attraction in the first position and repulsion in the second, and the result is a tendency to aggregate in filaments. On the foundation of the analysis of Kirchhoff, Konig has C) o o 1 o calculated the forces operative in the case of two spheres which are not too close together. If Oi, Oj be the radii of the spheres, r their distance asunder, the angle between the line of centres and the direction of the current taken as axis of z (Fig. 54 d), W the velocity of the current, then the components of force upon the sphere B in the direction of z and of a Fig. 54(2. 1 wad. Aim. t. xui. pp. 858, 549, 1891. t- »S6.] STRIATIONS IX KUKDTS Tl'BES. 47 drftwii perpendicular to z in the plane containing s and the line if centres, are given by _ Z = -^^''-"''"='^'cose(3-5co8'e) t3), „_ 3Trpa,*ai*W' sin ^ (1 - 5 cos'^) (4), the thii-d component Y vanishing by virtue of the symmetry. In the case of Fig. .54 6 = 0. and there Is repulsion equal to Qirpa,*a^*W'/r*; in the case of Fig. 54 e ^ = iTr, and the force is an attraction 3irpai*a,'WV»^- In oblique positions the direction of the force does not coiucide with the line of centres. If the spheres be rigidly connected, the forces upon the system reduce to a couple (tending to increase d) of moment given by -Zsil.« + A'co.«-?^l'— sm2S (5X When the current is alternating, we are to take the mean value of IT' in (3). (4), (5). 264 The exact experimental determination of the velocity of sound is a matter of greater difficulty than might have been expected. Obser\'ations in the open air are liable to eiTors from the etfecls of wind, and from uncertainty with respect to the exact condition of the atmosphere as to temperature and dryness. On the other hand when sound is propagated through air con- tained in pipes, disturbance arises from friction and from transfer of heat ; and, although no great errors from these aoui-cea are to be feared in the case of tubes of considerable diameter, such aa some of those employed by Regnault, it is difficult to feel sore that the ideal plane waves of theorj' are neaily enough realized. The following Table' contains a list of the principal experi- me ntal determinations which have been made hitherto. KftmBH ot Obsercers, Velocity of Sound at 0° Cent, in Melrei. Acad^mie des Sciences (1738) 332 Bemenberg (1811) |gg2.g Goldingham (1821) 3311 Bureau des Longitudes (1822) 330-6 Uoll and van Beek 3322 ' BoBiiiiqael, Phil. Mag. AptU, 1877. I 48 VELOCITY OF SOUND. [254. Names of Observers. Velodty of Sonnd at OP Cent, in Metres. Stampfer and Myrback 332*4 Bravais and Martins (1844) 3324 Wertheim 3316 Stone (1871) 332*4 Le Roux 330-7 Regnault 3307 In Stone's experiments" the course over which the sound was timed commenced at a distance of 640 feet from the source, no that any errors arising from excessive disturbance were to a great extent avoided. A method has been proposed by Bosscha' for determining the velocity of soynd without the use of great distances. It depends upon the precision with which the ear is able to decide whether short ticks are simultaneous, or not. In Konig's' form of the experiment, two small electro-magnetic counters ai*e controlled by a fork-interrupter (§ 64), whose period is one-tenth of a second, and give synchronous ticks of the same period. When the counters are close together the audible ticks coincide, but as one counter is gradually removed from the ear, the two series of ticks fall asunder. When the difference of distances is about 34 metres, coincidence again takes place, proving that 34 metres is about the distance traversed by sound in a tenth part of a second. [On the basis of experiments made in pipes Violle and Vautier* give 33110 as applicable in free air. The result includes a cor- rection, amounting to 0*68, which is of a more or less theoretical character, representing the presumed influence of the pipe (0*7™ in diameter).] 1 Phil. Trans. 1872, p. 1. « Pogg. Ann. xcii. 486. 1864. • Pogg. Ann. cxviii. 610. 1868. * Ann. de Chim. t. xix.; 1890. 265. We have already (§ 2i5) considered the solution of our fuDdanientiil equation, when the velocity- potential, in an unlimited fluid, ia a function of one space co-ordinate only. In the abBence uf friction no change would be caused by the introduction of any number of fixed cylindrical auifacea, whose generating lines are parallel to the co-ordinate in question ; for even when the surfaces are absent the fluid has no tendency to move across them. If one of the cylindrical surfaces be closed (in respect to its transverse sectiun), we have the important problem of the axial motion of air within a cylindrical pipe, which, when once the mechanical condi- tions at the ends are given, is independent of anything that may happen outside the pipe. Considering a simple harmonic v-ibration, we know (§ 245) that, if 4> varies as e*"', S+'=*='' (i)' where -^== «■ The solution may be written in two forms — tt> = (A cosfce + fisin Au*)e''"] ,t> = {Ae'^+Be-^)e'"' \ 1 which finally only the real parts will be retained. The first fiunnrill be most convenient when the vibration is stationary, or -(3), HARMONIC WAVES IN ONE DIMENSION. [255. nearly bo, aod the second when the motion reduces itself to a positive, or nej^ative, progressive undulatioa. The constants A and B in the symbolical solution may be complex, and thus the final expression in terms of real qiiantities will involve four arbi- trary constants. If we wish to use real quantities throughout, we must take ^ = {A cos kx + Bsia kx) cos nt •^■{Ccoskx + D!nakx)&\ant (4), but the analytical work would genemlly be longer. When no ambiguity can arise, we shall sometimes for the sake of brevi^ drop, or restore, the factor involving the time without express mention. Equations such as (1) are of course equally tnie whether the factor be understood or not. Taking the first form in (-?), we have ^= Acoakx+ Bsinkx i^ = — kA sin kx + kB cos kn: ax If there be any point at which either tf> or d<f>/dx is permaneutly zero, the ratio A : B must be real, and then the vibration is «ftj- tionary, that is, the same in phase at all points simultaneously. Let us suppose that there is a node at the origin. Then ythaa a = 0, dtpjdx vanishes, the condition of which 18^0 = 0. Thus ..(5). <f> = A cos kx c' d^_ dx~ - kA sin kx e'"' ..(6), from which, if we substitute Pe' imaginary part, 4,= P cos ib: cos {fit + 6) d^ dx = — kP siu kx cos {nt + 0) A, and throw away the P). From these equations we learn that d(f}/dx vanishes wherever sin foe = ; that is, that besides the origin there are nodes at tha points X = i[m\, m being any positive or negative integer. At aoi of these places infinitely thin rigid plane barriers normal to 1 might be stretched across the tube without in any way alte^ ing the motion, Midway between each pair of consecutive node there is a loop, or place of no pressure variation, since Sp = ~pi (6) § 244. At any of these Voopa a commviaication with th< pitemal atmosphere might be opened, without causing auy diaturb- wce of the tiiotioii from air passing in or out. The loops are the places of maximum velocity, aud the nodes those of maximum pressure variation. At intervals of X everything is exactly re- peated. If there be a node at a! = l,aa well as at the origin, sin kt = 0, or X = 2//m, where m is a 4)0sitive integer, The gravest tone which can be sounded by air contained in a doubly closed pipe of length / is therefore that which has a wave-length equal to 2/, This statement, it will be observed, holds good whatever be the gas with which the pipe is filled; but the frequency, or the place of the tone in the musical scale, depends also on the nature of the particular gas. The periodic time is given by ..(8). The other tones possible for a doubly closed pipe have periods which are submultiples of that of the gravest tone, and the whole system forms a harmonic scale. Let us now suppose, without stopping for the moment to in- quire how such a condition of things can be secured, that there is a loop instead of a node at the point a: = l. Equation (6) gives cosW = 0. whence X = 4i-e-(2nt+ 1), where m is zero or a positive integer. In this case the gravest tone has a wave-length equal I to four times the length of the pipe reckoned from the node to the loop, and the other tones form with it a harmonic scale, from which, however, all the members of even order are missing. 266. By means of a rigid barrier there is no difficulty in jecuring a node at any desired point of a tube, but the condition for a loop, i-e. that under no circumstances shall the pressure vary, can only be realized approximately, In most coses the variation of pressure at any point of a pipe may be made small by allowing a free communication with the external air. Thus Euler and Lagrange assumed constancy of pressure as the condition to be istlisfied at the end of an open pipe. We shall afterwards return 10 the problem of the open pipe, and investigate by a rigorous jirucess the conditions to be sati.sfied at the end. For our im- w mediitte purpose it will be sufficient to know, what is indeed KbJ^ obvious, that the open end of a pipe may \ie ttealed a& 52 CONDITION FOR AN OPEN END. ] a loop, if the diameter of tht pipe be neglected in comparisdl with t^e wave-length, provided the extemnl pressure in the neigifl bourhood of the open end be not itself variable from some caun independent of the motion within the pipe. When there is lU independent source of sound, the pressure at the end of the piptd is the same an it would be in the same place, if the pipe weM away. The impediment to securing the fulfilment of the conditin for a loop at any desired point lies in the inertia of the machineiV required to sustain the pressure. For theoretical purposes we may overlook this difficulty, and imagine a maasleas piston backed by a compressed spring also without mas». The assumption of a loop at an open end of a pipe is tantamount to neglecting tl inertia of the outside air. We have seen that, if a node exist at any point of a pi there must be a series, ranged at equal intervals J\, that mid' between each pair of consecutive nodes there must be a loop, that the whole vibration must be stationary. The same conclusic follows if there be at any point a loop ; but it may perfectly well happen that there are neither nodes nor loops, as for example in the case when the motion reduces to a positive or negative pro- gressive wave. In stationary vibration there is no transference of energy along the tube in either direction, for energy caimot pass a node or a loop. 267. The relations between the lengths of an open or closed pipe and the wave-lengths of the included column of air may aW be investigated by following the motion of a pulse, by which i» understood a wave confined witfiin narrow liniit«« and ct>mpuaed of uniformly condensed or rarefied fluid. In looking at the matter from this pomt of view it is necessary to take into account care- fully the circumstances under which the various rejections take place. Let us first suppose that a condensed pulse travels in the positive direction towards a barrier fixed across the tube. Since the energy contained in the wave cannot escape from the tube, there inust be a reflected wave, and that this reflected wave is also a wave of condensation appears from the fact that there is no loss of fluid. The same conclusion may be arrived at in another way. The efiect of the barrier may be imitated by the introdw tion of a similar and equidistant wave of condensation moviog ij the negative direction. Since the two waves are bftth conder propagated in contrary directions, the \ i e is no mother . viogifl 1 HKbU composing them are equal aud opposite, and therefore neu- H ■ tmlise one another when the waves are superposed. V If the progress of the negative reflected wave be interrupted hy a second barrier, a similar reflection takes place, and the wave, still remaining condensed, regains its positive character. When a distance has been travelled equal to twice the length of the pipe, the original state of things is completely restored, and the same Ugrde of events repeats itself indefinitely. We learn therefore that ) period within a doubly closed pipe is the time occupied by a Ise in travelling twice the length of the pipe. The case of an open end is somewhat different. The supple- entary negative wave necessary to imitate the effect of the open CTid must evidently be a wave of rarefaetioa capable of neutralising ihe positive pressure of the condensed primary wave, and thus in ihe act of reflection a wave changes its character from condensed (o rarefied, or from rarefied to condensed. Another way of con- ■Bdering the matter is to observe that in a positive condensed P^nlse the momentum of the motion is forwards, and in the absence of the necessary forces cannot be changed by the reflection. But forward motion in the reflected negative wave is indissolubly Ctinnecied with the rarefied condition. When both ends of a tube are open, a pulse travelling back- wards and forwards within it is completely restoi-ed to its original litatti after traversing twice the length of the tube, suffering in the process two reflections, and thus the relation between length and I-eriod ia the same as in the case of a tube, whose ends are both iUwed ; but when one end of a tube is open and the other closed, t doable passage is not sufficient to close the cycle of changes. B <mginal condensed or rarefied character cannot be recovered al after two reflections from the open end, and accordingly in the case contemplated the period ia the time required by the pulse 111 travel ovur/oiir times the length of the pipe. After the full discussion of the corresponding problems l>khe chapter on Strings, it will not be necessary to say much on B conpti'und vibratiorw of columns of air. As a simple example lay take the case of a pipe open at one end and closed at the r, which is suddenly brought to rest at the time t = 0, after ome time in motion with a uniform velocity parallel to The initial state of the coatained air ia then oue of 54 PBOBLRM. [258. uniform velocity u, parallel to x, and of freedom from compressioQ and rarefaction. If we suppose that the origin is at the closed end, the general solution is by (7) § 255, ^ = (-4i cos n^t + Bi sin iiit) cos k^ + {A^ cos n^ + ^1 sin ti^) cos k^ + (1), where kr^ir — ^yw/l, Ur^akr, and -4,, £,, A^, jB,... are arbitrary constants. Since ^ is to be zero initially for all values of x, the coeffi- cients B must vanish ; the coefficients A are to be determined by the condition that for all values of x between and I, ^kr Ar sin krX^^ — TlQ (2), where the summation extends to all integral values of r from 1 to 00 . The determination of the coefficients A from (2) is effected in the usual way. Multiplying by mik^dx, and inte- grating from to Z, we get ^IkrAr^-Uo/kr, or ^r^-j^l (3)- The complete solution is therefore . 2t/o^'""*oo8A:^ ,.. * = - X \^i kT- ^^^ W- 269. In the case of a tube stopped at the origin and open at x = l,\etif> = cos nt be the value of the potential at the open end due to an external source of sound. Determining P and in equation (7) § 265, we find cosi'^ = ;, cosnf (1). ^ cosA;^ ^ ^ It appears that the vibration within the tube is a minimum, when cos kl = ± 1, that is when i is a multiple of ^X, in which case there is a node s,t x = L When I is an odd multiple of l\ cos it/ vanishes, and then according to (1) the motion would become infinite. In this case the supposition that the pressure at the open end is independent of what happens within the tube breaks down ; and we can only infer that the vibration is very laigei in "-59.J FORCED VIBRATION. 55 consequeDcci of the isochronisin. Since there is a node a.b a: = 0, tbere iDitst be a loop when a; is an odd multiple uf \\, and we conclude that in the case of isochronism the variation of pressure at the open end of the tube due to the external cause is exactly neutralised by the variation of pressure due to the motion within the tube itself. If there were rfially at the open end a variation of pressure on the whole, the motion must increase without limit in the absence of dissipative forces. If we suppose that the origin is a loop instead of a node, the solution is ..(2), where = cos nt is the given value of tp at the open end x = l. In this case the expression becomes infinite, when kl = vi-tt, or We will next consider the case of a tube, whose ends are both opec and exposed to disturbances of the same period, making ^ equal to He'"', ife"" respectively. Unless the disturbances at the ends are in the same phase, one at least of the coefficients H, K must be complex. Taking the first form iu (3) § 255, we have as the general expreHsiou for ^ ^ = e"" (A cos kr + B sin kx). If we take the origin in the middle of the tube, and assume that the values He"". A'e""' con-espond respectively to x = l, x=—l, we get to determine A and B, H = Acqs kl+B sin kl, K = Acoskl-BsiQkl, P«. wHcl that Um t iTbifl result might also be deduced fi-om (2), if we consider that p required mutioo arises from the supei-position of the motion, which is <lue to the disturbance He"" calculated on the hypothesis that the other end x = — l is a loop, on the motion, which is ! to Ki^^ on the hypothesis that the end x = l is & loop. 56 BOTH ENDS OPEN. [2o9. The vibration expte^sed hy (4) cannot be statioaari/, uiii<Me itm ratio H : Khe real, that is unless the disturbances at the ends be in similar, or in opposite, phases. Hence, except in the case* reserved, there is no loop anywhere, and therefore no place at which a branch tube can be connected along which sound will not be propagated'. At the middle of the tube, for which x = 0, *.|!i|^"' (5), shewing that the variation of pressure (proportional to 0) vanishes if ir+^ = 0, that is, if the disturbances at the ends he equal and in oppoaite phases. Unless this condition be satisfied, the expres- sion becomes infinite when H = ^ (2m + 1) X. At a point distant ^X Irom the middle of the tube the expression for ^ is *'^,^-' («)■ vanishing when H = K, that is, when the disturbances at the ends are equal and in the same phase. In general ^ becomes infinite, when sin kl ~ 0, or 11 = rtiK. If at one end of an unlimited tube there be a variation of pressure due to an external source, a train of progressive waves will be propagated inwards from that end. Thus, if the length along the tube measured from the open end be y, the velocity- potential is expressed by = cos(n(— ny/a), corresponding to ^ E cos nf at y ~ ; so that, if the cause of the disturbance within the tube be the passage of a train of progressive waves across the open end, the intensity within the tube will be the same aa in the space outside. It mu»t not be forgotten that the diameter of the tube is supposed to be infinitely small in comparison with the length of a wave. > An UTUigeiDeDt of this kind hu been propoied by Prot Mi^er (Phil. Mag. XLv. p. 90, 1S7B) for comparing the inteDBities of louroe* of Mond of the luiie pitch. E»ob end of tha tube is eipoaed to the Mtion of one of the ioium* to Iw mmpucd, and the diatuioei are mdjnated nntil the mmplitade* of the vjhntimi dmotad b; H tnd K u« eqiuJ. The branoh tube ii led to a muMmetiia e^aok (1 30i), u>d the method uiiunM tbftt b? Tujing the point of jnaeiion the dWoA- ■noe of Dm flame eu be itopped. Fiom the dieanuioa in the tnt it apfma tkrt 4ii ■iiii>|illiiii li not duontlnaUj MneoL ^.] FORCED VIBRATION OF PISTON. 57 Let U8 next suppose that the source of the motion is withiu the tnbe ileelf. due for example to the inexorable motion of a piston tit the origin'. The constants in (■')) § 255 are to be deterniineti by the conditions that when x = 0, d^jdx = cos nt (say), and that, when x = l, <^ = 0. Thus Aj4= — tanW, kB=\, and the ex- jresaion for ^ is w Tbe in( icosH ^^^' Te motion is a mimrnum. when cosW= + 1, that is, when the length of tbe tube is a multiple of ^X. When i is an odd multiple of \\, the place occupied by the piston would be a notie, if the open end were really a loop, but in this case the solution fails. The escape of energy from the tube prevents the energy from accumulating beyond a certain point ; but no account can be taken of this so long as the open end is treated rigorously as a loop. We shall resume the questiun of resonance after we have considered in greater detail the theory of the open end, when we shall be able to Heal with it more satis- fiiclorily. tbi like manner if the point ir = ^ be a node, insteail of a loop, expression for ^ is , COB k (I ■hi k siu kL («t; ihus the motion is a minimum when I is an odd multiple- of \\, ■hich case the origin is a loop. When i is an even multiple of \\. the origin should be a node, which in forbidden by the condi- tions of the question. In this case accoi-diug to (8) the motion becomes infinite, which means that in the absence of dissipative forces the vibration would increase without limit. 260. The experimental investigation of aerial waves within pipes \ia» been effected with considerable success by Kundt'. To generate waves is easy enough ; but it is not so easy to invent a method by which they can be effectually examined. Kundt dis- covered that the nodes of stationary waves can be made evident by dust. A little tine sand or lycopodiutii seed, shaken over the nor of a fjlaws tube containing a vibrating column of air problema are oonsideted b; Puiuwn, MIm. dc VliutUut, t. ii. p. SOS, ISID. ixv. p. 3ST, imm. f 58 kundt's experiments. [260. disposes itself in recurring patterns, by meaDS of which it is easy to determine the positions of the nodes and to measure the intervals between them. In Kundt's experiments the origin of the sound was in the longitudinal vibration of a glass tube called the sounding-tube, and the dust-figures were formed in a second and larger tube, called the wave-tube, the latter being provided with a moveable stopper for the purpose of adjusting its length. The other end of the wave- tube was fitted with a cork through which the sounding-tube passed half way. By suitable friction the sounding-tube was caused to vibrate in its gravest mode, so that the central point was nodal, and its interior extremity (closed with a cork) excited aerial vibrations in the wave-tube. By means of the stopper the length of the column of air could be adjusted so as to make the vibrations as vigorous as possible, which happens when the interval between the stopper and the end of the sounding-tube is a multiple of half the wave-length of the sound. With this apparatus Kuudt was able to compare the wave- lengths of the same sound in various gases, from which the rela- tive velocities of propagation are at once deducible, but the results were not entirely satisfactory. It was found that the intervals of recurrence of the dust-patterns were not strictly equal, and, what was worse, that the pitch of the sound was not constant from one experiment to another. These defects were traced to a communication of motion to the wave-tube through the cork, by which the dust-figures were disturbed, and the pitch made irregular in consequence of unavoidable variations in the mounting of the apparatus. To obviate them, Kundt replaced the cork, which formed too stiflF a connection between the tubes, by layers of sheet indiarubber tied round with silk, obtaining in this way a flexible and perfectly air-tight joint ; and in order to avoid any risk of the comparison of wave-lengths being vitiated by an alteration of pitch, the apparatus was modified so as to make it possible to excite the two systems of dust-figures simultaneously and in response to the same sound. A collateral advantage of the new method con- sisted in the elimination of temperature-corrections. In the improved " Double Appai-atus " the sounding-tube was caused to vibrate in it^ second mode by friction applied near the middle : and thus the nodes were formed at the points distant from the ends by one-fourth of the length of the tube. At each A.I 2(50.] KUNDT8 KXPERIMENT8. 59 of these poiate connection was made with an independent wave- tube, provided with an adjustable stopper, and with branch tubes and stopcocks suitnbte for adniitting the various gases to be experimented upon. It is evident that dust-figures formed in the two tubes correspond rigorously to the same pitch, and that there- fore a comparison of the intervals of recurrence leads to a correct determination of the velocities of propagation, under the circum- stances of the experiment, for the two gases with which the tubes lire filled. The results at which Kuadt arrived were as follows : — (a) The velocity of sound in a tube diminishes with the diameter. Above a certain diameter, however, the change is not perceptible. (6) The diminution of velocity increases with the wave- length of the tone employed. (c) Powder, scattered in a tube, diminishes the velocity of HOund in narrow tubes, but in wide ones is without effect, <d) In narrow tubes the e£Fect of powder increases, when it i-s very finely divided, and is strongly agitated in consequence. (e) Roughening the interior of a narrow tube, or increasing its sur&ce, diminishes the velocity. (/) In wide tubes these changes of velocity are of no im- purtanve. so that the method may be used in spite of them for exact determinations. (ff) The intluence of the intensity of sound on the velocity cannot bo proved. (&> With the exception of the first, the wave-lengths of a ■iiie aa shewn by dust are not affected by the mode of excitation. (») Jn wide tubes the velocity is independent of pressure, 'Lit iu small tubes the velocity increases with the pressure. (j) All the observed changes in the velocity were due to friction and especially to exchange of heat between the air and the sides of the tube. (k) The velocity of sound at 100" agrees exactly with that given by theory', FltHn Willie eipreSHiims in the roemolt alrernl; cited, from which the naCioe U prindpally derived, Kuudt appeon in h&ve oauteraplated Ul iDTeatigatiouH ; but I am uoable tu find nay later publication ()0 KrXDTS KXrKKIMENTS. [260. We shall return to the (juestion of the propagation of sound in narrow tubes as aflFected by the causes mentioned above (j), and shall then investigate the formulae given by Helmholtz and Kirchhoff. [The genesis of the peculiar transveree striation which forms a leading feature of the dust-figures has already been considered § 2536. According to the observations of DvoMk* the powerful vibrations which occur in a Kundt's tube are accompanied by certain mean motions of the gas. Thus near the walls there is i flow from the loops to the nodes, and in the interior a return flow from the nodes to the loops. This is a consequence of viscosity acting with peculiar advantage upon the parts of the fluid con- tiguous to the wallsl We may perhaps return to this subject in a later chapter.] 261. In the experiments described in the preceding section the aerial vibrations are forced, the pitch being determined by the external source, and not (in any appreciable degree) by the length of the column of air. Indeed, strictly speaking, all sustained vibrations are forced, as it is not in the power of free vibrations to maintain themselves, except in the ideal case when there is absolutely no friction. Nevertheless there is an important prac- tical distinction between the vibrations of a column of air as excited by a longitudinally vibrating rod or by a tuning-fork, and such vibrations as those of the organ-pipe or chemical harmonicon. In the latter cases the pitch of the sound depends principally on the length of the aerial column, the function of the wind or of the flame' being merely to restore the energy lost by friction and by comnmnication to the external air. The air in an organ-pipe is to be considered as a column swinging almost freely, the lower end, across which the wind sweeps, being treated roughly as open, and the upper end as closed, or open, as the case may be. Thus the wave-length of the principal tone of a stopped pipe is four times the length of the pipe ; and, except at the extremities, there is neither node nor loop. The overtones of the pipe are the add ^ Pogff, Ann, t. clvii. p. 61, 1S76. 3 On the Circulation of Air observed in Eundt's Tubes, and on Bome allied Acoustical Problenui, Phil. Trans, vol. clzzv. p. 1, 1884. > The subject of sensitive flames with and without pipes is treated in eon- siderable detail by Prof. Tyndall in his work on Sound; but the meohanies of this class of phenomena is still very imperfectly understood. We shall return to it in a subsequent chapter. EXPERIMENTS OF 8AVART AND KONIG. 61 ■nonics, twelfth, higher third, &c, correspond] og to the various nbdhisions of the column of air. In the case of the twelfth, for (ample, there is a node at the point of trisection nearest to the peo end, and a loop at the other point of trisection midway letween the fin^t and the stopped end of the pipe. In the case of the open organ-pipe both ends are loops, and here must be at least one internal node. The wave-length of the principal tone is twice the length of the pipe, which is divided into two similar parts by a node in the middle. From this we see tb« foundation of the ordinary rule that the pitch of an open pipe 1b the same as that of a stopped pipe of half its length. For reasons t) be more fiilly esplaiued iu a subsequent chapter, connected nith our present imperfect treatment of the open end, the rule is uBiy approximately correct. The open pipe, differing in this re- Sjn'ct from the stopped pipe, is capable of sounding the whole series iif tones forming the harmonic scale founded upon its principal i'lic. In the case of the octave there is a loop at the centre of the :_•<■ and nodes at the points midway between the centre and the n mi ties. Since the frequency of the vibration in a pipe is proportional to the velocity of propagation of sound in the gas with which the ppe is filled, the comparisou of the pitches of the notes obtained ■r^m the same pipe in different gases is an obvious method of ' nnining the velocity of propagation, in cases where the impos- ility of obtaining a sufficiently long column of the gas precludes use of the direct method. Id this application Chtadui with his u»ual sagacity led the way. The subject was resumed at a later date by Dulong' and by Wertheim', who obtained fairly satisfac- tory results. The condition of the air in the interior of an organ-pipe iras investigated experimentally by Savart*, who lowered into the wpe a small stretched membrane on which a little sand was icattbred. In the neighbourhood of a node the sand remained lensibly undisturbed, but, as a loop was approached, it danced with Dore and more vigour. But by far the most striking form of the ■ Beoheralieit <ar lea cltKlears tp^fiqnea des fluidea itutiques. Ann. ile Chim., IX. p. 113. \>fVi. • Ann. dr Chim., S"™ iftie, t. mn, p, *3<. 184^. • Ami. -It Chim.. I. xxtT. p. 66, 1S33. 62 CUBVED PIPE. [262. experiment is that invented by Konig. In this method the vibra- tion is indicated by a small gas flame, fed through a tube which is in communication with a cavity called a manometric capsule This cavity is bounded on one side by a membrane on which the vibrating air acts. As the membrane vibrates, rendering the capacity of the capsule variable, the supply of gas becomes un- steady and the flame intermittent. The period is of course too small for the intermittence to manifest itself as such when the flame is looked at steadily. By shaking the head, or with the aid of a moveable mirror, the resolution into more or less detached images may be efifected ; but even without resolution the altered character of the flame is evident from its general appearance. In the application to organ-pipes, one or more capsules are mounted on a pipe in such a manner that the membranes ai*e in contact with the vibrating column of air ; and the difference in the flame is very marked, according as the associated capsule is situated at a node or at a loop. 263. Hitherto we have supposed the pipe to be straight, but it will readily be anticipated that, when the cross section is small and does not vary in area, straightness is not a matter of impor- tance. Conceive a curved axis of x running along the middle of the pipe, and let the constant section perpendicular to this axis be 8, When the greatest diameter of 8 is very small in comparison with the wave-length of the sound, the velocity-potential ^ becomes nearly invariable over the section; applying Green's theorem to the space bounded by the interior of the pipe and by two cross sections, we get ///v.*.r.s.^0. Now by the general equation of motion ///v.*.r.^.///*.r.i^///*.r-|*/*.fe. and in the limit, when the distance between the sections is made to vanish, so that I 263.] BRANCHED PIPES. 63 -Ii-^wing that 4> depends upon .c in the same way aa if the pipe .1 re straight. By means of equation (1) the vibrations of air in iirvt-d pipiis of unifoTTU section may be easily iuvestigated, and thi; Ffsults are the rigorous consequences of our fiindamental equations I which take no account of friction), when the section is suppose-d to be infinitely amall. In the case of thin tubes such aa would be used in experiment, they suffice at any rate to give a very good representation of what actually happens. 264. We now pass on to the consideration of certain cases of lected tubes. In the accompanying figure AD represents a pipe, which divides at D into two branches DB, DO. At E the branches reunite and form a single tube EF. The sections of the single tubes and of the branches are assumed to be uniform a» well as very small. Fig. 55. In the first instance let us suppose that a positive wave of »rbitniry type is advancing in A. On its arrival at the fork D, it wiW give rise to positive waves in B and C, and, unless a certain I'Liuditiou be satisfied, to a negative reflected wave in A. Let the potential of the positive waves be denoted by/j./«,/p,/being in i-(ich case a function of x— at; and let the reflected wave be F{x+ at). Then the conditions to be satisfied at Z) are first that the pressures shall be the same for the three pipes, and secondly I'lat ihf whole velocity of the fluid in A shall be equal to the sum !j!" the whole velocities of the fluid in B and G. Thus, using \,S.CU) denote the are.as of the sections, we have, § 244, A(fj+F') = Bf, + Cf, I ^ '• B + C-A ^ -B+C^Af' *^'' >-.-('- W^Af- <'>■■ • fommlft, u applied to detennine the reflwted and re&Boted wnvea of seetioQH JV + C, and A renpee lively, are given by 64 BRANCHED PIPES. [264. It appears that/^ and/^ are always the sama There is no reflec- tion, if B + C^A (4), that is, if the combined sections of the branches be equal to the section of the trunk ; and, when this condition is satisfied, /,=/o=/. (5). The wave then advances in B and C exactly as it would have done in A, had there been no break. If the lengths of the branches between D and E be equal, and the section of Fhe equal to that of A , the waves on arrival at E combine into a wave pro- pagated along F, and again there is no reflection. The division of the tube has thus been absolutely without effect ; and since the same would be true for a negative wave passing from F to Ay we may conclude generally that a tube may be divided into two, or more, branches, all of the same length, without in any way influencing the law of aerial vibration, provided that the whole section remain constant. If the lengths of the branches from D to E be imequal, the result is different. Besides the positive wave in Fy there will be in general negative reflected waves in B and C. The most interesting case is when the wave is of harmonic type and one of the branches is longer than the other by a multiple of ^ X. If the difference be an even multiple of J X, the result will be the same as if the branches were of equal length, and no reflection will ensue. But suppose that, while B and C are equal in section, one of them is longer than the other by an odd multiple of ^ X. Since the waves arrive at E in opposite phases, it follows from symmetry that the positive wave in F must vanish, and that the pressure at E, which is necessarily the same for all the tubes, must be constant. The waves in B and C are thus reflected as from an open end. That the conditions of the question are thus satisfied may also be seen by supposing a barrier taken across the tube F in the neighbourhood of E in such a way that the tubes B and C communicate without a change of section. The wave in each tube will then pass on into the other without interruption, and the pressure-variation at E, being the resultant of equal and opposite components, will vanish. This being so, the barrier may be removed without altering the conditions, and no wave will be propagated along F, whatever its section may be. The arrange- PoiBson, Mim. de Plmtitut, t. u. p. 805, 1819. The reader will not forget that both diameten must be small in oompariflon with the wave-length. BRANCHED PIPES. meat ngw under ci>ii8ideration was invented by Herechel, and has been employed by Quincke and others for experimental pui-poses, — an iipplication that we shall afterwai'ds have occasion to describe. The phenomenon itself is often referred to as an example of inter- ference, to which there can be no objection, but the same cannot be said when the reader is led to suppose that the positive waves neutralise each other lu F, and that there the matter ends. It must never be forgotten that there is no loss of energy in interference, but only a different distribution ; when energy is diverted from one place, it reappears in another. In the present case the positive wave in A conveys energy with it. If there is no wave along F, there are two possible alternatives. Either energy accumulates in the branches, or else it passes back along A in the form of a negative wave. In order to see what really happens, let ua trace the progress of the waves reflected back at E. These waves are equal in magnitude and start from E in opposite phases ; in the passage from E to D one has to travel a greater distance than the other by an odd multiple of JX; and therefore on ai-rival at 1) they wiil be in complete accordance. Under these circumstances they combine into a single wave, which travels negatively along A, and there is no reflection. When the negative wave reaches the end of the tube A, or is otherwise dis- turbed in its course, the whole or a part may be reflected, and then the process is repeated. But however often this may happen there »ill be no wave along F. unless by iiccumulation, in consequence of a coincidence of periods, the vibration in the branches becomes so great that a small fraction of it can no longer be neglected. Or we may reason thus. Suppose the tube F cut off by a harrier as before. The motion in the Fin. 56. ring being due to forces acting at D is Qece«Barily symmetrical with respect to D, and 1/ — the point which divides DBCD into equal parts. Hence D' is a node, and the vibration is stationary. nils being the case, at a point E distant 4X tmm D' on either side, there must be V loop ; and if the barrier be removed will still be no tendency to produce mtioit in F. If the perimeter of the ; be a multiple of \, th ere may be R. 11. 66 BRANCHED PIPE& [264. vibration within it of the period in question, independently of any lateral openings. Any combination of connected tabes may be treated in a similar manner. The general Fig. 57. principle is that at any junction a space can be taken large enough to include all the region through ^-^___ which the want of uniformity affects the law of the waves, and yet so small that its longest dimension may be neglected in comparison with X. Under these circumstances the fluid within the space in question may be treated as if the wave-length were infinite, or the fluid itself incompressible, in which case its velocity-potential would satisfy V»^ = 0, following the same laws as electricity. 266. When the section of a pipe is variable, the problem of the vibrations of air within it cannot generally be solved. The case of conical pipes will be treated on a future page. At present we will investigate an approximate expression for the pitch of a nearly cylindrical pipe, taking first the case where both ends are closed The method that will be employed is similar to that used for a string whose density is not quite constant, §§ 91, 140, depending on the principle that the period of a free vibration fulfils the stationaiy condition, and may therefore be calculated fi-om the potential and kinetic energies of any hypothetical motion not departing far fix)m the actual type. In accordance with this plan we shall assume that the velocity normal to any section S is constant over the section, H8 must be very nearly the case when the variation of S is slow. Let X represent the total transfer of fluid at time t across the section at x, reckoned from the equilibrium condition ; then i reproHents the total velocity of the current, and X-r-S represents the actual velocity of thi; particles of fluid, so that the kinetic energy of the motion within the tube is expressed by da: (1). '^'-yjs The potential energy § 24fi (IS) is expressed in general by V~ia*pjfji'dV, WBSb.] TAItlABLE SECTION. 67 or, since dV= Sdx, by F = ia»pjss»<tc (2). Agaio, by the condition of continuity, 1 ^^ -'=3d« W' aod thus ^^-i«■'/s(S)"<^■ (*>• If we uow assume for X an expression of the same form as uld obtain if S were constant, viz. Z = ain -^ co8h( (5), we obtain from the values of T and V in (1) and (4), h'ti^ [' „ TTX die f ■ . wx dx "-"FrTs*v"'Ts («'• or, if we write S = 5i, + ASand neglect the square of AS, "= I' v-''ln--s;T\ '^)- The result may be expressed conveniently in terms of Al, the cor- rvction that must be made to I in order that the pitch may be • filculated from the ordinary formula, as if jS were constant. For :Ke value of Af we have Al.l'««^^%^d^ (8). The effect of a variation of section is greatest near a nodti or near a loop. An enlargement of section in the first case lowers the [litcb, and in the second case raiises it. At the points midway ■iween the nodes and loops a slight variation of section is with- in effect. The pitch is thus decidedly altered by an enlargement ■ r ooutmction near the middle of the tube, but the influence of a -:i|^ht conic&lity would be much less. The cxpressioa for Al given by (8) is applicable aa it stands to I g»iivi«t tone only; but we may apply it to the vi"' tone Kionic scale, if we modify it by the substitution of \ for cos{2-Trj.;'l\ 0—2 ^^ 68 TABIABLB SECTION. [265. In the case of a tube open at both ends (5) is replaced by I which leads to X = co6 -f cosnt (9), . , n iirx AS, .,^, ^0 i o% instead of (8X The pitch of the sound is now raised by an enlargement at the ends, or by ar contraction at the middle, of the tube ; and, as before, it is unaffected by a slight general oonicalitj (§ 281X 266. The case of progressive waves moving in a tube of vari- able section is also interesting. In its general form the problem would be one of great difficulty ; but where the change of section is very gradual, so that no considerable alteration occurs within a distance of a great many wave-lengths, the principle of energy will guide us to an approximate solution. It is not difficult to see that in the case supposed there will be no sensible reflection of the wave at any part of its course, and that therefore the energy of the motion must remain unchanged ^ Now we know, § 245, that for a given area of wave-front, the energy of a train of simple waves is as the square of the amplitude, from which it follows that as the waves advance the amplitude of vibration varies inversely as the square root of the section of the tube. In all other respects the type of vibration remains absolutely unchanged. From these results we may get a general idea of the action of an ear-trumpet. It appears that according to the ordinary approximate equations, there is no limit to the concentration of sound producible in a tube of gradually diminishing section. The same method is applicable, when the density of the medium varies slowly from point to point. For example, the amplitude of a sound-wave moving upwards in the atmosphere may be determined by the condition that the energy remains unchanged. From § 245 it appears that the amplitude is in- versely as the square root of the density'. 1 Phil. Mag. (5) i. p. 261, 1876. ' A delicate question arises as to the ultimate fate of sonorous wayes propagate! upwards. It should be remarked that in rare air the deadening inflwence of Tisoosi^ is much inereased. CHAPTER XIII. SPECIAL PROBLEMS. REFLECTION AND REFRACTION OF PLANE WAVES. 267. Before undertaking the discus!iion of the geueral equa- tions for aerial nbrations we may conveniently turn our attention to a few special problems, relating principally to motion in two dimensions, which are susceptible of rigorous and yet compara- tively simple solution. In this way the reader, to whom the subject is new, will acquire some familiarity with the ideas and tthods employed before attacking more formidable difficulties. In the previous chapter (§ 255) we investigated the vibrations in dimension, which may take place parallel to the axis of a tube, of which both ends are closed. We will now inquire what vibrations are possible within a closed i-ectangular box, dispensing with the restriction that the motion is to be in one dimension only. For each simple vibration of which the system is capable, ^ varies aa a circular function of the time, say co» kat, where k is some constant ; hence ^ = — t^a?^, and therefore by the general differen- tial equation (9) § 244 V'0 + i'i^ = O (1). Equation (1) must be satisfied throughout the whole of the included volume. The surface condition to be satisfied over the I sides of the box is simply ^^Bre<ifl rep ^^Bily for sp< d<t> ■{iX dn represents an element of the normal to the surface. It ly for special values of k that it is possible to satisfy (1) and leously. 70 AERIAL VIBRATIONS Taking three edges which meet as axes of rectangular co-ordi- nates, and supposing that the lengths of the edges are respectively a> /9> % we know (§ 255) that ^l-coifl^p^), * = cos(5^), ^^coe^r^), where p, q, r are integers, are particular solutions of the problem. By any of these forms equation (2) is satisfied, and provided that k be equal to p-rr/a, qir/fi, or nr/7, as the case may be, (1) is also satisfied. It is equally evident that the boundary equation (2) is satisfied over all the surface by the form <^ = coe^jp^jco8^5^jcos^r— j (3), a form which also satisfies (1), if it be taken such that '— (M-?) <♦)• where as before p, 5, r are integers ^ The general solution, obtained by compounding all particular solutions included under (3), is ^sSSS(il cosita^ + ^sinita^) xcos^p^jcos^y^jcos^r^j (5). in which A and B are arbitrary constants, and the summation is extended to all integral values of p, 5, i\ This solution is suflSciently general to cover the case of any initial state of things within the box, not involving molecular rotation. The initial distribution of velocities depends upon the initial value of ^, or /(u^da; + Vocfy + tc'^cl^), and by Fourier's theorem can be represented by (5), suitable values being ascribed to the coefficients A, In like manner an arbitrary initial distribu- tion of condensation (or rarefaction), depending on the initial value of ^, can be represented by ascribing suitable values to the coefficients B. The investigation might be prdsented somewhat differently by commencing with amumiug in aoooitlaiioe with Fourier^a •^^HHMBBi^^B^a vv^^^^^Vv^^V ^f^W^»^^ ^B^^^^^%^ t^^B^ ^^^^ •^p^^^* 267.] rN A RECTANGULAR CHAMBER. 71 theorem that the general value of ^ at time t can be expressed in the form * = SSSCco8 / trxS f ■n-y\ I ■irz\ in which the coefficients G may depend upon (, but not upon X, y, z. The expressions for T and V would then be formed, and »hewa to involve only the squares of the coefficients C, and from the:<e expressions would follow the normal equations of motion connecting each normal co-ordinate C with the time. The gravest mode of vibration is that in which the entire motion is parallel to the longest dimension of the box, and there ia ui) internal node. Thus, if a be the greatest of the three sides a. A 7- *^ ^f* 'o take ;> = 1 , 9 = 0, )■ = 0. In the case of a cubical box, a = ^=7, and then instead of 1 4) we have A^=J'(y+9'+^) (6). iir, if X be the wave-length of plane waves of the same period, >. = 2a^V(p' + 9» + r') (7). For the gravest mode ;) = 1, 5 = 0, r = 0, or p = 0, 5 = 1. r=0, &c., iuid X = 2a. The next gravest is whenp= 1, 5 = 1, r = 0, &c, and then X = V2«. When yi=l, 5 = 1, r=l, X = 2a/V3. For the fourth gravest mode p = 2, ^ = 0, r = 0, &c., and then X = ia. As in the case of the membrane (| 197), when two or more primitive modes have the same period of vibration, other modes of liltf period may be derived by composition. The trebly infinite series of i>o,ssible simple component vibra- tions \a not necessarily completely represented in particular coses uf ootupound vibrations. If, for example, we suppase the contents of the box in its initial condition to be neither condensed nor mreBed in any part, and to have a uniform velocity, whose components pm-allel to the axes of co-ordinates are respectively ti,. r,, u\, no simple ribrations are generated for which : than one of the thi-ee numbers ^, (y, r is finite. In fact each component initial velocity may be considered separately, and the »blem is similar to that solved in § 258. I future chapt«n we shall meet with other examples of the adthin completely closed vessels. 72 NOrra OF NARROW PA88AQEB. [267. Some of the natural notes of the air ocmtained within a room may generally be detected on singing the scale. Probably it is somewhat in this way that blind people are able to estimate the size of rooms\ In long and narrow passages the vibrations parallel to the length are too slow to affect the ear, but notes due to transverse vibrations may often be heard. The relative proportions of the various overtones depend upon the place at which the disturbance is created*. In some cases of this kind the pitch of the vibrations, whose direction is principally transverse, is influenced by the occurrence of longitudinal motion. Suppose, for example, in (3) and (4), that ^ = 1, r = 0, and that a is much greater than fi. For the principal transverse vibration p = 0, and k = ir//3. But besides this there are other modes of vibration in which the motion is principally transverse, obtained by ascribing to p small integral values. Thus, whenp= 1, shewing that the pitch is nearly the same as before*. 268. If we suppose 7 to become infinitely great, the box of the preceding section is transformed into an infinite rectangular tube, whose sides are a and fi. Whatever may be the motion of the air within this tube, its velocity-potential may be expressed by Fourier's theorem in the series 4> = llApgCos^^ cos^ (1), where the coefficients A are independent of x and y. By the use of this form we secure the fulfilment of the boundary condition ^ A remarkable instance is quoted in Young*8 Natural Philo$ophy, 11. p. 272, from Darwin's Zoonomia^ u, 487. ** The late blind Justice Fielding walked for the first time into my room, when he once visited me, and after speaking a few words said, * This room is about 22 feet long, 18 wide, and 12 high * ; all which he gnessed by the ear with great accuracy.*' « Oppel, Die hamumiiehen OherUine de$ durch parallele Wdnde erregten Re- JUxiontUmes, ForUehritU der Phyrik, xx. p. 180. * There is an underground passage in my house in which it is possible, by singing the right note, to excite free vibrations of many seconds* duration, and it often happens that the resonant note is affected with distinct beats. The breadth of the passage is abont 4 feet, and the heifl^t aboat ^ fsei. •268.] RECTANGULAR TUBE. 73 that there is to be no velocity across the sides of the tube ; the nature of A &s a. function of z and t depends upon the other conditioDs of the problem. Let us consider the caee in which the motion at every point is harmonic, and due to a normal motion imposed upon a barrier Satoetching across the tube at a = 0. Assuming to be proportional Ip 8*" at all p^jints, we have the usual differential equation -(2), daf df dz* ^ which by the conjugate property of the functions must be satisfied separately by each terra of (1). Thus to determine Aj^ as a function of z, we get d?A^ ds' ■[^'--•$-f)Wo ..(3). The solution of this equation differs in form according to the sign of the coefficient of Ap^. When p and q are both aero, the coeffi- cient is necessarily positive, but as p and q increase the coefficient changes sign. If the coefficient be positive and be called ;**, the general value of Ap^ may be written = S™ c*'*^*'^ + Cm e** ..(*), where, as the factor e"^ is expressed. Bp^, C^ are absolute constants. However, the first terra in (4) expresses a motion propagated in the negative direction, which is excluded by the conditions of the problera, and thus we are to take simply as the t«rm correspoudiug to p, q. piTX 'IJ^, ^Tn tliL« expression Cj^ may be complex ; passing to real quantities and taking two new real arbitrary constants, we obtain ^fc[i)„cofl(fatf-/w) + £^^sin(A-a(-/i«)]cos^coB^...(5). ^^HWe have now to consider the form of the solution in cases ^^^nre the coefficient of A,^ in (3) is negative. If we call it — t^, ^^^k aoliition corresponding to (4) is «(B„e"+0„8-«).. 74 RECTANGULAR TUBE. [268. of which the first term is to be rejected as becoming infinite with z. We thus obtain corresponding to (5) ^ = e-« [Bpq cos hat + E^q sin kai] cos ^^ cos 2^ (7). The solution obtained by combioing all the particular solutions given by (5) and (7) is the general solution of the problem, and allows of a value of d^jdz over the section z^O, arbitrary at every point in both amplitude and phase. At a great distance from the source the terms given in (7) become insensible, and the motion is represented by the terms of (5) alone. The effect of the terms involving high values of p and q is thus confined to the neighbourhood of the source, and at moderate distances any sudden variations or discontinuities in the motion at ^ » are gradually eased off and obliterated. If we fix our attention on any particular simple mode of vibra- tion (for which p and q do not both vanish), and conceive the frequency of vibration to increase from zero upwards, we see that the effect, at first confined to the neighbourhood of the source, gradually extends further and further and, after a certain value is passed, propagates itself to an infinite distance, the critical frequency being that of the two dimensional free vibrations of the corresponding mode. Below the critical point no work is required to maintain the motion ; above it as much work must be done at z = as is carried off to infinity in the same time. 268 a. If in the general formula; of § 267 we suppose that r = 0, we fall back upon the case of a motion purely two-dimen- sional. The third dimension (7) of the chamber is then a matter of indifference ; and the problem may be supposed to be that of the vibrations of a rectangular plate of air bounded, for example, by two parallel plates of glass, and confined at the rectangular boundary. In this form it has been treated both theoretically and experimentally by Kundt\ The velocity-ix)tential is simply * = cos(p^)cos(5^) (1), where p and q are integers ; and the frequency is determined by *« = 7r« (!>»/«• -h3-/i8») (2). ^ Pogg, Afm. vol. xl. pp. 177, 887» 1878. RECTANCULAR PLATE. If the plate be open at the boundary, an approximate solution may be obtained bj siippoaing that ^ ia there evanescent. In this case the expression for i^ is derived from (1) by writing sines instead of cosines, while the frequency equation retains the same form (2). This has already been discussed under the head of membranes in § 197. If a = ;3, so that the rectangle becomes a square, the various normal modes of the same pitch may be combined, as explained in § 197. In Kundt's experiments the vibrations were excited through a perforation in one of the glass plates, to which was applied the extremity of a suitably tuued rod vibrating longitodinally, and the division into segments was indicated by the behaviour of cork filings. As regards pitch there was a good agreement with calculation in the cose of plates closed at the boundary. When the rectangular boundaiy was opeti, the observed frequencies were too small, a discrepancy to be attributed to the merely approxi- mate character of the assumption that the pressure is there invarisble (see § 307). The theory of the circular plate of air depends upon Bessel's functions, and is considered in § 339. 269. We will now examine the result of the composition of two trains of plane waves of harmonic tj-pe, whose amplitudes and wave-lengths are equal, but whose directions of propagation are inclined to one another at an angle 2a. The problem is one of two dimensions only, inasmuch as everything is the same in planes perpendicular to the lines of intereection of the two sets of wave-fronts. At any moment of time the positions of the planes of maximum ifition for each train of waves may be represented by pa- I Itnea drawn at equal intervals \ on the plane of the paper, t these lines must be supposed to move with a velocity a in a ) perpendicular to their length. If both setst of lines be t, the paper will be divided into a system of equal parallelo- jfl, which advance in the direction of one set of diagonals. At t comer of a parallelogmm the condensation is doubled by the asition of the two trains of waves, and in the centre of each [kllelograra the rarefaction is a maximum for the same reason. ■^Mch diagonal there is therefore a series of maxima and minima I, advancing without change of relative position and 76 TWO EQUAL TRAINS OF WAVES. [269. with velocity a/coaoL Between each adjacent pair of lines of maxima and minima there is a parallel line of zero condensation, on which the two trains of waves neutraliise one another. It is especially remarkable that, if the wave-pattern were visible (like the corresponding water wave-pattern to which the whole of the preceding argument is applicable), it would appear to move for- wards without change of type in a direction different from that of either component train, and with a velocity different firom that with which both component trains move. In order to express the result analytically, let us suppose that the two directions of propagation are equally inclined at an angle a to the axis of x. The condensations themselves may be denoted by cos — {at — w cosa — y sina) and cos —(at-x cos a + y sin a) respectively, and thus the expression for the resultant is 27r iir « = cos — (at — a: cos a — y sin a) + cos — (at — xcosa + y sin a) At At = 2 cos —{at — xcosa) cos — (ysina) (1). It appears from (1) that the distribution of 8 on the plane xy advances parallel to the axis of x, unchanged in type, and with a uniform velocity a/cos a. Considered as depending on y, 8 is a maximum, when y sin a is equal to 0, X, 2X, 3X, &c., while for the intermediate values, viz. ^ X, f X, &c., 8 vanishes. If a = ^ TT, so that the two trains of waves meet one another directly, the velocity of propagation parallel to x becomes infinite, and (1) assumes the form 8—2 cos (— at) cos f— y] (2); which represents 8tatianary waves. The problem that we have just been considering is in reality the same as that of the reflection of a train of plane waves by an infinite plane wall Since the expression on the right-hand side of equation (I) is an even function of y, ^ is symmetrical with respect to the axis of x, and consequently there is no moti<m REFLECTION FROM FIXED \rALL. 77 269.] across that axis. Under these circumstances it is evident that the motion could in no way be altered by the introduction along the axis of X of an absolutely immovable wall. If a be the angle between the surface and the dii-ection of propagation of the inci- dent waves, the velocity with which the places of maximum con- densation (corresponding to the greatest elevation of water-waves) move along the wall is a/cos a. It may be noticed that the aerial pressures have no tendency to move the wall as a whole, except in the case of absolutely perpendicular incidence, since they are at r moment as much negative as positive. 269 a. When sound waves proceeding from a distant source *»re reflected perpendicularly by a solid wall, the superposition of the direct and reflected waves gives rise to a system of nodes and loops, exactly as in the case of a tube considered in § 2.55. The nodal planes, viz. the surfaces of evanescent motion, occur at distances from the wall which are even multiples of the quarter wave length, and the loops bisect the intervals between the nodes. Id exploring experiraeatally it is usually best to seek the places of aiinimum effect, but whether these will be nodes or loops depends upon the apparatus employed, a consideration of which the neglect has led to some confusion'. Thus a resonator will cease to respond when its mouth coincides with a loop, so that H tioB method of experimenting gives the loops whether the ^■Monator be in connection with the ear or with a " manometric Bl^teule" (§ 282). The same conclusion applies also to the use of the unaided ear, except that in this case the head is an obstacle large enough to disturb sensibly the original distribution of the loop and nodes'. If on the other hand the indicating apparatus ^tte a small stretched membrane exposed upon both sides, or a ^HDsitive smoke jet or flame, the places of vanishing disturbance Hpn the nodes*. The complete establishment of stationarj- vibrations with Dodes and loops occupies a certain time during which the sound is to be tnaintained. When a harmonium reed is sounding steadily ID a room free from carpets and curtains, it is easy, listening with a resonator, to hnd places where the principal tone is almost rely subdued. But at the tirst moment of putting down the [ly subdui I N. 8»v » Fhil. J ' Phil, i K. Skv&tt, ^RR, .1. Chim. LIU. p. 20, 1839 : i itari. vn. p. 160, 1879. Fhit. Slnj, he. eit. p. ISA. 78 REFRACTION OF PLANE WAVES. [269 a. key, or immediately after letting it go, the tone in question asserts itself, often with surprising vigour. The formation of stationary nodes and loops in front of a reflecting wall may be turned to good account when it is desired to determine the wave-lengths of aerial vibrations. The method is especially valuable in the case of very acute sounds and of vibrations of frequency so high as to be inaudible. With the aid of a high pressure sensitive ilame vibrations produced by small "bird-calls'* may be traced down to a complete wave-length of 6 mm., corresponding to a frequency of about 55,000 per second. 270. So long as the medium which is the vehicle of sound, continues of unbroken uniformity, plane waves may be propagated in any direction with constant velocity and with tjrpe unchanged ; but a disturbance ensues when the waves reach any part where the mechanical properties of the medium undergo a change. The general problem of the vibrations of a variable medium is probably quite beyond the grasp of our present mathematics, but many of the points of physical interest are raised in the case of plane waves. Let us suppose that the medium is uniform above and below a certain infinite plane (x = 0), but that in crossing that plane there is an abrupt variation in the mechanical properties on which the propagation of sound depends — namely the coniprem- bility and the density. On the upper side of the plane (which for <listinctness of conception we may suppose horizontal) a train of plane waves advances so as to meet it more or less obliquely ; the problem is to determine the (refracted) wave which is propagated onwards within the second medium, and also that thrown back into the first medium, or reflected. We have in the first place to form the equations of motion and to express the boundary conditions. In the upper medium, if p be the natural density and s the <iondensation, density = p (1 -h «), and pressure = P (1 + -4 «), where il is a coefficient depending on the compressibility, and P is the undisturbed pressure. In like manner in the lower medium density = pi (1 + «i), pressure 8P(1 +-4i«i)i .CTION OF PLANE WAVES. 9^B the undisturbed pressure being the same on both sides of ic = 0. Taking the axis of e parallel to the line of intersection of the plane of the waves with the surface of separation a; = 0. we have for the upper medium (§ 24-*), '^=F'f^'* + '^'*^ (1) df ^ \dj?^ df) "^'' and #+Ps = (2). where V^ = PA^p (3). Similarly, in the lower medium, d?<P,_yjd^^ d^4>\ dp-^'[dj^^ df) ^*'' and ^^^- +V;^s, = (5). where K,' = P.d,-=-p, (6). These equations must be satiatieii at all points of the fluid. Further the boundary conditions require (i) that at all poiuts of the sur&ce of sepavation the velocities perpendicular to the sur&ce shall be the same for the two fluids, or d^ldx = d^ildx, when ^■ = (7); (ii) that the pressures shall be the same, whence -4|«i = ^s, or by (2), (3), (5) and (6), p dil>ldt = fhdipildt, when x=0 (8). In order to represent a train of waves of harmonic type, we may assume ^ and 0, to be proportional to £''('"+*''+'*', where at:-¥hy=' const, gives the dii'ection of the plane of the waves. If we assume for the incident wave, = ^' e''i''«-^'*+'='t (9), L^e i-tjflected and refracted waves may be represented respectively P = (^ V'-"'^+*='+'" (10), I 0, = 0,e'>.'+»f+'") (11). The coefficient of t is necessarily the same in all three waves ^^ account of the periodicity, and the coefficient of y must be the ^Htae, since the traces of all the waves on the plane of separation 80 green's investigation [270. must move together. With regard to the coefficient of «, it ap- pears by substitution in the differential equations that its sign is changed in passing fix)m the incident to the reflected wave ; in fact c>=F«[(±a)» + 6»]=F,«[a,«+6»] (12). Now 6 -s- V(a' + 6") is the sine of the angle included between the axis of X and the normal to the plane of the waves — in optical language, the sine of the angle of incidence, and 6 -;- V(^*+ &*) is in like manner the sine of the angle of refraction. If these angles be called 0, 0i, (12) asserts that sin ^ : sin ^i is equal to the con- stant ratio V: Vi, — the well-known law of sines. The laws of refraction and reflection follow simply from the &ct that the velo- city of propagation normal to the wave-fronts is constant in each medium, that is to say, independent of the direction of the wave- front, taken in connection with the equal velocities of the traces of all the waves on the plane of separation (F-T-sin^= Fj-s-sin^x). It remains to satisfy the boundary conditions (7) and (8). These give />(f+f0=/>i^j ^ ^' whence ^'-(^a*i ^"-(?-?)* (")• This completes the sjrmbolical solution. If Oi (and ^i) be real, we see that if the incident wave be ^ s cos (ax + 6y + c<), or in terms of F, X, and 0, * 2*rr ^ = cos — (a?cos ^ + ysin^+ Vt) (15), the reflected wave is pi cot ^1 = P — £^ eos ^(-a?cos ^ + y sin^ + Vt) ... (16), ^ Pi cot^i X p cot ^ and the refracted wave is <Ih = =— 2-co8=^(a?co8^i + ysin^x + FiO...(17). Pi, COttfi Ki P QOt$ 270.] OF REFLECTION AND KEFRACTIOK. The foniiuia for the amplitude of the reflected wave, viz. pi cot 6, <f>" p cot tf ■(18). ^' pi cottf," p cotff is here ohtaioed on the supposition that the waves are of harmonic tj'pe ; but since it does not involve \, and there is no change of pha£e, it may be extended by Fouriei-'s theorem to waves of any type whatever. If there be no reflected wave, cot^i :cot9 = p,: p. from which an<f (1 +cot»^,) : (1 +cot'(9)= 7"' : F;» we deduce (5i-^)'=«*'^=^I-l (19). which Bhewa that, provided the refractive index Fi : F be inter- mediate in value between unity and p : p^ there is always an angle of incidence at which the wave is completely intromitted ; but otherwise there is no such angle. Since (18) is not altered (except as to sign) by an interchange of 0, 6i ; p, pi', &c., we infer that a wave incident in the second medium at an angle ^, is reflected in the same proportion as a wave incident in the firyt medium at an angle 6. Aa a numerical example let us suppose that the upper medium - air at atmospheric pressure, and the lower medium water. >i:betitutiDg for cot^i its value in terms of B and the refractive iijdex, we get o^4V'-(^'-')-^- (»>■ or, »ince F, : F = 4'3 approximately, cot 0,lcote=2S V(l - 17-5 tan' $), which shews that the ratio of cotangents diminishes to zero, as d iitcreases from zero to about 13°, afler which it becomes imaginary, indicating total reflection, as we shall see presently. It must be DDembered that in applying optical tenns to acoustics, it is the r that must be conceived to be the ' rai'e ' medium. The ratio ^densities is about 770 : 1 ; so that " ^ 1- 0003 Vfl- 17-5 tan ' g) ^ ~ 1 + 0003 Vt 1-17-5 tan' 0) = 1 - 0006 V(l - 17o tan> 0) very nearly, iDdiciUar incidence the reflection ia sensibly pel 82 fbbsnel's expressions. [270. If both media be gaseous, Ai^A, if the temperature be con- stant ; and even if the development of heat by compression be taken into account, there will be no sensible difference between A and Ai in the case of the simple gases. Now, if Ai^A, pi : p = sin'tf : sin'^i, and the formula for the intensity of the reflected wave becomes ^ Bin 20 - sin 20^ ^ tan (0 - 0^) .^i \ f "8in2d + sin2dx"tan(d+d0 ^ ^' coinciding with that given by Fresnel for light polarized perpen- dicularly to the plane of incidence. In accordance with Brewster's law the reflection vanishes at the angle of incidence, whose tangent is V/Vi. But, if on the other hand p^ = p, the cause of disturbance being the change of compressibility, we have <^'' ^ tan gi - tan g sin (0^ - 0) . ^x <f>' "" tan ^1 + tan tf " sin (0, + 0) ^ '' agreeing with Fresnel's formula for light polarized in the plane of incidence. In this case the reflected wave does not vanish at any angle of incidence. In general, when ^ = 0, *" = '^' = ?-F/7 + F, <23); SO that there is no reflection, if pj : p = F : F^. In the case of gases V^ : Vi^ = pi : p, and then Suppose, for example, that after perpendicular incidence re- flection takes place at a surface separating air and hydrogen. We have p = 001 276, pi = -00008837 ; whence Vp • Vf>i = 3*800, giving <^" = - -5833 if)'. The ratio of intensities, which is as the square of the amplitudes, is '3402 : 1, so that about one-third part is reflected. If the difference between the two media be very small, and we write Fi= V+BV, (24) becomes ^ = -*-F (2^>- '270.] REFLECTION DUE TO TEMPERATTIRE AND MOISTURE. 83 If the first medium be air at 0° Cent., and the second medium be air at f Cent., V"+ tV = P"V(1 + -00366 () ; so that 070' = --OOO9U The ratio of the ioteosities of the reflected and incident sounds is therefore -83 x 10"* x (= : 1. As another example of the same kind we may take the case in which the first medium is dry air and the second is air of the same temperature saturated with moisture. At 10° Cent, air saturated with moisture is tighter than dry air by about one part in 220,80 that 81^ = ^15^ nearly. Hence we conclude from (25) that the reflected sound is only about one 774,000"" part of the incident sound. From these calculations we see that reflections from warm or moist air must generally be very small, though of course the effect may accumulate by repetition. It must also be remembered that in practice the transition from one state of things to the other would be gradual, and not abrupt, as the present theory supposes. If the spare occupied by the transition amount to a considerable fraction of the wave-length, the reflection would be materially Ifssened. On this account we might expect grave sounds to travel through a heterogeneous medium less freely than acute sounds. The reflection of sound from surfaces separating portions of gas of different densities has engaged the attention of Tyndall, who has devised several striking experiments in illustration of the ■■object'. For example, sound from a high-pitched reed was con- ■jucted through a tin tube towards a sensitive flame, which served ,1- an indicator. By the interposition of a coal-gas flame issuing from an ordinary bat's-wing burner between the tube and the -.joaitive flame, the greater part of the effect could be cut oft Xot only so, but by holding the flame at a suitable angle, the Bound could be reflected through another tube in sufficient quantity to excite a second sensitive flame, which hut for the interposition of the reflecting flame would have remained undisturbed. ^B [The refraction of Sound has been demonstrated experimentally ^R Sondhauss' with the aid of a collodion balloon charged with carbonic add.] ' S«imi, Brd edition, p. 282. 1873. ^H * Fogg. Jnn. t. 85, p. H7S, 1863. PUX. itag. ToL v, p. 73, 1853. 6—2 84 TOTAL REFLBCnON. [270. The preceding expressions (16), (17), (18) hold good in ev&rj case of reflection from a 'denser' medium; but if the velocity of sound be greater in the lower medium, and the angle of incidence exceed the critical angle, Oi becomes imaginary, and the formuls require modification. In the latter case it is impossible that a refracted wave should exist, since, even* if the angle of refraction were 90'', its trace on the plane of separation must necessarily outrun the trace of the incident wave. If — lOi' be written in place of aj, the symbolical equations are Incident wave Reflected wave p a Refracted wave A = ? ^ ^(-ia,'x+by+et) • Pl .Oi ^~ % p a from which by discarding the imaginary parts, we obtain Incident tvave <^ = cos(cw; + 6y + cO (26), Reflected wave <^ = cos(-cw? + 6y + c^ + 26) (27), Refracted wave <l>=77A;piri'^''<^(h + ct+^) (28), [Pi I ^1 y where tan e = - (29). api ^ These formulae indicate total reflection. The disturbance in the second medium is not a wave at all in the ordinary sense, and at a short distance from the surface of separation (x negative) be- comes insensible. Calculating a^' from (12) and expressing it in terms of 6 and X, we find «a' = X\/'^*^-'S ^^^' shewing that the disturbance does not penetrate into the second medium more than a few wave-lengtha 270.'] LAW OF ENERGY VERIFIED. 85 The difference of phase between the reflected and the iucident waves is 2<, where UiTie=^/Jtaa'e-~se-i'8 (31). If the media have the same compressibilities, p : p,= V,' : V', and ' VV V' tan'5-fiec'? (32). Since there is no loss of energy in reflection and refraction, the work transmitted in any time across any area of the front of the incident wave must be equal to the work transmitted in the same time across corresponding areas of the reflected and refracted waves. These corresponding areas are plainly in the ratio and thus by § 245 (t being the same for all the waves), cos (9 ^ (^'' - 0">) = cos 0i Y 4>i\ or since P" : Fi = sin 5 : sin 6, , p cot ^(^'^ -<#."=) = /!, cot ^i^i' (33),' which is the energy condition, and agrees with the result of multi- plying together the two boundary equations (13). When the velocity of propagation is greater in the lower than in the upper medium, and the angle of incidence exceeds the critical angle, no energy is transmitted into the second medium ; in other words the reflection is total. The method of the present investigation ia substantially the same as that employed by Green in a paper on the Reflection and Refraction of Sound'. The case of perpendicular incidence was first investigated by Poiason', who obtained formula corresponding p (23) and (2*). which hatl however been already given by Young ' the reflection of Light. In a subsequent memoir' Poisson idered the general case of oblique incidence, limiting himself, rcTcr, to gaseous media for which Boyle's law holds good, and I very complicated analysis anived at a result equivalent to Aridgt Tramaetiom, vol. vi. p. 403, 1^38. L df l-IiutUut. t. a. p. 305, ISIO. t le moonmeat de deux flnidw Alaatiquu mipeiposiB." iUm. 86 BEFLECnON FROM A (21). He also verified that the eoergies of the reflected and re- fracted waves make up that of the incidcDt wave^ 271. If the secood medium be indefinitely extended down- wards with complete uniformity in its mechanical properties, the transmitted wave is propagated onwards continually. But if at a;ss — 2 there be a further change in the compressibility, or density, or both, part of the wave will be thrown back, and on arrival at the first surface (^ = 0) will be divided into two parts, one trans- mitted into the first medium, and one reflected back, to be again divided at ^ » — ^, and so on. By following the progress of these waves the solution of the problem may be obtained, the resultant reflected and transmitted waves being compounded of an infinite convergent series of components, all parallel and harmonic. This is the method usually adopted in Optics for the corresponding problem, and is quite rigorous, though perhaps not always 8uf- ficiently explained ; but it does not appear to have any advantage over a more straightforward analysis. In the following investi- gation we shall confine ourselves to the case where the third medium is similar in its properties to the first medium. In the first medium In the second medium In the third medium with the conditions c«=F»(a« + 6«)=Fi«(ai« + 6») (1). At the two surfaces of separation we have to secure the equality of normal motions and pressures ; for a? = 0, f>(*' + f ') = />! (^' + ^")i (2); for a; = — I, -iozr (3), ^ [It is interesting and encouraging to note Laplace's remark in a correepondence with T. Young. The great analyst writes (1817) **Je persiste 4 oroire que le probldme de la propagation des ondes, lorsqu'elles traversent di£f6rens milieux, n*a jamais «t6 r^aolii, et qu'n surpaaae peul-Mre lee forces actoelles de ramUyie*' (Young^ ITorib, vol x. p. 874).] 271.] PLATE OF FINITE THICKNESS, from which ^' and ■^" are to be eliminated. We get (tf,' — ^") cos tt,i - 1 -^ (0' + if)") siau,l = <t>,e-*^ {<!>' + 4>") COB a,l-i^ (0' - 0") sin a^l = *,<r ■''^ and from these, if for brevity ap,/a,p = a, A" a—tr' -w. a + a'' — 2t cot a,r* 2«" ••(5). ..(6). 4>' 2 COS Oi^ + 1 sin Oi i (a + or') In order to pass to real quantities, these expressions must be put into the fonn Re'*. If a, he real, we find corresponding to f incident wave B reflected wave (a~' — a) sin (~ ax + bt/ + ct — e) -m. (8). (9). !, and the V{*cDt'Oii + (K + a"')'} the transmitted wave , _ _ _ 2 COB {a^ + by + ct+ai'~e) VI* C03'ai( + 8in'a]i (a + «'')*! where tan * = i (a + a~' ) tan a,l If a = pi cot OjpQot ^1 = 1, there is no reflected \ transmitted wave is represented by <^ = cos iam + Jy + c( + ai — a, 0. shewing that, except for the alteration of phase, the whole of the medium might as well have been uniform. If i be small, we have approximately for the reflected wave ^ = i a, i (ar' - a) sin (— ax + by + ct), i\ formula applying when the plate is thin in comparison with the wave-length. Since a, = {'lir/X,) cos 8, , it appears that for a given angle of incidence the amplitude varies inversely as \i, ot asX. In any case the refiection vanishes, if cot'a,! =x , that is, if 2fcos^, = mX,. fjt. The wave is then wholly transmitted. 88 REFLECTION FROM A PLATE [271. At perpendicular incidence, the intensity of the reflection is expressed by {^.-^h-J^^n^i^^y o»> Let us now suppose that the second medium is incompressible, so that Fi = 30 ; our expression becomes ^[i + ir^ifHi/pxy] ^''^' shewing how the amount of reflection depends upon the relative masses of such quantities of the media as have volumes in the ratio of 2 : X. It is obvious that the second medium behaves like a rigid body and acts only in virtue of its inertia. If this be suf- ficient, the reflection may become sensibly total. We have now to consider the case in which Oi is imaginary. In the symbolical expressions (5) and (6) cosoi^ and tsina,{are real, while a, a + a"^ a — a~^ are pure imaginaries. Thus, if we suppose that ai = ia/, a = ia\ and introduce the notation of the hyperbolic sine and cosine (§ 170), we get V 2 cosha,7 - i (a' - a'-^ sinh a,'l ' ^_ 2i^ <^' " 2 cosh a/i -i (a' - a^-^ ^^^^ (hi ' Hence, if the incident wave be ^ = cos {ax + 11/ + ct), the reflected wave is expressed by - _ (a' + a'~0 sinh a,7 cos (—cuc + by + ct + e) ,- ^. ^ V{4 co8h»a,7 + (a' - a'-^ sinh«a,7} ^ ^' where cot e = ^ (a'~^ — a') tanh a^'l (13), and the transmitted wave is expressed by _ 2sin(aj? + 6y + c^ + aZ + e) . "^ - V{4^h^a;7 + (a' - a'-^* sinh* a//} ^^*^- It is easy to verify that the energies of the reflected and transmitted waves account for the whole energy of the incident wave. Since in the present case the corresponding areas of wave- front are equal for all three waves, it is only necessary to add the squares of the amplitudes given in equations (7), (8), or in equa- tions (12), (14). OF FINITE THICKNESS. 89 272. These calculatioue of reflection and refi-action under various circumstances might be carried further, but their interest would be rather optical than acoustical. It is important to hear in mind that no energy is destroyed by any number of refiections and refractions, whether partial or total, what is lost in one direc- 1 always reappearing in another. 1 account of the great difference of densities reflection is ijly nearly total at the boundary between air and any solid or iqnid matter. Sounds produced in air are not easily communi- cated to water, and vice verad sounds, whose origin is under water, are heard with difficulty in air. A beam of wood, or a metallic wire, acts like a speaking tube, conveying sounds to considerable distances with very little loss. 172 a. In preceding sections the surface of separation, at 1 reflection takes place, is supposed to be absolutely plane. p of interest, both from an acoustical and from au optical point «f view, to inquii'e what effect would be produced by roughnesses, or corrugations, in the reflecting surface; and the problem thus presented may be solved without difficulty to a certain extent by the method of § 268, especially if we limit ourselves to the case of perpendicular incidence. The equation of the reflecting surface fbe supposed to be e = f, where f is a periodic function of x e mean value is zero. As a particular case we Tnay take f = c cos pa; { I ) ; 3 general we should have to supplement the first term of the series expressed in (1) by cosines and sines of the multiples of px. de velocity- potential of the incident wave (of amplitude unity) e written = e'*«^+'' (2). Bop the regularly reflected wave we have </> = A„e~''", the time r being dropped for the sake of brevity ; but to this must be I tenDB in cosjxr, cos 2^, &c. Thus, as the complete value a the upper medium, ^ + A^e-^'" + AiB-'-'^' w^px -^ A^e-''^'' QQs2p.v + (3), bi«h ^ = li?-p', /4' = A>-V. (4). |he HX^^ssion (3), in which for simplicity sines of multiples 1 from the first, would be sufficiently 90 REFLECTION FROM [2720. general even though cosines of multiples of px accompanied ccoBpx in (1). As explained in § 268, much turns upon whether the quanti- ties /ii, /is,... are real or imaginary. In the latter case the corresponding terms are sensible only in the neighbourhood of z = 0. If all the values of fi be imaginary, as happens when p>k, the reflected wave soon reduces itself to its first term. For any real value of /a, say /l^, the corresponding part of the velocity-potential is representing plane waves inclined to ^ at angles whose sines are ±rplk. These are known in Optics as the spectra of the rth order. When the wave-length of the corrugation is less than that of the vibration, there are no lateral spectra. In the lower medium we have <f>i = BoB^^' + B^e^^' cos px + B^e^' cos ipx -h (5), where A^'* = *i* -;>', /*>'* = Ai' - 4p*, (6). In each exponential the coefficient of ^ is to be taken positive; if it be imaginary, because the wave is propagated in the negative direction; if it be real, because the disturbance must decrease, and not increase, in penetrating the second medium. The conditions to be satisfied at the boundary are (§ 270) that p<f>-pi<f>i (7), and that cUfy/dn = d<f>i/dn, where dn is perpendicular to the surftu^e z=^. Hence dz dx dx Thus far there is no limitation upon either the amplitude (c) or the wave-length (iirlp) of the corrugation. We will now suppose that the wave-length is very large, so that p* may be neglected throughout. Under these conditions, (8) reduces to d{<l>-<tH)ldz = (9). In the differentiation of (3) and (5) with respect to s, the rarioua terms are multiplied by \.\xe co^flic\«cto ^k^ fk«**«Mt'i /%'»••.; 272 a.] A CORRUGATED SURFACE. 91 but when p* is neglected these quantities may be identified with k, k, respectively. Thus at the boundary '^£=ik{e^i-A,e-'*i-A^e-^i cos pa;- I; aud '^ = ik,^,JM^^ ds ^ p. In' (7), Accordingly, k,p{e'^ + A, e-'*^ + Ait- '^ cos px + j = kp,{e'^-A,e-'^i-A,e-'^icoaptc- }, or 'l^^^^'^^ + A, + A,coap!x^ + A,coe2pj!+... = (10). By this equation A,, A^ &c. are determined when ^ is known. If we put f = 0, we fall back on previous results (23) § 270 for a truly plane surface. Thus j4,, A„... vanish, while ift-fcjp • kp, + k,p ^ '' expressing the amplitude of the wave regularly reflected. We will now apply (10) to the case of a simple corrugation, as expressed in (1), and for bi-evity we will denote the right hand member of (11) by R. The determination of A^, A„... requires the expression of e**** in Fourier's series. We have (compare 1343) ■*">"»- = J, (2*c) - 27, (2ic) cos ipx + ZJ, (Uc) cos 4 jaa: 4- . . . t[2*/",(2A'c)co3pa; — 2/,(2^c)co3 Zpx + 2 J^ (ike) cos 5 px — ...} (12), ; Jt, J,.... are the Bessel's functions of the various orders. A,!R= J,{2kc). A,/R= 2tJ,(2A;c), ^^it = -2/,(2fa;), A,/R=-2iJ,{2kc). > AJR= 2/,(2A-c). ^JR= 2iV.(atc), ' ^ '" the coefficients of even order being real, and those of odd order pure imaginaries. The complete solution of the problem of I'fCdectioD, under the restriction that p is small, is then obtained If snbetitution in (3); aud it may be remarked that it is the same would be furnished by the usual optical methods, which take ' of phage retardations. Thus. i\a xfega.tA» ^\\e "«wjft 92 CASE WHERE THE SECOND [272 a. reflected parallel to z, the retardation at any point of the sur&ce due to the corrugation is 2^, or 2cco8^. The influence of the corrugations is therefore to change the amplitude of the reflected vibration in the ratio /cos {2kc cospx)dx : fdx, or Jo(2kc). In like manner the amplitude of each of the lateral spectra of the first order is J^ (2A:c), and so on. The sum of the intensities of all the reflected waves is i2«{/o» + 2/i« + 2j;« + ...}=i? (14) by a known theorem ; so that, in the case supposed (ptp infinitely small), the fraction of the whole energy thrown back is the same as if the surface were smooth. It should be remarked that in this theory there is no limitation upon the value of 2kc. If 2kc be small, only the earlier terms of the series are sensible, the Bessel's function Jn(2kc) being of order (2ic)'*. When on the other hand 2 Arc is large, the early terms are small, while the series is less convergent. The values of Jo and Ji are tabulated in § 200. For certain values of 2kc individual reflected waves vanish. In the case of the regularly reflected wave, or spectrum of zero order, this first occurs when 2fe = 2*404, § 206, or c = •2X. The full solution of the problem of the present section would require the determination of the reflection when k is given for all values of c and for all values of p. We have considered the case of p infinitely small, and we shall presently deal with the case where |)>i. For intermediate values of p the problem is more difficult, and in considering them we shall limit ourselves to the simpler boundary conditions which obtain when no energy pene- trates the second medium. The simplest case of all arises when Pi = 0, so that the boundary equation (7) reduces to <^ = (16), the condition for an " open end," § 256. We may also refer to the case of a rigid wall, or "closed" end, where the surface condi- tion is d<f>/dn^O (16). By (3) and (15) the condition to be satisfied at the sorfiM^ is e^-^Ac + Ai^*-*^^* co82)x + A»e^*-»^* w»^p«4p ...-©•..(16X 1^2 Ct.] MEDIUM IS IMPENETRABLE. 93 I In our problem t is given by (1) as a function of x\ and the ■qnatioQs of condition are to be found by equating to zero the DOeSicients of the various terms involving CQ&j>x, cos 'i.px, &c., when the left hand member of (16) is expanded in Fourier's series, The development of the various exponentials ia effected as in (12); and the resulting equations are H J,(2fc) + vl, + iJ,y,{/t-/^)-^,/,(A--^)-...=0...(17), H 2iV,(2fc) + J4Jp(i--M,)-/,a--^,)l ^ft +J,[»V,(t-rt)-tV,(i-^)l + ...=0 (18), ^ +^,|/.(A--M,) + /,{fc-^)}+... = (19). Lind BO on, where for the sake of brevity c has been made equal to unity. So far as (k — fi) may be treated as real, as happens for a large number of terms when p is small relatively to k, the various Besael's functions are all real, and thus the A's of even order are real and the A's of odd order are pure imaginaries. Accordingly ihe phaite of the perpendicularly reflected wave is the same as if (- = 0; but it must be remembered that this conclusion is in reality only approximate, because, however small p may be, the /j/s end by becoming imaginary. From the above equatioua it is easy to obtain the value of A^ aa far as the term in p*. From (19) A, = 2J^('2k); from (18) iA, = 2 J, (2k) + (k - n,)J, (2k): finally from (17) -A, = J,(2k) + {k-^)J,m) + !i(fc~^,)(i--^.)-i(t-/4)'K.(2i)+ (-20). From (4) ''-'"-a+ &+■■■■■ from iO ihaC, as expandi-'d in powers of p with re introduction of c, ~A, = J,(2kc)+^.^kc.J,(2kc) +^[^kc.J,{2kc)-^k'd'.J,(2kc)] (21)'. > Srit. Asi. Rep. 189S, p. 691. 94 FIXED WALL. [272 a. This gives the amplitude of the perpendicularly reflected wave, with omission of p* and higher powers of p. The case of reflection from a fixed wall is a little more compli- cated. By (8) the boundary condition is d<f>ldz +pc siapx. d^/dx^O^ which gives 6^ - A - ^ -4i«*<*"''»^' cos px - ^ -4,e<<*-'^' cos 2pj: ~ . . . (22) as the equation to be satisfied when z^c cospx. The first approxi- mation to A I gives A,=^2iJ,{2kc) (23); whence to a second approximation A = /o(2A:c) + {-KA:-/ii)+^}i^i = /o(2A:c)-^.A:c./i(2i-c) (24). The first approximation to the various coefficients may be found by putting -K = -|-l in (13). When p>k, there are no diflfracted spectra, and the whole energy of the wave incident upon an impenetrable medium must be represented in the wave directly reflected. The modulus of Ao is therefore unity. When p<k, the energy is divided between the various spectra, including that of zero order. There is thus a relation between the squares of the moduli of -4o, Ai, -4,, ..., the series being continued as long as ^ is real. A more analytical investigation may be based upon v. Helm- holtz*s theorem (§ 293), according to which where 8 is any closed surface, and y^ and x satisfy the equation In order to apply this we take for '^ and x ^he real and ENERGY EQUATION. 95 imaginaiy parts respectively of <^ as given by (3). Thus repre- senting each complex coefficient A^ in the form C„ + iD„, we get ijf = cos A'z + Cicoa A'i + i), sin ii + {CiC0SfLti + D,sinfi,2)cospj:+ (25), ;( = siQii — Cjsin kt + D^coskz +(— Cisin/i]2+iJjCos/ii«)coa^aj+ (26), In (25 ), (26), when the series are carried sufficiently tar, the WTiuf change their form on account of fi becoming imaginary ; but for the present purpose these terms will not be required, as they disappear when i is very great. The surface of integration N is made up of the reflecting surface and of a plane parallel to it jI ii great distance. Although this surface is not strictly closed, ;; may be treated as snch, since the part still remaining open laterally at infinity does not contribute sensibly to the result. Now the part of the integral corresponding to the reflecting rarfece vaDishes, either because Lir el^ because diff-jdn = dx'dn = : iiuil we conclude that when z is great [u^x.^m rf;r = . ..(27). The application of (27) to the values of ^ and x in (25), (26) u«l,, i' + W + S(C,' + £,')+■ = 1 . ..(2S), ibe series in (28) being continued so far as to include every real valae o{ ft. In (28) J (C„' + D„') represents the intensity of each spectrum of the nth order. The coefficient ii„/k is equal to cos &„, where 0„ is the obliquity of the diffracted raj-s. The meaning of this factor will be eWdent when it is remarked that to each unit of area of the waves iocident and directly reflected, there corresponds an area cos ^„ of the waves which constitute the spectrum of the nth order. If all the values of /j, are imaginary, as happens when p>k, (28) reduces to C„' + A'=l (29), or the intensity of the wave directly reflected is unity. It is of 96 OBLIQUE INCIDENCE. [272 a. importance to notice the fall significance of this result. However deep the corrugations may be, if only they are periodic in a period less than the wave-length of the vibration, the regular reflection is total. An extremely rough wall will thus reflect sound waves of moderate pitch as well as if it were theoretically smooth. The above investigation is limited to the case where the second medium is impenetrable, so that the whole energy of the incident wave is thrown back in the regularly reflected wave and in the diflracted spectra. It is an interesting question whether the conclusion that corrugations of period less than X have no efiect can be extended so as to apply when there is a wave regularly transmitted. *^'is. evident that the principle of energy does not suffice to decide the question, but it is probable that the answer should be in the negative. If we suppose the corrugations of given period to become very deep and involved, it would seem that the condition of things would at last approach that of a veiy gradual transition between the media, in which case (§ 148 b) the reflection tends to vanish. Our limits will not allow us to treat at length the problem of oblique incidence upon a corrugated surface; but one or two remarks may be made. If p* may be neglected, the solution corresponding to (13) is Ao=^R Jo (2kc cos 0) (30), being the angle of incidence and reflection, and R the value of Aot § 270, corresponding to c = 0. The factor expressing the effect of the corrugations is thus a function of c cos ; so that a deep corrugation when is large may have the same effect as a shallow one when is small. Whatever be the angle of incidence, there are no reflected spectra (except of zero order) when the wave-length of the corrugation is less than the half of ihat of the vibrations. Hence, if the second medium be impenetrable, the regular reflection under the above condition is total. The reader who wishes to pursue the study of the theory of gratings is referred to treatises on optics, and to papers by the Author^ and by Prof. Rowlands ^ The MannfiAotare and Theozy of Diffraction Gratings, Phil, Mag, vol. zlvii. pp. Sly 198, 1S74 ; On Copying Dilbaotion Gratinge, and on some Phenomena eon- neoted therewith, PkiL Mag. vol. zz. p. 196, 18S1 ; Ene. Brit. Wave Theory of U^U > Grating! In Theory and Pnelioe, PkiU Mag. yd. nxy. p. 897, 1898. I J CHAPTER XrV. GENERAL EQUATIONS. In counection with the general problem of aerial ratious in three dimensiooB one of the first questiona, which / offers itself, is the determination of the motion in an nitod atmosphere consequent upon arbitrary initial dis- ces. It will be assumed that the disturbance is evvill, so e ordinary approximate equations are applicable, and further e initial velocities are such as can be derived from a velocity- intial, or (§ 240) that there is no circulation. If the latter con- a be violated, the problem is one of vortex motion, on which D not enter. We shall alao suppose in the tirst place that no forces act upon the fluid, su that the motion to be tigated is due solely to a disturbance actually existing at J (( = 0), previous to which we do not push our inquiries. ) method that we shall employ is not very different from that if Poisson', by whom the problem was first Bucceasfully attacked. If u,, Vg, w, be tbe initial velocities at the point ir, y, z, and «„ ht initial condensation, we have (§ 244), -/' ^ = (ii.ii» + «,% + w.iz) . (1), *.--«■«. (2). 7 which the initial values of the velocity-potential ^ and of its I differential coeflScient with respect to time ^ are determined, I The problem before us is to determine ^ at time ( from the above ' Sot I'int^ration de qnelques iqustjons liniairea am diffirence* partieUea, ' [»rtieuljf>r«nieiit de t'^astion gindrale da tnouvemeiit des fluidea ^IsBtiquei. '■I., fir rintlihtr. t. in. p. 121. 1B20. 7 98 ARBITBABT INITIAL DI8TUBBANGE. [273. initial values, and the general equation applicable at all times azid places, (*-a«V.)* = (3). When 4> is known, its derivatives give the component velocities at any point. The symbolical solution of (3) may be written if> = mi (iaVt).e + COS {iaVt).x (4), where 6 and x *re two arbitrary functions of x, y, z and i = V(" !)• To connect and x ^^^^ ^^^ initial values of ^ and ^, which we shall denote by / and F respectively, it is only necessary to observe that when f = 0, (4) gives so that our result may be expressed /• Tj^K ^. sin(iaVO o /-x = cos(iaV^)./+ — : ^ \ F (5), in which equation the question of the int.erpretation of odd powers of V need not be considered, as both the symbolic functions are wholly even. In the case where <f> was a function of x only, we saw (§ 245) that its value for any point x at time t depended on the initial values of <f> and <^ at the points whose co-ordinates were x — at and X -f at, and was wholly independent of the initial circumstances at all other points. In the present case the simplest supposition open to us is that the value of ^ at a point depends on the initial values of <f> and <^ at points sititated on the surface of the sphere, whose centre is and radius at ; and, as there can be no reason for giving one direction a preference over another, we are thus led to investigate the expression for the mean value of a function over a spherical surface in terms of the successive diflTer- ential coeflScients of the function at the centre. By the symbolical form of Maclaurin's theorem the value of F{x, y, z) at any point P on the surface of the sphere of radius r may be written F{x, y, z) = e'^^'^*^'^* .F(xo. yo, z,\ the centre of the sphere being the origin of oo-ordinatea Xq ITRABY INITIAL DISTURBANCE. the integratioD over the surface of the sphere rf/(ir„, d/rfy„, d/dz, behave as constants ; we may denote them temporarily by I, m, n, 80 that V' = /' + m" + n'. Thus, r being the radius of the sphere, and dS an element of its surface, since, by the symmetry of the sphere, we may replace any function of —^ — ^- ^ by the same function of e without altering the result of the integration, = jfe"dS = Sttt rV'rfi =^-'' (e" The mean value of F over the surface of the sphere of i-adius r ia thus expressed by the result of the operation on F of the symbol siu (jVr)/»Vr, or, if Jjdo- denote integration with respect to augular space, By comparison with (5) we now see that so f on the initial values of ^, it is expressed by (6). <f> depends * = Sr//^ <"'>''' <'>• or in words, <f> at any point at time ( is the mean of the initial values of <j) over the surface of the sphei-e described round the point in question with radius at, the whole multiplied by (, By Stokes' rule (§ 95), or by simple inspection of (5), we see that the part of depending on the initial values of <f> may be derived from that just written by differentiating with respect to t and changing the arbitrary function. The complete value of <f> at time ( is therefore ,jJF(«i)d,+ l-J^tjj/(at)d, (8), which is Poisson's result '. On account of the importance of the present problem, it may ' Anotluir inveBtJgation will be foniid in KirclihoS'B VorUiungen llher Uatht- maxinke Pkyrik, p. 317. 1S76. [See &!■□ Note to g 27B at the end of Eliia volume.] ^'i.VV^'d 100 VBRIFICATION OF SOLUTION. [273. be well to verify the solution a posteriori. We have first to prove that it satisfies the general differential equation (SX Taking for the present the first term only, and bearing in mind the general symbolic equation d» _1 d d dt'^'ldi^di ^^^' yre find fix)m (8) dS being the surface element of the sphere r » at But by Green 8 theorem and thus Now IjV^Fda' is the same as V» jJFdc; and thus (3) is in feet satisfied. Since the second part of ^ is obtained from the first by differen- tiation, it also must satisfy the fundamental equation. With respect to the initial conditions we see that when t is made equal to zero in (8), ^ = ^///(a<)d«r« = 0)=/(0); 4> = :^//^(«<) ^'^ (« = 0) + 4^ S *//-^^"*^ '^ ^^ ^ ^^' of which the first term becomes in the limit F{0). When ^ = 0, J, tjjfiat) da = 2 jjjf(at) d<r (t = 0) = 2ajjf (at) da (^ = 0) = 0, since the oppositely situated elements cancel in the limit, when the i*adius of the spherical surface is indefinitely diminished. The expression in (8) therefore satisfies the prescribed initial oon* ditions as well as the general differential equation. 274i If the inilial disturbance be confined to a space T, the tntegrals in (8) § 273 are zero, uuless some part of the surface of the sphere r = at be included within T. Let be a point external to T, r, and r, the radii of the least and greatest spheres described about which cut it. Then so long as ti( < rj, ^ remains equal to zero. When at lies between r, and r,, may be finite, but for values greater than r, i^ is again zero. The disturbance is thus at any moment confined to those parts of space for which at is inter- mediate between rj and j-,. The limit of the wave is the envelope of spheres with radius at, whose centres are situated on the surface of T. " When t is small, this ' system of spheres will have an exterior envelope of two sheets, the outer of these sheets being ffltterior, and the inner interior to the shell formed by the as- wmbtage of the spheres. The outer sheet forma the outer limit to the portion of the medium in which the dilatation is different from zero. As ( increases, the inner sheet contracts, and at last its opposite sides cross, and it changes its character from being ex- tenor, with reference to the spheres, to interior. It then expands, and forms the inner boundary of the shell in which the wave of condensation is comprised'." The successive positions of the boundaries of the wave are thus a series of parallel surfaces, and '?ach boundary is propagated normally with a velocity equal to a. If at the time ( = there be no motion, so that the initial tli^lurbance consists merely in a variation of density, the subse- ijueut condition of things is expressed by the first term of (8) § 273. Let us suppose that the original disturbance, still limited to a tioite region T, consists of condensation only, \vithout rarefaction. h might be thought that the same peculiarity would attach to the resulting wave throughout the whole of its subsequent course; but, as Prof Stokes has remarked, such a conclusion would be erroneous. For values of the time less than rja the potential at is zero ; it then becomes negative (a„ being positive), and continues nega- tive until it vanishes again when ( = rja. after which it always remains equal to zero. While is diminishing, the medium at is in a state of condensation, but as i^ increases again to zero, the state of the medium at is one of rarefaction. The wave propa- gated outwards consists therefore of two parts at least, of which the first is condensed and the last rarefied. Whatever may be the character of the original disturbance within T, the final value of <p ' Stokes, "Dynamioal Tliaory of Diffraction," 102 CASE OF PLANE WAVES. [274. at any external point is the same as the initial value, said there- fore, since a*8 » — ^, the mean condensation during the passage of the wave, depending on the integral fsdt, is zero. Under the head of spherical waves we shall have occasion to return to tius subject (§ 279). The general solution embodied in (8) § 273 must of course embrace the particular case of plane waves, but a few words on this application may not be superfluous, for it might appear at first sight that the effect at a given point of a disturbance initially confined to a slice of the medium enclosed between two parallel planes would not pass off in any finite time, as we know it ought to do. Let us suppose for simplicity that ^ is zero throughout, and that within the slice in question the initial value ^ is constant. From the theory of plane waves we know that at any arbitrary point the disturbance will finally cease after the lapse of a time t, such that a^ is equal to the distance (d) of the point under consideration fix)m the further boundary of the initially disturbed region; while on the other hand, since the sphere of radius at continues to cut the region, it would appear from the general formula that the disturbance continues. It is true indeed that <f> remains finite, but this is not inconsistent with rest It will in fjEU^t appear on examination that the mean value of ^ multiplied by the radius of the sphere is the same whatever may be the position and size of the sphere, provided only that it cut completely through the region of original disturbance. If at>d, <f) is thus constant with respect both to space and time, and accordingly the medium is at rest. [The same principles may find an application to the phenomena of thunder. Along the path of the lightning we may perhaps suppose that the generation of heat is uniform, equivalent to a uniform initial distribution of condensation. It appears that the value of ^ at the point of observation can change rapidly only when the sphere r = at meets the path of the discharge at its extremities or very obliquely.] 276. In two dimensions, when ^ is independent of z, it might be supposed that the corresponding formula would be obtained by simply substituting for the sphere of radius at the circle of equal radius. This, however, b not the case. It may be proved that i] TWO DIMENSIONS. 103 the laeaa value of a iunction F(ie, y) over the circumference of a circle of radius r IB i/'„(iVV)f„, where i = V(— 1). and J„ is Beead's function of zero order; so that ■ differiag from what is required to satisfy the fundamental equation. The correct result applicable to two dimensions may be obtained from the general formula. The element of spherical surface dS m«y be replaced by rdrd$/cnsy}r, where r, 6 are plane polar co-ordinates, and ^ is the angle between the tangent plane and that in which the motion takes place. Thus y(aH^-i^) -(1). ^V(af) is replaced by F{r, 0). and so B f[F{r,0)rdrdtt B^rhere the integration extends over the area of the circle r= at. The other terra might be obtained by Stokes' rule. This solution is applicable to the motion of a layer of gas between two parallel planes, or to that of an unlimited stretched membrane, which depends upon the same fundamental equation. 276. From the solution in terms of initial conditions we may, as usual (§ 66), deduce the effect of a continually renewed dis- turbance. Let us suppose that throughout the space T (which will ultimately be made to vanish), a uniform disturbance <^. equal to * {t')dt', is communicated at time ('. The resulting value of ^ at tirae t is * ft') dt'. e S denotes the part of the surface of the sphere r = a{t — t') 5epted within T. a quantity which vanishes, unless a((— (') be nprised between the narrow limits r, and r,. Ultimately ( — (' ■ay be replaced by r/a, and ^ ((') by <t> (i - r/a) ; and the result I the integration with respect to dt' is found by writing T (the yiar JaSdt: Hence -4> (- .■{1). 104 SOURCES OF SOUND. [276. shewing that the disturbance originating at any point spreads itself symmetrically in all directions with velocity a, and with amplitude varying inversely as the distance. Since any number of particular solutions may be superposed, the general solution of the equation ^ = a«V«^ + 4> (2) may be written ^-^IIK'-^T « r denoting the distance of the element (2 K situated at x, y, t from (at which ^ is estimated), and 4> (f — rfa) the value of 4> for the point X, y, z at the time t — r/a. Complementary terms, satisfying through all space the equation ^ = a'V^, may of course occur independently. In our previous notation (§ 244) <I> = |- [(Zeir 4- Fdy + Zdz) ; and it is assumed that Xdx + Ydy + Zdz is a complete diflTerential. Forces, under whose action the medium could not adjust itself to equilibrium, are excluded; as for instance, a force uniform in mag- nitude and direction within a space T, and vanishing outride that space. The nature of the disturbance denoted by 4> is perhaps best seen by considering the extreme case when 4> vanishes except through a small volume, which is supposed to diminish without limit, while the magnitude of 4> increases in such a manner that the whole effect remains finite. If then we integrate equation (2) through a small space including the point at which 4> is ulti- mately concentrated, we find in the limit = a«//gd^-.///*dF (4). shewing that the eflfect of 4> may be represented by a proportional introduction or abstraction of fluid at the place in question. The simplest source of sound is thus analogous to a focus in the theory of conduction of heat, or to an electrode in the theory of electricity, 277. The preceding expressions are general in respect of the relation to time of the functions concerned ; but in almost all the applications that we shall have to make, it will be convenient to analyse the motion by Fourier's theorem and treat separately the HARMONIC TYPE. siiDple harmonic motions of various periods, afterwards, if necessary, cumpounding the results. The values of <t> and "t*, if simple har- monic at every point of apace, may be expressed in the form R cos (nt + e), R and e being independent of time, but variable Irom point to point. But as in such cases it often conduces to simplicity to add the term i R sin (nt + e), making altogether R^'"+", or Re" .e'"*, we will assume simply that all the functions which enter into a problem are proportional to e'"', the coeffi- cients being in general complex. After our operations are com- pleted, the real and imaginary parte uf the expressions can be separated, either of them by itself constituting a solution of the question. Since 4> is proportional to e'"', 4> = ^'"-"<i>'> ind the differential e<i nation becomes V'^ + fc'^-(-a-'<I> = (1), where, for the sake of brevity, k is written in place of nja.. If X Kleaote the wave-length of the vibration of the period in question, k=n/a=27rl\ (2). To adapt (3) of the preceding section to the present case, it is lily neccssarj' to remark that the substitution of ( — rfa for t is lilected by introducing the factor e"'"''''', or e~*: thus *((-r/«) = e-'*^*(0. and the solution of (1) is *=Ty/f"?*'"'- ("• (o which may be added any solution of V''ip + k''<j> = 0. If the disturbing forces be all in the same phase, and the t through which they act be very small in comparison with ^length, c~*' may be removed from under the integral Lmid at a sufficient distance we may take ■ in real quantities, on restoring the time factor and replacing 'by *,. 13 {nt — kr+e) ,.(4). 106 VERIFICATION OP SOLUTION. [277. In order to verify that (S) satisfies the difFerential equation (1), we may proceed as in the theory of the common potential Con- sidering one element of the integral at a time, we have first to shew that * = ^ (5) satisfies V"^ + A:»^ = 0, at points for which r is finite. The simplest course is to express V* in polar co-ordinates referred to the element itself as pole, when it appears that r \dr^ r dr) r ^ r dr* ' r r ' We infer that (3) satisfies V»^ + ifc»^ = 0, at all points for which 4> vanishes. In the case of a point at which 4> does not vanish, we may put out of account all the elements situated at a finite distance (as contributing only terms satisfying V^ + A*0 = 0), and for the element at an infinitesimal distance replace e~^^ by unity. Thus on the whole exactly as in Foisson's theorem for the common potential*. 278. The effect of a force 4>i distributed over a surface S may be obtained as a limiting case from (3) § 277. 4>dF is replaced by 4> hdS, h denoting the thickness of the layer ; and in the limit we may write 4> 6 = 4>i. Thus *-Ki-//*'^'^- »>■ The value of ^ is the same on the two sides of S, but there is discontinuity in its derivatives. If dn be drawn outwards &om S normally, (4) § 276 gives [t)At}r->' <^>-- If the surface S be plane, the integral in (1) is evidently symmetrical with respect to it, and therefore (d4>ldn\^{d4>ldn\. 1 See ThomBon and Tait's IfaJtmdl PAOoMp^, § 491. t HefanhoUs. OrvUf , i. 57. p. SI« I860. ^D78.] StTRPACE DISTRIBUTIONS. 107 Hence, if d^/dn be the given normal velocity of the fluid in contact with the plane, the value of <#i is determined by *=-^Jlt^^^ <"). which IB a result of considerable importance. To exhibit it in terms of real quantities, we may take dj>ldn = Pe""'*" (4), P and ( being real functions of the position of dS. The symbolical Bolution then becomes 0._±.||pe<».-k*.i#' (5), from which, if the imaginary part be rejected, we obtain ^*=-,y/p"'*"'-;-^*><^s (6), esponding to d<^/rfn = Pco3(n(+e) (7). The same method is applicable to the general case when the motion ia not rtstricted to be simple harmonic. We have *-^F('-3-v <«>■ where by V{t — rja) is denoted the normal velocity at the plane for the element dS at the time t — rja. that ia to say, at a time r, a antecedent to that at which ^ is estimated. In oinJer to complete the solution of the problem for the unlimited mass of fluid lying on one side of an infinite plane, we have to add the most general value of 0, consistent with F = 0. This part of the question is identical with the general problem of tieflectiou from an infinite rigid plane*. \ It is evident that the effect of the constraint will be represented Of the introduction on the other side of the plane of fictitious initial displacements and forces, forming in conjunction with those actually existing on the first side a systt-m perfectly symmetrical «'ith respect to the plane. Whatever the initial values of and ^ may be belonging to any point on the first aide, the same must be Ascribed to its image, and in like manner whatever function of ^^^B ' PoisEnn, Journal de Vfcolt polytcchtiiijue. I. vii. 1808. ^^ 108 INFINITE PLANB WALL. [278. the time 4> may be at the first point, it must be conceived to be the same function of the time at the other. Under these circumstances it is clear that for all future time <f> will be symmetrical with respect to the plane, and therefore the normal velocity zero. So far then as the motion on the first side is concerned, there will be no change if the plane be removed, and the fluid continued indefinitely in all directions, provided the circumstances on the second side are the exact reflection of those on the first. This being understood, the general solution of the problem for a fluid bounded by an infinite plane is contained in the formul® (8) § 273, (3) § 277, and (8) of the present section. They give the result of arbitrary initial conditions (^o and ^o)i arbitrary applied forces (4>), and arbitrary motion of the plane (F). Measured by the resulting potential, a source of given magni- tude, i.e. a source at which a given introduction and ¥dthdrawal of fluid takes place, is thus t>vice as eflective when close to a rigid plane, as if it were situated in the open ; and the result is ulti- mately the same, whether the source be concentrated in a point close to the plane, or be due to a corresponding normal motion of the surfietce of the plane itself. The operation of the plane is to double the effective pressures which oppose the expansion and contraction at the source, and therefore to double the total energy emitted ; and since this energy is diffused through only the half of angular space, the intensity of the sound is quadrupled, which corresponds to a doubled amplitude, or potential (§ 245). We will now suppose that instead of d<f>/dn = 0, the prescribed condition at the infinite plane is that ^ = 0. In this case the fictitious distribution of ^o> ^oi ^> on the second side of the plane must be the opposite of that on the first side, so that the sum of the values at two corresponding points is always zero. This secures that on the plane of sjonmetry itself <f> shall vanish throughout. Let us next suppose that there are two parallel surfaces iSi, S^y separated by the infinitely small interval dn, and that the value of 4>i on the second surface is equal and opposite to the value of 4>i on the first. In crossing Si, there is by (2) a finite change in the value of d4>/dn to the amount of 4>i/a', but in crossing 8^ the same finite change occurs in the reverse direction. When dn is reduced without limit, and ^dn replaced by <E>u, d<l>/dn will be ihe same on the two sides of the double sheet, but there will be fontinuitj in the value of ^ to the amount of *„/«'. At the e time (1) becomes ^^i^./Z^C?)*.."- ('>• If the surfaoe S be plane, the values of tf) on the two sidee of it are numerically equal, and therefore close to the surface itself i^ = ± i «-'*„. Hence (9) may be written *'-im^W^ <'0); I under the integral sign represents the surface- potential, ' positive on the one side and negative on the other, due to the action of the forces at S. The direction of dn must be under- [vatood to be towards the side at which if> is to be estimated. I 279. The problem of spherical waves diverging from a point B already been forced upon us and in some degree considered, but on account of its importance it demands a more detailed treatment. If the centre of symmetry be taken as pole the velo- city-potential is a function of r only, and (§241)V= reduces to v- + - T- , or to - J-, »■■ Tlie equation of free motion (3) S 273 dH r ar i- dr^ ' ^ thus becomes dHr4>)_ dUf^ ... dt' "" dr^ ^^*' whence, as in § 24-5, tr^=/(af-r) + f(ai + r) (2). The values of the velocity and condensation are to be found by trentiation in accordance with the formulae «=S. -i^f <^'- As in the case of one dimension, the first term represents a wave adifancing in the direction of r increasing, that is to say, a diver- gent wave, and the second term represents a wave converging upon the pole. The latter does not in itself possess much interest. If we confine our attention to the divergent wave, we have ^^ /(«f-r) f{at-r) _ ^^ fj at-r) ^^^ 110 SPHERICAL WAVES. [279, When r is very great the term divided by r* may be neglected, and then approximately U = 08 (5), the same relation as obtedns in the case of a plane wave, as might have been expected. If the type be harmonic, r^ = ii c*<«*-^^> (6), or, if only the real part be retained, r<f>=^ A cos -—{at + 6" r) (7). At If a divergent disturbance be confined to a spherical shell, within and without which there is neither condensation nor velocity, the character of the wave is limited by a remarkable re- lation, first pointed out by Stokes^ From equations (4) we have (as — u) r^=f((tt — r), shewing that the value of /{at - r) is the same, viz. zero, both inside and outside the shell to which the wave is limited. Hence by (4), if a and fi be radii less and greater than the extreme radii of the shell, srdr^O (8), f.' which is the expression of the relation referred to. As in § 274, we see that a condensed or a rarefied wave cannot exist alona When the radius becomes great in comparison with the thickness, the variation of r in the integral may be neglected, and (8) then expresses that the mean condensation is zero. [Availing himself of Foucault's method for rendering visible minute optical differences, Topler" succeeded in observing spherical sonorous waves originating in small electric sparks, and their reflection fi:'om a plane wall. Subsequently photographic records of similar phenomena have been obtained by Mach*.] In applying the general solution (2) to deduce the motion resulting from arbitrary initial circumstances, we must remember that in its present form it is too general for the purpose, since it covers the case in which the pole is itself a source, or place where i Phil. Mag. zzzir. p. 68. 1849. * Pogg. Amu vol. ozzzi. pp. 88, 180. 1867. « SiMff. tfir Wimm Akad.^ 1889. 279.] CONTINUITY THROUGH POLE. HI Huid is introduced or withdrawn in violation of the equation of continuity. The total ciitrent across the surfiice of a sphere of radius r is tTn-'w, or by (2) and (3) b - 4ir {/{at - r) + F(at + r)} + iirr {F' (at + r) -f {at - r)], Kb thai ^B equ By the known initial circumstances the values of u and s are determined for the time ( = 0, and for all (positive) values of r. If these initial values be represented by m„ and Sj, we obtain from (2) and (3) /(_r) + F(r) = rKdr } ^^^^^ I that, if the pole be not a source, /(a( — r) + ^(a( + r), or r<^, : vanish with r. Thus /(,.;) + f{</() = (9), I equation which must hold good for all positive values of the f(-r-)-F(r-)~a\,.rir by which the function / is determined for all negative arguments, and the fuoction F for all positive arguments. The form of / for iwsitive arguments follows by means of (9), and then the whole subsequent motion is determined by (2), The form of F for negative arguments is not required. The initial disturbance divides itself into two parts, travelling in opposite directions, in each of which r0 is propagated with constant velocity a, and the inwards travelling wave ia continually reflected at the pole. Since the condition to be there satisfied is r^ = 0, the case is somewhat similar to that of a parallel tube terminated by an open end, and we may thus perhaps better anderstand why the condensed wave, arising from the liberation of a mass of condensed air round the pole, is followed immediately by a wave of rarefaction. [The compoBite character of the wave resulting from an initial H condenaation may be invoked to explain a phenomenon which has ^■BAen occasioned surprise. When windows are broken by a violent ^HjfplosioD in their neighbourhood, they are frequently observed to ^B ) Tlu e: * Tb* Milntion (or spliericul TibralioDs may be obtained without the use of (I) f troiai or plane waTes, related limilarl; to the pole, and tra- K onlvards in &1I direotioaa aynnnetrjcoll;. 112 SIMPLE 80URCB. [279. have fallen outwards as if from ezposore to a wave of larefiu^tioD. This effect may be attributed to the second part of the eompoand wave; but it may be asked why should the second part preponderate over the first? If the window were freely suspended, the momentum acquired from the waves of condensation and rare- faction would be equal. But under the actual conditions it may well happen that the force of the condensed wave is spent in overcoming the resistance of the supports, and then the rarefied wave is left free to produce its full effect] 280. Returning now to the case of a train of harmonic waves travelling outwards continually from the pole as source, let us investigate the connection between the velocity-potential and the quantity of fluid which must be supposed to be introduced and withdrawn alternately. If the velocity-potential be <^ = -4^cos^-(a<-^) (1)» we have, as in the preceding section, for the total current crossing a sphere of radius r, 4nrr*-^ = A {cos k (at — r) — kr sink (at — ryi^ A coakat, where r is small enough. If the maximum rate of intixxluction of fluid be denoted by A, the corresponding potential is given by (1)* It will be observed that when the source, as measured by A, is finite, the potential and the pressure-variation (proportional to ^) are infinite at the pole. But this does not, as might for a moment be supposed, imply an infinite emission of energy. If the pressure be divided into two parts, one of which has the same phase as the velocity, and the other the same phase as the acceleration, it vdll be found that the former part, on which the work depends, is finite. The infinite part of the pressure does no work on the whole, but merely keeps up the vibration of the air immediately round the source, whose effective inertia is indefinitely great We will now investigate the energy emitted from a simple source of given magnitude, supposing for the sake of greater generality that the source is situated at the vertex of a rigid cone of solid angle o). If the rate of introduction of fluid at the source be ^ COB JxLt, we have mf*d^ldr^A coshat l] ENERGY EMITTED PKOM GIVES SOURCE. 113 lately, corresponding to = COS k-(at~r) (2); whence ^ = ^ sin h {at -r) (3), and o»r= < — -^ [coa k {at — r) — /LTsia k {at — r)} (4). Thus, as in § 245, if dW be the work transmitted in time dt, we get, since Sfi = — pj>, dW pkaA' . , , , . , , . , -, =— - sm A-((x( — r)ciis t(a(— r) + p sin' K {at — r). Mf the right-hand member the first terra is entirely periodic, and !i the second the mean value of sin' k (at — r) is ^. Thus in the ; iig run , ^=-2^'^ (5)'- L It will be remarked that when the source is given, the ampli- ^^He varies inversely as tu, and therefore the intensity inversely ^^Hh'. For an acute cone the intensity is greater, not only on ^^pDUnt of the diminution in the solid angle through which the I sound is distributed, but also because the total energy emitted 'liiin the source is itself increased. When the source is in the open, we have only to put tii = 4Tr, ■ifi'i when it is close to a rigid plane, w = 2ir, The results of this article find an interesting application in the iheoiy of the speaking trumpet, or (by the law of reciprocity J§ 109. 29+) hearing trumpet. If the diameter of the large open end be small in comparison with the wave-length, the waves on irnval suffer copious reflection, and the ultimate result, which ii»t depend largely on the precise relative lengths of the tube I'i «f the wave, requires to be determined by a different process. i. jI by sufficiently prolonging the cone, this reflection may be niiuished, and it will tend to cease when the diameter of the - 11 end includes a large number of wave-lengths. Apart from -lioa it would therefore be possible by diminishing w to obtain ■rn a given source any desired amount of energy, and at the k' Csinhriilge JJathematioal Tripoi ExomlnatioD, 1S7S. 114 SPEAKING TRUMPET. [280. same time by lengthening the cone to secure the unimpeded transference of this energy from the tube to the surrounding air. From the theory of diffraction it appears that the sound will not fall off to any great extent in a lateral direction, unless the diameter at the large end exceed half a wave-length. The ordinary explanation of the effect of a common trumpet, depending on a supposed concentration of rays in the axial direction, is thus untenable. 281. By means of Euler s equation, d^irii>)_ d}{r<t>) we may easily establish a theory for conical pipes with open ends^ analogous to that of Bernoulli for parallel tubes, subject to the same limitation as to the smallness of the diameter of the tubes in comparison with the wave-length of the sounds Assuming that the vibration is stationary, so that r<f> is everywhere proportional to cos kat, we get from (1) ^^^*'.r* = (2). of which the general solution is r<f> = A cos At -h 5 sin At (3). The condition to be satisfied at an open end, viz., that there is to be no condensation or rarefaction, gives r^ = 0, so that, if the extreme radii of the tube be Vi and rj, we have A cos h'l + B sin Atj = 0, A cos kr^ 4- 5 sin kr^ = 0, whence by elimination o{ A : B, sin k (r^ — r^) = 0, or rj — ri = J wX, where m is an integer. In fact since the form of the general solution (3) and the condition for an open end are the same as for a parallel tube, the result that the length of the tube is a multiple of the half wave-length is necessarily also the same. A cone, which is complete as far as the vertex, may be treated as if the vertex were an open end, since, as we saw in § 279, the condition r<f> = is there satisfied. The resemblance to the case of parallel tubes does not extend to the position of the nodes. In the case of the gravest vibration ^ D. BemonUi, M€m. d, VAead. d. SeL 1762 ; Dahamel, LiouTille Jaunu Math, Tol. uv. p. 98, 1849. of a paraltel tube open at both ends, the node occupies a central posiiiou, aud the two halves vibrate synchronously as tubes open M uiie end and stopped at the other. But if a conical tube were 'iiviil«i by a partition at its centre, the two parts would have different periods, as is evident, becanse the one part differs from a |]arallel tube by being contracted at its open end where the effect of n contraction is to depress the pitch, while the other part is oontracled at its stopped end, where the effect is to raise the pitch, la order that the two periods may be the same, the partition must approach nearer to the narrower end of the tube. Its actual pisition may be determined analytically from (3) by equating to zero the value of d^jdr. When both ends of a conical pipe are closed, the corresponding notes are determined by eliminating A : B between the equations, A (cos h; + At, ain h;) + B (sin Ati — kr, cos h;) = 0, A (coa hi; + kr, sin kr,) + B (sin kr^ — In; cos kr,) = 0, nf which the result may be put into the form kr, — tan~' kr, = h\ — tan"' kr, (4). If 1-1 = 0, we have simply tan kr, = kri (5)'; if r, and r, be very great, tan^'^r, and tan"' At, are both odd multiples of ^tt, so that r, — r, is a multiple of J \, as the theory of parallel tubes requires, [If r, — 7-1 = /, rj + 7', = )■, (4) may be written _ kl l+^i''{r'-l') comparison with ^ the approximate solution tan kl = ..(6). When r is great ij of (6) gives ^::(-.^) (". ifliience of conicality upon the pitoh is m being an integer. The thus of the second order. Experiments upon conical pipes have been made by Boutet' ud by Blaikley'.] > Fat the roots of this equation ue g 207. ' Aon. d. Chim, toI. in. p, 150, 1870. ' Fhii. Mag. VI. p. 119, 1878, 116 TWO SOURCES OF LIKE PITCH. [2^ 282. If there be two distinct sources of sound of the pitch, situated at Oi and 0„ the velocity-potential ^ at a P whose distances from d, 0, are r, and r., may be expressed ^^ cos A(af-r,) ^ ^cos <'(a ^ -r,-tt) ^ where A and B are coefficients representing the magnitudes the sources (which without loss of generality may be supposed have the same sign), and a represents the retardation (consii as a distance) of the second source relatively to the first The trains of spherical waves are in agreement at any point P, ra+ a — ri= ± mX, where m is an integer, that is, if P lie on one of a system of hj-perboloids of revolution having foci it Oi and 0,. At points lying on the intermediate h\-perboloidllj represented by r, + a — r, = ± J(2m + 1)X, the two sets of wafdl are opposed in phase, and neutralize one another as far as their actual magnitudes permit. The neutralization is complete, 3 r^w^^A '. B, and then the density at P continues permanently unchanged. The intersections of this sphere with the system of hyperboluids will thus mark out in most cases ?»everal circles ol absolute silence. If the distance 0^0^ between the sources be greal in comparison with the length of a wave, and the sources themselvei be not ver}' unequal in power, it will be possible to depart from the sphere r^ :r^ = A:B for a distance of several wave-IengthflL without appreciably disturbing the equality of intensities, and thus to obtain over finite surfaces several alternations of sound and of almost complete silence. There is some difficulty in actually realising a satisfactory interference of two independent sounds. Unless the unison be extraordinarily perfect, the silences are only momentary and are consequently difficult to appreciate. It is therefore best to employ sources which are mechanically connected in such a way that the relative phases of the sounds issuing from them cannot var)^ The simplest plan is to repeat the first sound by reflection from a flat wall (^ 269, 278), but the experiment then loses something in directness owing to the fictitious character of the second source. Perhaps the most satisfactory form of the experiment is that described in the Philosophical Magazine for June 1877 by myaeIC ''An intermittent electric current, obtained from a fork interrupts making 128 vibrations per second, excited by means of electro- magnets two other forks, whose fr^n^ncv was 256, (§§ 63, 64) 118 EXPERIMENTAL METHODS. [282. state of things which he wishes to examine. Among indicators of sound may be mentioned membranes stretched over cups, the agita- tion being made apparent by sand, or by small pendulums resting lightly against them. If a membrane be simply stretched across a hoop, both its faces are acted upon by nearly the same forces, and consequently the motion is much diminished, unless the membrane be large enough to cast a sensible shadow, in which its hinder face may be protected. Probably the best method of examining the intensity of sound at any point in the air is to divert a portion of it by means of a tube ending in a small cone or resonator, the sound so diverted being led to the ear, or to a manometric capsule. In this way it is not difficult to determine places of silence with considerable pijecision. By means of the same kind of apparatus it is possible to examine even the phase of the vibration at any point in air, and to trace out the surfaces on which the phase does not vaiy'. If the interior of a resonator be connected by flexible tubing with a manometric capsule, which influences a small gas flame, the motion of the flame is related in an invariable manner (depending on the apparatus itself) to the variation of pressure at the mouth of the resonator ; and in particular the interval between the lowest drop of the flame and the lowest pressure at the resonator is independent of the absolute time at which these efiects occur. In Mavers experiment two flames were employed, placed close together in one vertical line, and were examined with a revolving mirror. So long as the associated resonators were undisturbed, the serrations of the two flames occupied a fixed relative position, and this relative position was also maintained when one resonator was moved about so as to trace out a surface of invariable phase. For further details the reader must be referred to the original paper. 283. When waves of sound impinge upon an obstacle, a portion of the motion is thrown back as an echo, and under cover of the obstacle there is formed a sort of sound shadow. In order, however, to produce shadows in anything like optical perfection, the dimensions of the intervening body must be considerable. The standard of comparison proper to the subject is the wave- length of the vibration ; it requires almost as extreme conditions to produce rays in the case of sound, as it requires in optics to avoid producing them. Still, sound shadows thrown by hills, or 1 Mayer, Phil Mag. (4), xut. p. 821. 187S. 283.] SOUND SHADOWS. U9 buildings, are often tolerably complete, and must be within the experience of all. For closer examination let us take first the case of plane waves of harmonic type impinging upon an immovable plane screen, of indnitesinial thickness, in which there is an aperture of any form, the phvne of the screen {x = 0) being pamllel to the fronts of the waves. The velocity-potential of the undisturbed train of waves may be taken, <f> = C08(7lt-kx) (1). If the value of d<ft/da! over the aperture be known, formula (6) and (7) § 278 allow us to calculate the value of (ft at any point on the further side. In the ordinary theory of diffraction, as given in works on optics, it is assumed that the disturbance in the plane of the aperture is the same as if the screen were away. This hypotheaia, though it can never be rigorously exact, will suffice when the aperture is very large in comparison with the wave- length, as is usually the case in optics. ror the undisturbed wave we have #(ar = 0) = tsinn( (2), and therefore on the further side, we get ^__^iJJriM??^)^ (3), ^B integration extending over the area of the aperture. Since T=2ir/X, we see by comparison with (1) that in supposing a primary wave broken up, with the view of applying Huygens' principle, dS must be divided by \r, and the phase must be accelerated by a quarter of a period. When r is large in comparison with the dimensions of the aperture, the composition of the integral is beat studied by the aid of Fresnel's' zones. With the point 0, for which <i> is to be estimated, aa centre describe a series of spheres of radii increasing by the constant difference JX,, the first sphere of the series being of such radius (c) as to touch the plane of the screen. On this plane are thus marked out a aeries of circles, whose radii p are > ITh«<B zones are aHnallj spoken ot m Sajgeae' Eonea b; optical writers (eg. BiUei. Tniti d'Optique phijiique, vol. i. p. 103. Paris. 1868); bot, na has been IKliQtrf otit bj SobUBtet (PAi'I. Mag. vol. ii£i. p. 8S, 18U1), tt is more correct to - a.] 120 fresnel's zones. [283. given by /J* + c* = (c + i nX)*, or p' = nc\, very nearly ; so that the rings into which the plane is divided, being of approximately equal area, make contributions to (f> which are approximately equal in numerical magnitude and alternately opposite in sign. If lie decidedly within the projection of the area, the first term of the series representing the integral is finite, and the terms which follow are alternately opposite in sign and of numerical magnitude at first nearly constant, but afterwards diminishing gradually to zero, as the parts of the rings intercepted within the aperture become less and less. The case of an aperture, whose boundary is equidistant from 0, is excepted. In a series of this description any term after the first is neutralized almost exactly [that is, so far as first differences are concerned] by half the sum of those which immediately precede and follow it, so that the sum of the whole series is represented approximately by half the first term, which stands over uncom- pensated. We see that, provided a sufficient number of zones be included within the aperture, the value of (f> at the point is independent of the nature of the aperture, and is therefore the same as if there had been no screen at all. Or we may ceJculate directly the effect of the circle with which the system of zones begins; a course which will have the advantage of bringing out more clearly the significance of the change of phase which we found it necessary to introduce when the primary wave was broken up. Thus, let us conceive the circle in question divided into infinitesimal rings of equal area. The parts of <]> due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is there- fore midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle. The amplitude of the resultant will be less than if all its components had been in the same phase, in the ratio f^ sinxdx : ir, or 2 : tt ; and therefore since the area of the circle is ttXc, half the effect of the first zone is ^ = — ^ . - . - - - — -— . ttXc = cos {7it — kc), the same as if the primary wave were to pass on undisturbed. When the point is well away from the projection of the aperture, the result is quite different. The series representing the integral then converges at both ends, and by the same reasoning k-] FRESSELS ZONES. ■re its Bura is seen to be approximately zero. We conclude the projection of on the plane x = fall within the iure. and be nearer to by a great many wave-lengths than point of the boundary of the aperture, then the irbance at is nearly the same as if there were no obstacle at if the projection of fall outside the aperture and be rer to by a great many wave-lengths than the nearest point of I boundary, then the disturbance at practically vanishes. 1 is the theory of sound rays in ita simplest form, i argument is not verj" different if the screen be oblique to b plane of the waves. As before, the motion on the further side f the screen may be regarded as due to the normal motion of the particles in the plane of the aperture, but this normal motion now varies in phase from point to point. If the primarj' waves proceed from a source at Q, Fresnel's zones for a point P are the series of elli]>ses represented by Ti-I- r, = i'Q-l-^ nX, where r, and r^ are the distances of any point on the screen from Q and P respectively, 1 n 19 an integer On account of tlie assumed smsllness of \ in tarison with r, and r-,, the zones aie at first of equal area and pe equal and opposite contributions to the value of ift ; and B by the same reasoning as before we may conclude that at any pt decidedly outside the geometrical projection of the aperture I disturbance vanishes, while at any point decidedly within the metrical projection the disturbance is the same as if the iry wave had passed the screen unimpeded. It may be Marked that the increase of area of the Fresnel's zones due to ob1i<iiiity is compensated in the calculation of the integral by the correspond iugly diminished value of the normal velocity of the The enfeeblement of the primary wave between the screen I the point P due to divergency is represented by a diminution • area of the Fresnel's zones below that corresponding to ! incident waves in the ratio r, + n. : i\. There is a simple relation between the transmission of sound ragh an aperture in a screen and its reflection from a plane Wtor of the same form as the aperture, of which advantage may letimes be taken in experiment. Lot us imagine a source r to Q and in the same phase to be placed at (/, the image of ) plane of the screen, and let us suppose that the screen is d and replaced by a plate whose form and position is exactly ; then we kitow th^it tlie elTeft at P nf the two 122 CONDITIONS OF COMPLETE REFLECTION. [283. sources is uninfluenced by the presence of the plate, so that the vibration from Q[ reflected from the plate and the vibration from Q transmitted round the plate together make up the same vibra- tion as would be received from Q if there were no obstacle at all Now according to the assumption which we made at the begin- ning of this section, the unimpeded vibration from Q may be regarded as composed of the vibration that finds its way round the plate and of that which would pass an aperture of the same form in an infinite screen, and thus the vibration from Q as transmitted through the aperture is equal to the vibration from Q as reflected from the plate. In order to obtain a nearly complete reflection it is not neces- sary that the reflecting plate include more than a small number of Fresners zones. In the case of direct reflection the radius p of the first zone is determined by the equation p«(l/c,-hl/c,) = X (4), where Cj and c% are the distances from the reflector of the source and of the point of observation. When the distances concerned are great, the zones become so large that ordinary walls are insufiicient to give a complete reflection, but at more moderate distances echos are often nearly perfect. The area necessary for complete reflection depends also upon the wave-length ; and thus it happens that a board or plate, which would be quite inadequate to reflect a grave musical note, may reflect very fairly a hiss or the sound of a high whistle. In experiments on reflection by screens of moderate size, the principal diflSculty is to get rid sufficiently of the direct sound. The simplest plan is to reflect the sound from an electric bell, or other fairly steady source, round the comer of a large build ing\ 284. In the preceding section we have applied Huygens' principle to the case where the primary wave is supposed to be broken up at the surface of an imaginary plane. If we really know what the normal motion at the plane is, we can calculate the disturbance at any point on the further side by a rigorous process. For surfaces other than the plane the problem has not been solved generally ; nevertheless, it is not difficult to see that when the radii of curvature of the surface are very great in com- parison with the wave-length, the effect of a normal motion of an 1 PkiL Mag. (6), m. p. 468. 1877. 284.] DIVERGING WAVES. 123 element of the siirface must be very nearly the same aa if tbe surface were plane. Od this understandiDg we may employ the same integral as before to calculate the aggregate result. As a matter of convenience it is usually best to suppose the wave to be broken up at what is called in optics a wave-surface, that is, a _ anrface at every point of which the p)uue of the disturbance is the ^r Let us consider the application of Huygens' principle to cal- culate the progress of a given divergent wave. With any point P,at which the disturbance ia required, as centre, describe a series of spheres of radii continually increasing by the constant difference ^ V the first of the series being of such radius (c) as to touch the given wave-surface at C. If R be the radius of curvature of the eur&ce in any plane through P and G, the corresponding radius p ^^^ tbe outer boundary of the n'" zone is given by the equation ^Hfelttn which we get approximately w K^i)- • •(I). If the surface be one of revolution round PC, the area of the first n zones is n-p', and since p' is proportional to n, it follows that the zt'ues are of equal area. If the surface be not of revolution, the area of the first n zones is represented ^Jp'd0, where is the azimuth of the plane in which p is measured, but it still remains true that the zones are of equal area. Since by hj'pothesia the normal motion does not vary rapidly over the wave-surface, the disturbances at P due to the various zones are nearly equal in magnitude and alternately opposite in sign, and we conclude that, as in the case of plane waves, the aggregate effect is the half of that due to the first zone. The phase at F is accordingly retarded behind that prevailing over the given wave-surface by an amount corresponding to the distance c. The intensity of the disturbance at P depends upon the area of S FresneVs zone, and upon the distance c. In the case of metry, we have Trp' _ irXR c'~ R + c' bi shews that the disturbance is less than if R wei-e infinite ia This diminution is the effect of 124 VARIATION OP INTENSITY. [284. and is the same as would be obtained on the supposition that the motion is limited by a conical tube whose vertex is at the centre of curvature (§ 266). When the surface is not of revolution, the value of \S^p^d0 -i- c may be expressed in terms of the principal radii of curvature Ri and B^, with which 12 is connected by the* relation 1/iJ = cos»tf/i2i + sin*^/ii,. We obtain on effecting the integi-ation 1 r*' (2), so that the amplitude is diminished by divergency in the ratio V(iJi + c) (-fi, + c) : ^RiRt, a result which might be anticipated by supposing the motion limited to a tube formed by normals drawn through a small contour traced on the wave-surface. Although we have spoken hitherto of diverging waves only, the preceding expressions may also be applied to waves converging in one or in both of the principal planes, if we attach suitable signs to Ri and iZ,. In such a case the area of the first Fresnel's zone is greater than if the wave were plane, and the intensity of the vibration is correspondingly increased. If the point P coincide with one of the principal centres of curvature, the expression (2) becomes infinite. The investigation, on which (2) was founded, is then insufficient ; all that we are entitled to affirm is that the disturbance is much greater at P than at other points on the same normal, that the disproportion increases with the frequency, and that it would become infinite for notes of infinitely high pitch, whose wave-length would be negligible in comparison with the distances concerned. 285. Huygens* principle may also be applied to investigate the reflection of sound from curved surfaces. If the material surface of the reflector yielded so completely to the aerial pressures that the normal motion at every point were the same as it would have been in the absence of the reflector, then the sound waves would pass on undisturbed. The reflection which actually ensues when the surface is unyielding may therefore be regarded as due to a normal motion of each element of the reflector, equal and opposite to that of the primary waves at the same pointy and may be investigated by the formula proper to plane sarfi^oea in the manner of the preceding section, and subject to a ■H**^!!^ 283.] REFLECTION FROM CURVED SURFACES. 125 liraitatitin as to the relative magnitudes of the wave-leogth and of the other distances concerned. ^H The most interesting case of reflection occurs when the ^^priace is so shaped as to cause a concentration of rays upon a ^Hblicular point {P). If the sound issue originally from a simple ^Tource at Q, and the surface be an ellipsoid of revolution having its foci at P and Q, the concentration is complete, the vibration reflected from every element of the surface being in the same phase on arrivnl at Q. If Q be infinitely distant, so that the incident waves are plane, the surface becomes a paraboloid having its focus at P, and its asis parallel to the incident rays. We must not suppose, however, that a symmetrical wave diverging from Q is converted by reflection at the ellipsoidal surface into a spherical wave converging symmetrically upon P; in fact, it is Ey to see that the intensity of the convergent wave must be lerent in diBerent directions. Nevertheless, when the wave- gth is very small in comparison with the radius, the different porta of the convergent wave become approximately independent of one another, and their progress is not materially affected by the failure of perfect symmetry. The increase of loudness due to curvature depends upon the , of reflecting surface, from which disturbances of uniform base arrive, as compared with the area of the first Fresnel's me of a plane reflector in the same position. If the distances of B reflector from the source and from the point of observation be siderable, and the wave-length be not very small, the first ^nel's zone is already rather large, and therefore in the case ' a reflector of moderate dimensions but little is gained by making it concave. On the other hand, in laboratory experiments, when the distances are moderate and the sounds employed are of high pitch, e.g. the ticking of a watch or the cracking of electric , concave reflectors are very eflBcient and give a distinct sntiatioD of sound on particular spots. 286. We have seen that if a ray proceeding from Q passes lor reflection at a plane or curved surface through P, the point t at which it meets the surface is determined by the condition wt QR + RP is a minimum {or in some oases a maximum). ibe point R is then the centre of the sj'stem of Fresnel's zones; e lunpUtnde of the vibration at P depends upon the area of the I 126 fermat's principle. [286. first zone, and its phase depends upon the distance QR + RP. If there be no point on the surface of the reflector, for which QR + RP is a maximum or a minimum, the system of Fresnels zones has no centre, and there is no ray proceeding from Q which arrives at P after reflection from the surface. In like manner if sound be reflected more than once, the course of a ray is deter- mined by the condition that its whole length between any two poiuts is a maximum or a minimum. The same principle may be applied to investigate the refraction of sound in a medium, whose mechanical properties vary gradually from point to point. The variation is supposed to be so slow that no sensible reflection occurs, and this is not inconsistent with decided refraction of the rays in travelling distances which include a very great number of wave-lengtha It is evident that what we are now concerned with is not merely the length of the ray, but also the velocity with which the wave travels along it, inasmuch as this velocity is no longer constant. The condition to be satisfied is that the time occupied by a wave in travelling along a ray between any two points shall be a maximum or a minimum ; so that, if V be the velocity of propa- gation at any point, and ds an element of the length of the ray, the condition may be expressed, S J V~^ ds = 0. This is Fermat's principle of least time. The further developement of this part of the subject would lead us too far into the domain of geometrical optics. The funda- mental assumption of the smallness of the wave-length, on which the doctrine of rays is built, having a far wider application to the phenomena of light than to those of sound, the task of developing its consequences may properly be left to the cultivators of the sister science. In the following sections the methods of optics are applied to one or two isolated questions, whose acousticeJ interest is sufiicient to demand their consideration in the present work. 287. One of the most striking of the phenomena connected with the propagation of sound within closed buildings is that presented by "whispering galleries," of which a good and easily accessible example is to be found in the circular gallery at the base of the dome of St Paul's cathedral. As to the precise mode of Action acoustical authorities are not entirely agreed. In the _'S7.] WHISPEKDTG GALLERIES. 127 iipimon of the Astiononier Royal' the effect is to be ascribed to reflection from the surface of the dome overhead, and is to be observed at the point of the yallery diaraetrically apposite to the - 'Urce of sound. Every ray proceeding from a radiant point and ■ rtocted from the surface of a spherical reflector, will after ■ tii-ction intersect that diameter of the sphere which contains the I'iiant point. This diameter is in fact a degraded form of one of L- two caustic surfecea touched by systems of rays in general, ■ ing the loci of the centres of principal curvature of the surfece which the rays are noi-mal. The concentration of rays on one i.imeter thus effecte<i. doe.s not require the proximity of the I idiant point to the reflecting surface. Judging from some observations that I have made in St Paul's whispering gallery, I am disposed to think that the principal phenomenon is to be explained somewhat differently. The ab- normal loudness with which a whisper is heard is not confined to the position diametrically opposite to that occupied by the whisperer, and therefore, it would appear, does not depend materially upon the symmetry of the dome. The whisper seems to creep round the gallery horizontally, not necessarily along the shorter arc. but rather along that arc towards which the whisperer faces. This is a consequence of the very unequal audibility of a whisper in front of and behind the speaker, a phenomenon which may easily be observed in the open air". Let lis consider the course of the rays diverging fi'om a radiant point P, situated near the surface of a reflecting sphere, and let us denote the centre of the sphere by 0, and the diameter passing through P by AA'. so that A is the point on the surface nearest to P. If we fix our attention on a ray which issues from P at an u^W ± B with the tangent plane at A. we see that after any BOtnber uf reflections it continues to touch a concentric sphere of I OP co^0, 80 that the whole conical pencil of rays which ally make angles with the tangent plane at A numerically I than 6, is ever afterwards included between the reflecting e and that of the concentric sphere of radius OP cos $. The divergence in three dimensions entailing a dimiuishiug eaty varying as r~' is replaced by a divergence in two dinsen- I, like that of waves issuing from a source situated between AU7 On Souiul, 2Dd edition, ISTl, p, 145. Phil. Staff. (5), m. p. 458, 1877. WHI8PERISG GALLERIES. two parallel reflecting planes, with an intensity varying as r' The less rapid enfeeblement of sound by distance than that experienced is the leading feature in the phenomena of whit galleries, The thickness of the sheet included between the two sphered 1>ecomeB less and less as A approaches P, and in the limiting case of a radiant point situated on the surface of the reflector is expressed by OA{l-cos0), nr, i( $ be small, ^0A.$' approxi- mately. The solid angle of the pencil, which determines t whole amount of radiation in the sheet, is W^; so that as S diminished without limit the intensity becomes infinite, as com- pared with the intensity at a Bnite distance from a similar soiii in the open. It is evident that this clinging, so to speak, of sound to th4 surface of a concave wall does not depend upon the exactness o the spherical form. But in the case of a true sphere, or rather a any surfiice symmetrica! with respect to AA', there is in additicH the other kind of concentration spoken of at the commencement the present section which is peculiar to the point A' diametrical^ opposite to the source. It is probable that in the case of a near^ spherical dome like that of St Paul's a part of the observed e£Fecl depends upon the sjTnmetry, though perhaps the greater part i referable simply to the general concavity of the walls. The propagation of earthquake disturbances is probably affectet by the curvature of the surface of the globe acting like a whisper ing galleiy, and perhaps even sonorous vibrations generated at tlu surface of the laud or water do not entirL-Iy escape the same kiw of influence, In connection with the acoustics of public buildings there are many points which still remain obscure. It is important to bear in mind that the loss of sound in a single reflection at a smooth wall is very small, whether the wall be plane or curved. In order to prevent reverberation it may often be necessary to introduofl carpets or hangings to absorb the sound. In some cases tha presence of an audience is found sufficient to produce the deeireil effect. In the absence of all deadening material the prolongatioi of sound may be very considerable, of which perhaps the moa striking example is that afforded by the Baptistery at Pisa, wherfl the notes of the common chord sung consecutively may be hean] RESONANCE IN BUILDINGS. 129 ing on together for many seconds'. According to Henry' it is nptrittant to prevent the repeated reflection of sonud backwards lad forwards along the length of a hall intended for public apeak- Bg, which may be accomplished by suitably placed oblique RiriiMXS. In this way the number of reflections in a given time is ■te&eed, and the undue prolongation of sound is checked. ^^■8. Almost the only instance of acoustical refraction, which ^^^ practical interest, is the deviation of sonorous rays &om a netiUuear courae due to heterogeneity of the atmosphere. The Tiiriation of pressure at different levels does not of itself give rise to refraction, since the velocity of sound is independent of density; but, as was first pointed out by Prof Osborne Rejuolds', the case is different with the variations of temperature which are usually lobe met with. The temperature of the atmosphere is determined ally by the condensation or rarefaction, which any portion r must undergo in its passage from one level to another, and mat state is one of " convective equilibrium*,'' rather than of oity. According to this view the relfttion between pressure ■density is that expressed in (9) § 246, and the velocity of 1 is given by ^•=I-S©'"" <'^ 'To connect the pressure and density with the elevation {«), we Iiave the hydrostatical equation dp = -gpdi (2), from which and (1) we find F'-7,'-(7-l)ss (3), if K, be the velocity at the surface. The correspocding relation iHjpeen temperature and elevation obtained by means of equation (11)5246 is |-1-=^P^ W. E 6, is the temperature at the surface. i> obaeiraiione of my own, mode in IB83, gave the durattoD as IS Beconds. W ebftnses pitoh, both Bouudg are beard together and maj give rise to a, n-Xtsae, J 68. Bee HaberdJlul, Ueber die von Dvofiik beobBohteten Vori- Wien, Akad. Sitsbrr., 77, p. 304, 1B78.] I. Proc. 1866. p. 119. II 0/tbt Royal Society, Vol. »tt. p. 531. 1S74. \, On llu conveetivt tquilSyritim of ftmperature in tht atmorphtrt, , iBei—es. 1 130 ATMOSPHEBIC REFRACTION. [288. According to (4) the &11 of temperature would be aboat 1** Cent in 330 feet [100 m.], which does not differ much firom the results of Glaisher*s balloon observations. When the sky is clear, the &11 of temperature during the day is more rapid than when the sky is cloudy, but towards sunset the temperature becomes approximately constant\ Probably on clear nights it is often warmer above than below. The explanation of acoustical refraction as dependent upon a variation of temperature with height is almost exactly the same as that of the optical phenomenon of mirage. The curvature (/>"*) of a ray, whose course is approximately horizontal, is easily estimated by the method given by Prof James Thomson". Normal planes drawn at two consecutive points along the ray meet at the centre of curvature and are tangential to the wave-sur&ce in its two con- secutive positions. The portions of rays at elevations z and z-{'hz respectively intercepted between the normal planes are to one another in the ratio p : p — Bz, and also, since they are described in the same time, in the ratio V : V-\-SV. Hence in the limit ^=-^Mr (5). p dz ^ ' In the normal state of the atmosphere a ray, which starts horizontally, turns gradually upwards, and at a sufficient distance passes over the head of an observer whose station is at the sain& level as the source. If the source be elevated, the sound is heard at the surface of the earth by means of a ray which starts witb. a downward inclination; but, if both the observer and the source be on the surface, there is no direct ray, and the sound i^ heard, if at all, by means of diffraction. The observer may theri- be said to be situated in a sound shadow, although there may b^ no obstacle in the direct line between himself and the source^ According to (3) so that p=. —-^ =- . — (6); or the radius of curvature of a horizontal ray is about ten times the height through which a body must fall under the action of 1 Nature, Sept. 80, 1S77. * Bee Bveiett^ On the OpHee qf Mirage. Phil Mag. (4) zly. pp. 161, MS. V OONVECTIVE EQUILIBRIUM. gravity in order to acquire a velocity equal to the velocity of sound. If the elevations of the observer and of the source be «i and Zi, the greatest distance at which the sound can be heard otherwise than by difiraction is V(2^,f.) + V(2^>p) (7). It ia not to be supposed that the condition of the atmosphere ia always such that the relation between velocity and elevation is tliat expressed in (3), When the sun is shining, the variation of temperature upwards is more rapid ; on the other hand, as ProE Reynolds has remarked, when rain is falling, a much slower varia- tion is to be expected. In the arctic regions, where the nights are long and still, radiation may have more influence than convec- tion in determining the equilibrium of temperature, and if so the propagation of sound in a horizontal direction would be favoured by the approximately isothermal condition of the atmosphere. The general differential equation for the path of a ray, when tiie sorfaces of equal velocity are parallel planes, is readily obtained from the law of sines. If 6 be the angle of incidence, K/sin 6 ia not altered by a refracting surface, and therefore in the case supposed remains constant along the whole course of a ray. If x be the horizontal co-ordinate, and the constant value of F/sin B be called c, we get dxjdz = F/VCc" — V'), -/.-& <«>■ If the law of velocity be that expressed in (3), 2VdV (7-1)?' 2_ (■ V*dV or. on effecting the integration, {7-l)-7a! = constant-|- V"y(c»-V)-R»sin-'{r/c) (9), in which V may be expressed in terms of z by (3). A simpler result will be obtained by taking an approximate fonn of (3), which will he accurate enough to represent the cases of practical interest. Neglecting the square and higher powers of J, ve may take F-.-V.-. + Sf-r^' CO). di- and thus 132 PATH OF A BAT. [288. Writing for brevity fi in place oi\g{y-- l)/^o'» we have By substitution in (8) the origin of x being taken so as to correspond with V=c, that is at the place where the ray is horizontal. Expressing V in terms of Xy we find whence /Si^ = - Fp"^ + ^ (e^* + g-^) (12). The path of each ray is therefore a catenary whose vertex is 2F» downwards ; the linear parameter is — ; Vr- , and varies from ray to ray. 289. Another cause of atmospheric refr'action is to be found in the action of wind. It has long been known that sounds are generally better heard to leeward than to windward of the source ; but the f&ct remained unexplained until Stokes^ pointed out that the increasing velocity of the wind overhead must interfere with the rectilinear propagation of souud rays. From Format's law of least time it follows that the course of a ray in a moving, but otherwise homogeneous, medium, is the same as it would be in a medium, of which all the parts are at rest, if the velocity of propagation be increased at every point by the component of the wind-velocity in the direction of the ray. If the wind he horizontal, and do not vary in the same horizontal plane, the course of a ray, whose direction is everywhere but slightly inclined to that of the wind, may be calculated on the same principles as were applied in the preceding section to the case of a variable temperature, the normal velocity of propagation at any point being increased, or diminished, by the local wind-velocity, according as the motion of the sound is to leeward or to windward. Thus, when the wind increases overhead, which may be looked upon as the normal state of things, a horizontal ray travelling to windward is gradually bent upwards, and at a moderate distance passes over the head of an observer ; rays ti-avelling with the wind, on the ^ BriL Ajuoc. Sep, 1857, p. 22. J other hand, are bent dowDwards, so that an observer to leeward of the source hears by a direct ray which atarta with a slight upward ■ inclination, and has the advantage of being ont of the way of obstructions for the greater part of its course. The law of refraction at a horizontal surface, in crossing which the velocity of the wind changes discontinuously, is easily investi- gated. It will be sufficient to consider the case in which the direction of the wind and the ray are in the same vertical plane. If tf be the angle of incidence, which is also the angle between the plane of tlie wave and the surface of separation, U be the velocity of the air in that direction which makes the smaller angle with the ray, and V be the common velocity of propagation, the velocity of the trace of the plane of the wave on the surface of separa- tion ia • (1). which quantity is unchanged by the refraction. If therefore W be the velocity of the wind on the second side, and & be the angle of refraction, V V si^ + ^=si^ + ^' ^^'' which differs from the ordinary optical law. If the wind-velocity vary continuously, the courae of a ray may be calculated from the condition that the expression (1) remaiDS constant. If we suppose that U=0, the greatest admissible value of W is E7''=F|cosec5-l] (3). At a stratum where U' has this value, the direction of the ray which started at an angle has become pai-allel to the refracting surges, and a stratum where U' has a greater value cannot be penetrated at all. Thus a ray travelling upwards in still air at an inclination (Jtt — B) to the horizon is reflected by a wind overhead I f velocity exceeding that given in (3), and this independently of ihe velocities of intermediate strata. To take a numerical example, all rays whose upward inclination is less than 11", are totally reflected by a wind of the sauie azimuth moving at the moderate speed of 15 miles per hour. The effects of such a wind on the iDpagation of sound cannot fail to be very important. Over the ! of still water sound moving to leeward, being confined 134 TOTAL REFLECTION BY WIND. [289. between parallel reflecting planes, diverges in two dimensicHis only, and may therefore be heard at distances fiEu* greater than would otherwise be possible. Another possible effect of the reflector overhead is to render sounds audible which in still air would be intercepted by hills or other obstacles intervening. For the production of these phenomena it is not necessary that there be absence of wind at the source of sound, but, as appears at onoe from the form of (2), merely that the difference of velocities U'—U attain a suflicient value. The differential equation to the path of a ray, when the wind- velocity U is continuously variable, is ''V^*-"*^ «■ whonce "huciUy-V^ W In comparing (5) with (8) of the preceding section, which is the corresponding equation for ordinary re&action, we must remember that V is now constant. If, for the sake of obtaining a definite result, we suppose that the law of variation of wind at different levels is that expressed by U^a-^fiz (6), we have fix^ ^l ^{(c±ff-V^ <^>' which is of the same form as (11) of the preceding section. The course of a ray is accordingly a catenary in the present case also, but there is a most important distinction between the two problems. When the refraction is of the ordinary kind, depending upon a variable velocity of propagation, the direction of a ray may be reversed. In the case of atmospheric refraction, due to a diminu- tion of temperature upwards, the course of a ray is a catenary, whose vertex is downwards, in whichever direction the ray may be propagated. When the re&action is due to wind, whose velocity increases upwards, according to the law expressed in (6) with fi positive, the path of a ray, whose direction is upwind, is also along a catenary with vertex downwards, but a ray whose direction is downwind cannot travel along this path. In the latter case the vertex of the catenary along which the ray travels is directed upwards. 290. In the paper by Reyoolds already referred to. an account iLgiven of some ioterestiiig experiments especially directed to te&t : theory of refraction by wind. It was found that " In the rtion of the wind, when it was strong, the sound (of an electric I3t) could be heard as well with the head on the ground as when I, even when in a hollow with the bell hidden from view by B slope of the ground ; and no advantage whatever was gained either by ascending to an elevation or raising the bell. Thus, with the wind over the grass the sound could be heard 140 yards, and over snow 360 yards, either with the head lifted or on the ground ; whereas at right angles to the wind on all occasions the range was extended by raising either the observer or the bell." " Elevation was found to affect the range of sound against the wind in a much more marked manner than at right angles." '■ Over the grass no sound could be heard with the head on the ground at 20 yards from the bell, and at 30 yards it was lost with the head 3 feet from the ground, and its full intensity was lost wbeo standing erect at 30 yai-ds. At 70 yards, when standing erect, the sound was lost at long intervals, and was only faintly heard even then ; but it became continuous again when the ear was raised 9 feet from the ground, and it reached its full intensity at an elevation of 12 feet." Prof. Reynolds thus sums up the results of his experiments ; — 1. " When there is no wind, sound proceeding over a rough garbce is more intense above than below." 2. " As long as the velocity of the wind is greater above than betow, sound is lifted up to windward and is not destroyed." 3. "Under the same circumstances it is brought down to leeward, and hence its range extended at the surface of the ground." Atmospheric refraction has an important bearing on the audibility of fog^ignals, a subject which within the last few years has occupied the attention of two eminent physicists. Prof. Henry in America and Prof. Tyndall in this country. Henry' attributes ftltoost all the vagaries of distant sounds to refraction, and has tews how it is possible by various suppositions as to the motion e air overhead to explain certain abnormal phenomena which e come under the notice of himself and other observers, while * Boport of the Lightbouse Board of the United Stat^is tor the yeai 1874. 136 tyndall's observations [290. T}i)dall\ whose investigations have been equally extensive, coHRiders the very limited distances to which sounds are sometimes audible to be due to an actual stopping of the sound by a flooculent condition of the atmosphere arising from unequal heating or moisture. That the latter cause is capable of operating in this direction to a certain extent cannot be doubted. Tyndall has proved by laboratory experiments that the sound of an electric bell may be sensibly intercepted by alternate layers of gases of different densitieH ; and, although it must be admitted that the alternations of density were both more considerable and more abrupt than ciin well be supposed to occur in the open air, except perhaps in the immediate neighbourhood of the solid ground, some of the observations on fog-signals themselves seem to point directly to the explanation in question. Thus it was found that the blast of a siren placed on the summit of a cliff overlooking the sea was followed by an echo of ^adually diminishing intensity, whose duration sometimes amounted to as much as 15 seconds. This phenomenon was obs4'rviKl 'when the sea was of glassy smoothness/' and cannot appirently be attributed to any other cause than that assigned to it by Tyndall. It is therefore probable that refraction and m;oustical opacity are both concerned in the capricious behaviour of fog-signals. A priori we should certainly be disposed to attach the greater import^mce to refraction, and Reynolds has shewn that sonui of Tyndall's own observations admit of explanation upon this I)rineiple. A failure in recijyrociti/ can only be explained in accordance with theory by the action of wind (§ 111). According to the hypothesis of acoustic clouds, a difference might be expected in the behaviour of sounds of long and of short iluration, which it may be worth while to point out here, as it does not appear to have been noticed by any previous writer. Since «'U(.Tgy is not lost in reflection and refraction, the intensity of radiation at a given distance from a continuous source of sound (or light) is not altered by an enveloping cloud of spherical form and of uniform density, the loss due to the intervening parts of the cloud being compensated by reflection from those which lie beyond the source. When, however, the soimd is of short duration, the intensity at a distance may be very much diminished by the cloud on account of the different distances of its reflecting parts and the ^ PhU, Tram. 1874. Sound, 8xd edition, Ch. vn. ON FOG-SIGN AL3. coosequeut drawing out of the sound, although the whole intensity, as measured by the time-integral, may be the same as if there had been no cloud at all. This is perhaps the explanation of Tyndall's observation, that different kinds of signals do not always preserve the same order of effectiveness. In some states of the weather a " howitzer firing a 3-lb. charge commanded a larger range than the whistles, trumpets, or syren," while on other days " the inferiority "f the gun to the syren was demonstrated in the clearest manner." It should be noticed, however, that in the same series of experi- ments it was found that the liability of the sound of a gun " to be quenched or deflected by an opposing wind, so as to be practically useless at a very short distance to windward, is very remai-kable." The refraction proper must, be the same for all kinds of sounds, but for the reason explained above, the diffraction round the edge of an obstacle may be less effective for the report of a gun than for the sustained note of a siren. Another point examined by Tyndall was the influence of fog on propagation of sound. In spite of isolated assertions to the itrary', it was generally believed on the authority of Derham that the influence of fog was prejudicial. Tyndall's observations prove satisfactorily that this opinion is erroneous, and that the passage of sound is favoured by the homogeneous condition of the atmosphere which is the usual concomitant of foggy weather. When the air is saturated with moisture, the fall of temperature with elevation according to the law of convective equilibrium is much less rapid than in the case of dry air, on account of the condensation of vapour which then accompanies expansion. From a calculation by Thomson' it appears that in warm fog the effect of evaporation and condensation would be to diminish the fall of temperature by one-half The acoustical refraction due to tem- :r3ture would thus be lessened, and in other respects no doubt condition of the air would be favourable to the propagation of id, provided no obstruction were offered by the suspended icles themselves. In a future chapter we shall investigate the iturbance of plane sonorous waves by a small obstacle, and we find that the effect depends upon the ratio of the diameter of ie obstacle to the wave-length of the sound. The reader who is desirous of pursuing this subject may te dtr I'hyiik. a. p. 217. 186S. 1 138 LAW OF DIVERGENCE OF SOUND. [290. consult a paper by Reynolds " On the Refraction of Sound by the Atmo6phere\" as well as the authorities already referred to. It may be mentioned that Reynolds agrees with Henry in conside^ ing refraction to be the really important cause of disturbance, bat further observations are much needed. See also § 294. 291. On the assumption that the disturbance at an aperture in a screen is the same as it would have been at the same place in the absence of the screen, we may solve various problems respecting the diffraction of sound by the same methods as are employed for the corresponding problems in physical optics. For example, the disturbance at a distance on the further side of an infinite plane wall, pierced with a circular aperture on which plane waves of sound impinge directly, may be calculated as in the analogous problem of the diffraction pattern formed at the focus of a circular object-glass. Thus in the case of a symmetrical speaking trumpet the sound is a maximum along the axis of the instrument, where all the elementary disturbances issuing frx)m the various points of the plane of the mouth are in one phase. In oblique direc- tions the intensity is less; but it does not fall materially short of the maximum value until the obliquity is such that the difference of distances of the nearest and furthest points of the mouth amounts to about half a wave-length. At a somewhat greater obliquity the mouth may be divided into two parts, of which the nearer gives an aggregate effect equal in magnitude, but opposite in phase, to that of the further ; so that the intensity in this direction vanishes. In directions still more oblique the sound revives, increases to an intensity equal to *017 of that along the axis', again diminishes to zero, and so on, the alternations corresponding to the bright and dark rings which surround the central patch of light in the image of a star. If R denote the radius of the mouth, the angle, at which the first silence occurs, is sin~^(*610X/J?). When the diameter of the mouth does not exceed ^X, the elementary disturbances combine without any considerable antagonism of phase, and the intensity is nearly uniform in all directions. It appears that concentration of sound along the axis requires that the ratio R : X should be large, a condition not usually satisfied in the ordinary use of speaking trumpets, whose efficiency depends rather upon an increase in the original volume I Phil. Tram. YoL 166, p. 815. 1S76. * y«tdet, Le^OHi d^opHpu phifgique, 1. 1. p. 806. of souod (§ 280). When, however, the vibrations are of very short wave-length, a tnimpet of moderate size is capable of effecting a consideiable concentration along the axis, as I have myself verified in the case of a hiss, 292. Although such calculations as those referred to in the preceding section are useful as giving us a general idea of the phenomena of diSraction, it must not be forgotten that the auxiliary' assumption on which they are founded is by no means strictly and generally true. Thus in the case of a wave directly incident upon a screen the normal velocity in the plane of the aperture is not constant, as has beeo supposed, but increases from the centre towards the edge, becoming infinite at the edge itself In order to investigate the conditions by which the actual velocity is determined, let us for the moment suppose that the aperture is filled up. The incident wave tf> = cos (nt — kx) is then perfectly refiected, and the velocity-potential on the negative side of the screen (« = 0) is = cos {ji( — ifli) -t- cos (ni + kx) (1), giving, when a; = 0, ^ = 2 cos nt. This corresponds to the vanish- ing of the normal velocity over the area of the aperture; the completion of the problem requires us to determine a variable normal velocity over the aperture such that the potential due to it (§ 278) shall increase by the constant quantity 2 cos nt in crossing from the negative to the positive side ; or, since the crossing involves simply a change of sign, to determine a value of the normal velocity over the area of the aperture which shall give on the positive side ^ = cos nt over the same area. The result of superposing the two motions thus defined satisfies all the condi- tions of the problem, giving the same velocity and pressure on the two sides of the aperture, and a vanishing normal velocity over the remainder of the screen. ft If P cos {nt + e) denote the value of d4>/dx at the various points the area (S) of the aperture, the condition for determining and « is by (6) § 278, ljjp^<^±Z^t±l}aS = cosnt (2), ^—wbere r denotes the distance between the element dS and any ^■faed point in the aperture. When P and e are kuovm, t.kft 140 DIFFRACTION THROUGH SMALL APEBTUBa [292. complete value of ^ for any point on the positive side of the screen is given by ^^_^JJ^cos(n^-^M:e)^ (3). and for any point on the negative side by ^ = + 2" jjP ^^ -^dSf + 2co8n<co8ib? (4). The expression of P and € for a finite aperture, even if of circular form, is probably beyond the power of known methods ; but in the case where the dimensions are very small in comparison with the wave-length the solution of the problem may be effected for the circle and the ellipse. If r be the distance between two points, both of which are situated in the aperture, kr may be neglected, and we then obtain from (2) -»■ '=-s//J't- w- shewing that — P/Ztt is the density of the matter which must be distributed over S in order to produce there the constant potential unity. At a distance from the opening on the positive side we may consider r as constant, and take _ , , cos (nt — kr) r (6). where 3/= — ^l|PdS, denoting the total quantity of matter which must be supposed to be distributed. It will be shewn on a future page (§ 306) that for an ellipse of semimajor axis a, and eccentricity e, M^a^F{e) (7), where F is the symbol of the complete elliptic function of the first kind. In the case of a circle, F{e) = Jtt, c^nd Jf-^ (8). This result is quite different from that which we should obtain on the hypothesis that the normal velocity in the aperture has the value proper to the primary wave. In that case by (3) § 283 . Tra" sin (nt — kr) »"-X r (»>• i If there be several Hmall apertures, whose distances apart mch greater than their dimensions, the same method gives co8(h(— ^T,) ,, cos(nt — kr,) >}>=M,- \-M,-' (10). The diffraction of sound is a subject which has attracted but little attention either from mathematicians or experimentalists. Although the general character of the phenomena is well under- stood, and therefore no very startling discoveries are to be expected, the exact theoretical solution of a few of the simpler fooblema, which the subject presents, would be interesting ; and, e»en with the present imperfect methods, something probably might be done in the way of experimental examination. 292 a. By means of a bird-call giving waves of about 1 cm. wave-length and a high pressure sensitive flame it is possible to imitate many interesting optical experiments. With this apparatus the shadow of an obstacle so small as the hand may be made ^jparent at a distance of several feet. An experiment shewing the antagonism between the parts of a wave corresponding to the first and second Fresnel's zones (§ 283) U Fig. 57 a. i- very effective. A large glass screen (Fig. 57 a) is perforated "ith a circular hole 20 cm. in diameter, and is so situated between ibe source of sound and the burner that the aperture corresponds ro ihe firet two zones. By means of a zinc plate, he\d doae to ^ka 142 EXPERIMENTS ON DIFFRACTION. [292 <L glass, the aperture may be reduced to 14 cm., and then admiU only the first zone. If the adjustments are well made, the flame, unaffected by the waves which penetrate the lai^r apertnze, flares violently when the aperture is further restricted by the zinc plate. Or, as an alternative, the perforated plate may be replaced by a disc of 14 cm. diameter, which allows the second zone to be operative while the first is blocked ofL If a, 6 denote the distances of the screen from the source and from the point of observation, the external radius p of the nth zone is given by or approximately ah ''' = "^^6 0> When a = 6, p^^inTiM (2). With the apertures specified above, p* = 49 for n = 1 ; p" = 100 for 71 = 2 ; so that Xa=100, the measurements being in centimetres. This gives the suitable distances when X is known. In an actual experiment X = 1*2, a = 83. The process of augmenting the total effect by blocking out the alternate zones may be carried much further. Thus when a suitable circular grating, cut out of a sheet of zinc, is interposed between the source of sound and the flame, the effect is many times greater than when the screen is removed altogether'. As in Sorct's corresponding optical experiment, the grating plays the part of a condensing lens. The focal length of the lens is determined by (1), which may be written in the form rWi-y o^ 80 that f^p'InX (4). In an actual grating constructed upon this plan eight zones — the first, third, fifth &c. — are occupied by metal. The radius of the first zone, or central circle, is 7*6 cm., so that p^/n = 58. Thus, if X= 1*2 cm.,/= 48 cm. If a and b are equal, each must be 96 cm. > ** Diffraction of Sound," Proe. Boy, Intt, Jan. 20, 1886. The condition of things at the centre of the shadow of a circular disc is Btill more easily investigated. If we construct in imagination a system of zones beginning with the circular edge of the disc, we see, as in § 283, that the total effect at a point upon the axis, being represented by the half of that of the first zone, is the same as if no obstacle at all were interposed. This analogue of a famous optical phenomenon is readily exhibited'. In one experiment a glass disc 38 cm. in diameter was emjiloyed, and its distances from the source and from the flame were respectively 70 cm. and 25 cm. A bird-call giving a pure tone (X = 1'5 cm.) is suitable, but may be replaced by a toy reed or other source giving short, though not necessarily simple, waves. In private work the ear furnished with a rubber tube may be used instead of a sensitive 6ame. The region of no sensible shadow, though not confined to a mathematical point upon the axis, is of small dimensions, and a Tcry moderate movement of the disc in its own plane suffices to reduce the flame to quiet. Immediately surrounding the central spot there is a ring of almost complete silence, and beyond that again a moderate revival of effect. The calculation of the in- teDHtty of sound at points off the axis of symmetry is too com- plicated to be entered upon here. The results obtained by Lommel' may be readily adapted to the acoustical problem. With the data Bpocified above the diameter of the silent ring immediately surrounding the central region of activity is about 1-7 cm. 293. The value of a function^ which satisfies V'.^ = through- out the interior of a simply -connected closed space S can be expressed as the potential of matter distributed over the surface of S. In a certain sense this is also true of the class of functions with which we are now occupied, which satisfy V'0-(-A:'0 = O. The following is Helmhottz's proof*. By Green's theorem, if ^ aod ^ denote any two functions of x, y, z, lji''S''Hlli'''-i"''''SI*t''-llh'"i-'''- ' " Acomtical ObBervBtionB," Phil. Mag. Vol. n. p. 281, 1880 1 Proc. Soy. Iml. • Abh. dfT baycr. Akad. dtr H'iM. ii. CI., xv. Bd., li, Abtli. See Alio Eneyek- t4iu Hritanniea, Article ■■ Wiive Theory." * Thtori* drr Lufucliminiiiiniirn ia Riihren mi'C ofenin Eiidett. Cretle, Bd. Lni. EXTENSION OF GEEENS THEORI To each side add - jjlk'ttiyl^dV; then if a" {V'tf, + /.•»0) + <1> = 0, a' (V'l/r + k'^}r) + ■^ = 0. a-fJ4.''i<lS+liJ4,VdV=a-jj^^/jS+jjjf1>dV...W. If and ^ vanish within S, we have simply Iht-'HhP' «■ Suppose, however, that *--'* (3). where r represents the distance of any point from a fixed origin within S. At all points, except 0, <1> vanishes ; and the last term in (1) becomee jj f^0dV = -a' |'//t V' (i) dV= Wa'yjr. 1^ referring to the point 0. Thus in which, if "^ vanish, we have an expression for the value of ^ at any interior point in terms of the surface values of ^fr and of d'^jdn. In the case of the common potential, on which we fall back by putting fr = 0, -^ would be determined by the surface' values of dyfr/dn only. But with A; finite, this law ceases to bo' universally true. For a given space S there is, as in the case' investigated in § 267, a series of determinate values of k, corre- sponding to the periods of the possible modes of simple harmonic vibration which may take place within a closed rigid envelope, having the form of S. With any of these values of k, it is obvious. that ifr cannot be determined by its normal variation over S, and the fact that it satisfies throughout S the equation V^i^ + /r*^ But if the supposed value of k do not coincide with one of the series, then the problem is determinate ; for the difference of any two possible solutions, if finite, would satisfy the condition of giving no normal velocity over S, a condition which by hypothesis cannot be satisfied with the assumed value of k. HELMHOLTZS THEOREM. If the dimensions of the space S be very small in comparison tth \(=2ir/k), e~^ may be replaced by unity; and we leam tat ^ differs but little from a function which satisfies throughout i the equation V'0 = 0. 294. On his extension of Green's theorem (1) Helmholtz mds his proof of the important theorem contained in the following tateiuent; If in a spacefilled witfi air which is paHly bounded by hiUly extended fixed bodies and ia partly unbounded, sound waves il excited at any point A. tlie reaultinff velocity-potential at a second mint B IS the same both in magnitude and phase, as it would have n at A, had B been the source of the sound. If the equation '■m jds=.jjj(,)«i.-,f*)ir. (1), in which and ■^ are arbitrary functions, and 4> = - a'{V'0 + L-'<f>), * = - a' ( V'-^ + t-tfr). be applied to a space completely enclosed by a rigid boundary and containing any number of detached rigid fixed bodies, and if (f>, -^ he velocity-potentials due to sources within S, we get IJj(f<l>-4:f)dV=0 (2). L Ttus, if <t> he due to a source concentrated in ooe point A, "J>= Kucept at that point, and where I|l»t>d7 represents the intensity of the source. Similarly, if ^ be due to a source situated at B, Jlf^fdr.i,,jjjvdv. Accordingly, if the sources be finite and equal, so that jjj<t'dV=jjj^rdV (3), it follows that ki'.-'t'. (4). ch is the symbolical statement of Helmholtz'a theorem. B.U. 10 Uf) HELMHOLTZ'S THEOREM. [294. If tlio spaco S extuiid to intinity, the surface integral still vanishes, and the result is the sjime ; but it is not necessary to go into detail here, as this theorem is included in the vastly more general principle of reciprocity established in Chapter V. The investigation there given shews that the principle remains true io the presence of dissipative forces, provided that these arise fronci resistances varj'ing as the first power of the velocity, that the fluid need not be homogeneous, nor the neighbouring bodies rigid or fixed. In the application to infinite space, all obscurity is avoided by supposing the vibrations to be slowly dissipated after having escaped to a distance from A and B, the sources under contemplation. The reader must carefully remember that in this theorem ecjual sources of sound are those produced by the periodic intro- duction and abstraction of equal quantities of fluid, or something whose effect is the same, and that equal sources do not necessarily evolve equal amounts of energy in equal times. For instance, a source close to the surface of a large obstacle emits twice as much energy as an equal source situated in the open. As an example of the use of this theorem we may take the case of a hearing, or speaking, trumpet consisting of a conical tubej whose efficiency is thus seen to be the same, whether a sound pro- duced at a point outside is observed at the vertex of the cone, or a source of ecjual strength situated at the vertex is observed at th^ external point. It is important also to bear in mind that Helmholtz*s form of the reciprocity theorem is applicable only to simple sources of souudi which in the absence of obstacles would generate symmetricA^ waves. As we shall see more clearly in a subsequent chapter, it is possible to have sources of sound, which, though concentrated in an infinitely small region, do not satisfy this condition. It will be sufficient here to consider the case of double sources, for which the modified reciprocal theorem has an interest of its own. Let us suppose that ^ is a simple source, giving at a point B the potential - '^, and that ^' is an equal and opposite source situated at a neighbouring point, whose potential at £ is -^ + A*^. If both sources be in operation simultaneously, the potential at B is A*^. Now let us suppose that there is a simple source at B^ APPUOATION TO DOUBLE BOtTROBS. 147 intensity and phase arc the same as those of the soiirces at A' ; the resulting potential at A is i^, and at A' ■<^ + Ai^, distance AA' be denoted by k, and be supposed to diminish it limit, the velocity of the fluid at A in the direction AA' limit of A\}r/h. Hence, if we defioe a unit double source I limit of two equal and opposite simple sources whose dis- is diminished, and whose intensity is increased without in such a manner that the product of the intensity and idistance is the same as for two unit simple sources placed at distance apart, we may say that the velocity of the fluid in direction AA' due to a unit simple source at fi is numeri- eijual to the potential at B due to a unit double source al A, axis is in the direction AA'. This theorem, be it observed. i» Ime in spite nf any obstacles or reflectors that may exist in the Beighboorbood of the sources. Again, if A A' and BB" represent two unit double sources of the sune )ihase, the velocity at B in direction BB' due to the source ^-■1' is the same as the velocity at A in direction AA' due to the wiiree BR. These and other results of a like character may also be obtained on an immediate application of the general principle of S'OS. These examples will be sufficient to shew that in applying wf principle of reciprocity it is necessary to attend to the character *' the sources, A double source, situated in an open space, is in- •"Iible from any point in its equatorial plane, but it does not follow that a simple source in the equatorial plane is inaudible fnwn the position of the double source. On this principle, I believe, ""J be explained a curious experiment by Tyndall ', in which "iflre was an apparent failure of reciprocity'. The source of sound Wiployed was a reed of very high pitch, mounted in a tube, along *H09e a^is the intensity was considerably greater than in oblique 'ftf^ions. The kinetic energy T of the motion within a closed i 8 is expressed by -y-mt? ..(1); VctMingK of thf Rnynl Im I, Jan. 1375. Also Tyndall, On Sound. 3: "On the ApplicBlion of tha Principle of Bacipiocity to AcouEties." y Frotttdingi, VoL xxv. p. 118, 1876, or PhU. Mag. (6). ai. p. 300. 148 VARIATION OF TOTAL ENERGY. [295. ~f,Jl4.^ds-p,IIJ4^-i,ir (t\ by Green's theorem. For the potential energy Vi we have by (12) § 246 "-fe///*"^ <="■ by the general equation of motion (9) § 244. Thus, if £ denote the whole energy within the space S, dE f-4*S^^|:///f*^- ('^ of which the first term represents the work transmitted across the boundary S, and the second represents the work done by internal sources of sound. If the boundary £f be a fixed rigid envelope, and there be no internal sources, E retains its initial value throughout the motioa This principle has been applied by Kirchhoff* to prove the deter- minateness of the motion resulting from given arbitrary initial conditions. Since every element of ^ is positive, there can be no motion within S, if E be zero. Now, if there were two motions possible corresponding to the same initial conditions, their differ- ence would be a motion for which the initial value of E was zero; but by what has just been said such a motion cannot exist. ^ Vorltiungen Hher Math, Phy$ik, p. 811. CHAPTER XV. FDRTHER APPLICATION OF THE GENERAL EQUATIONS. 296. When a train of plane waves, otherwise unimpeded, impinges upon a apace occupied by matter, whose mechanical pro- pertiejs differ from those of the surrounding medium, secondary waves are thrown off, which may be regarded as a disturbance due to the change in the nature of the medium — a point of view more especially appropriate, when the region of disturbance, as well as the alteration of mechanical properties, ia small. If the medium and the obstacle be fluid, the mechanical properties ■poken of are two — the compressibility and the density: no account ia here taken of friction or viscosity. In the chapter on spherical harmonic analysis we shall consider the problem here proposed on the supposition that the obstacle is spherical, without any restriction as to the smallness of the change of mechanical properties; in the present investigation the form of the obstacle is arbitrary, but we assume that the squares and higher powers of the changes of mechanical properties may be omitted. If fi Vt K denote the displacements parallel to the axes of co-ordinates of the particle, whose equilibrium position is defined by X, y, X, and if o- be the normal density, and m the constant of compressibiKty so that Sp = ms, the equations of motion are d'Pd{7ns)_ ■■(1). and two similar equations in t] and ^. Ou the assumption that the whole motion is proportional to e"*", where as usual A- = 2ir/X, and (§ 244) a» = m/<r, (1) may be written <'-g^'-<r4'»-t-0 (2). 150 SECONDARY WAVES [296. The relation between the condensation «, and the displace- ments f, 77, f, obtained by integrating (3) § 238 with respect to the time, is (3). dx dy dz For the system of primary waves advancing in the directioi of —a;, 77 and f vanish; if f©, «o be the values of f and «, and mo, 0*0 be the mechanical constants for the undisturbed medium, we have as in (2) d (rrioSo) dx -<roA;«a*fo = (4); but fo, *o do not satisfy (2) at the region of disturbance on account of the variation in m and cr, which occurs there. Let us assume that the complete values are f + f . V> ?» ^o + * S and substitute in (2). Then taking account of (4), we get d (ms) f , . «• . / V dso . dm , v » « . .. ^ or, as it may also be written, ^ (ms) - ak'a?^ + ^ (Am.»o) - Acr.ifc^a'fo = (5), if Am, Act stand respectively for m — m©, <r — cto. The equations in 77 and ^ are in like manner -v- (m«) — al^a^t] + -1- (Am.^o) = -T- (ms) - <rA;'a'f + ^ (Am.So) = (6). It is to be observed that Am, Ao- vanish, except through a small space, which is regarded as the region of disturbance; f» ^» Ky *» being the result of the disturbance are to be treated as small quantities of the order Am, Act; so that in our ap- proximate analysis the variations of m and a* in the first two terms of (5) and (6) are to be neglected, being there multiplied by small quantities. We thus obtain from (5) and (6) by differ- entiation and addition, with use of (3), as the differential equation in s, V> ims) + y^ms = ]f?a^ j- (Act. fo) - V» (Am.«o) (7). > [This notation was adopted for bierity* It mi^t be (st^+AI, fsf«+Af, &0.; 80 that I, «, &o. should retain tlidr fdOMT dearav to tdB# 296.]] DUB TO VARIATION OP MEDIUM. 151 As in § 277, the solution of (7) is 4«n * =///^ j^' (Am.^o) - A^a* ^ (Act. f o) | dF (8), in which the integration extends over a volume completely in- cluding the region of disturbance. The integrals in (8) may be transformed with the aid of Green's theorem. Calling the two parts respectively P and Q, we have where 8 denotes the surfeice of the space through which the triple integration extends. Now on 5, Am and -i- (Am.*©) vanish, so that both the surface integrals disappear. Moreover \ r J rdr* r and thus ^Am.SodV' (9). -—Ill' If the region of disturbance be small in comparison with X, we may write P = -1(^80^ jjUmdF (10). In like manner for the second integral in (8), we find = i»a«f||A(r.fo^(^)eir=tfc»a«fo/i^ where fi denotes the cosine of the angle between x and r. The linear dimension of the region of disturbance is neglected in comparison with X, and X is neglected in comparison with r. If T be the volume of the space through which Am, Ao* are sensible, we may write jjUmdV= r.Am, [lTAa-dF= T.Acr, 152 LAW OF DEPENDENCE ON WAVE-UENOTH. [296. if on the right-hand sides Am, Ao* refer to the mean values d the variations in question. Thus from (8) « = - ^^^^^ |Am.«^-t4a«Ao-.foM| (12). To express fo i^ terms of «o, we have from (3), fo = ~/^c2a?; and thus, if the condensation for the primary waves be «^ = ^<*'*^, iA:^« = — «o> £^d (12) may be put into the form .(13), irTer^ (Am . Ao- \*r I m a- in which Sq denotes the condensation of the primary waves at the place of disturbance at time t, and a denotes the condensa- tion of the secondary waves at the same time at a distance r from the disturbance. Since the difference of phase represented by the factor er^ corresponds simply to the distance r, we may consider that a simple reversal of phase occurs at the place of disturbance. The amplitude of the secondary waves is inversely proportional to the distance r, and to the square of the wave-length \, Of the two terms expressed in (13) the first is symmetrical in all directions round the place of disturbance, while the second varies as the cosine of the angle between the primary and the secondaiy rays. Thus a place at which m varies behaves as a simple source, and a place at which a varies behaves as a double source (§ 294). That the secondary disturbance must vary as X~* may be proved immediately by the method of dimensions. Am and Act being given, the amplitude is necessarily proportional to T, and in accordance with the principle of energy must also vary inversely as r. Now the only quantities (dependent upon space, time, and mass) of which the ratio of amplitudes can be a function, are T, r, X, a (the velocity of sound), and cr, of which the last cannot occur in the expression of a simple ratio, as it is the only one of the five which involves a reference to mass. Of the remaining four quantities T, r, X, and a, the last is the only one which involves a reference to time, and is therefore excluded. We are left with r, r, and X, of which the only combination varying as 2V""*, and independent of the unit of length, is Tr~^ X~*.* An interesting application of the results of this section may be made to explain what have been called hamumic echoes*. 1 ** On the Light from the Sky," PhiL Mag. Feb. 1S71, and ** On the Hftttering of Light by smaU Partioles,*' Phil Mag. June, 1871. ' NtUwre^ 1S78, im, 819. SOUNDS ALTEKED IN CHARACTER. It' the primary sound be a compound niuaical note, the various ^nponeot tones are scattered in unlike proportions. The octave, : r example, is sixteen times stronger relatively to the funda- mental tone in the secondary than it was in the primary sound. There is thus no difficulty in understanding how it may happen that echoes relumed from such reflecting bodies as groups of trees may be raised an octave. The phenomenon has also a comple- mentary side. If a number of smalt bodie:! lie in the path of waves of sound, the vibrations which issue from them in all direc- : I'jns are at the expense of the energy of the main stream, and .\ here the sound is compound, the exaltation of the higher har- luunics in the scattered waves involves a proportional deficiency uf them in the direct wave after passing the obstacles, This is perhaps the explanation of certain echoes which are said to return a sound graver than the original ; for it is known that the pitch of a pure tone is apt to be estimated too low. But the evidence is conflicting, and the whole subject requires further careful expe- rimental investigation ; it may be commended to the attention of -■ liose who may have the necessary opportunities. While an altera- rmn in the character of a soimd is easily intelligible, and must indeed generally happen to a Umited extent, a change in the pitch of a simple tone would be a violation of the law of forced \-ibrations, and hardly to be reconciled with theoretical i<leas. In obtaining (13) we have neglected the effect of the vaiiable na.ture of the medium on the disturbance. When the disturb- ance on this supposition is thoroughly known, we might approxi- mate again in the same manner. The additional terms so obtained would be necessarily of the second order in Am, Aa, so that our expressions are in all cases correct as &r as the first powers of those quantities. Even when the region of disturbance is not small in com- ] arisen with X, the same method is applicable, provided the -ijuarea of Am, Ao- be really negligible. The total effect of any obstacle may then be calculated by integration from those of its piirts. In this way we may trace the transition from a small n?gion of disturbance whose surjace does not come into considera- tion, to a thin plate of a few or of a great many square wave- lengths in area, which will ultimately reflect according to the ri-gular optical law. But if the obstacle be at all elongated in the action of the primary rays, this method of calculati 154 SECONDARY SOURCES. [296. ceases to be practically available, because, even although the change of mechanical properties be very small, the inteiactioD of the various parts of the obstacle cannot be left out of account This caution is more especially needed in dealing with the case of light, where the wave-length is so exceedingly small in eompaiisoa with the dimensions of ordinary obstacles. 297. In some degree similar to the effect produced by a change in the mechanical properties of a small region of the fluid, is that which ensues when the square of the motion rises any- where to such importance that it can be no longer neglectei V'*^ + ^'8^ then acquires a finite value dependent upon the square of the motion. Such places therefore act like sources of sound; the periods of the sources including the submultiples of the ori- ginal period. Thus any part of space, at which the intensity jiccumulates to a sufficient extent, becomes itself a secondary source, emitting the harmonic tones of the primary sound. If there be two primary sounds of sufficient intensity, the secondary vibrations have frequencies which are the sums and differences of the frequencies of the primaries (§ 6S)\ 298. The pitch of a sound is liable to modification when the source and the recipient are in relative motion. It is clear, for instance, that an observer approaching a fixed source will meet the waves with a frequency exceeding that proper to the sound, by the number of wave-lengths passed over in a second of time. Thus if V be the velocity of the observer and a that of sound, the frequency is altered in the ratio a ±v : a, according as the motion is towards or from the source. Since the alteration of pitch is constant, a musical performance would still be heard in tune, although in the second case, when a and v are nearly equal, the fall in pitch would be so great as to destroy all musical character. If we could suppose v to be greater than a, a sound produced after the motion had begun would never reach the observer, but sounds previously excited would be gradually overtaken and heard in the reverse of the natural order. If v = 2a, the observer would hear a musical piece in correct time and tune, but backwards. Corresponding results ensue when the source is in motion and the observer at rest ; the alteration depending only on the relative motion in the line of hearing. If the source and the observer move with the same velocity there is no alteration of frequenqr, whether ^ Helmholti tlber CombinationstOne. Pogg. Atm. Bd. zcn. a. 487. 1851 ■:9s.] DOPPLERS PRINCIPLE. 155 mediam be in motion, or not. With a relative motion of I miles [64 kilometres] per hour the alteration of pitch is very .'i^picuous, amounting to about a semitone. The whistle of a loco- itive is heard too high ae it approaches, and too low as it recedes til an observer at a station, changing rather suddenly at the KiiQent o The principle of the alteration of pitch by relative motion was first enunciated by Doppler', and is often called Doppler's prin- ciple. Strangely enough its legitimacy was disputed by Petzval', whose objection was the result of a confusion between two perfectly distinct cases, that in which there is a relative motion if the source and recipient, and that in which the medium is in Tiiution while the source and the recipient are at rest. In the l:Htit-r case the circumstances are mechanically the same as if the ■I dium were at rest and the source and the recipient had a iiiraon motion, and therefore by Doppler's principle no change >.l pitch is to be expected. Doppler's principle haa been experimentally verified by Buijs Ballot* and Scott Bussell, who examined the alterations of pitch ! musical instruments carried on locomotivea A laboratory in- iiiment for proving the change of pitch due to motion has been :ivented by Much*. It consists of a tube six feet [183 cm.] in length, capable of turning about an axis at its centre. At one end is placed a small whistle or reed, which is blown by wind forced ■liong the axis of the tube. An observer situated in the plane of hition hears a note of fluctuating pitch, but if he places himself . I he prolongation of the axis of rotation, the sound becomes -.u.-;idy. Perhaps the simplest experiment is that described by Kiinig*. Two c" tuning-forks mounted on resonance cases are prepared to give with each other four beats per second. If the graver of the forks be made to approach the ear while the other retnaiiis at rest, one heat is lost for each two feet [61 cm.] of approach; if, however, it be the more acute of the two forks which Bifn>aches the ear, one beat is gained in the same distance. ^ Theoria des faibigan Uchtes der Doppelstenie. Fng, 1949. See PiBko, Die n AppariUc det Akaitik. Wien, ie«5. Witn. Ber. vm. 13*. 1862. Fortieliritte der Phyiik, vm. 107. E. Ann. un, p. 321. . p. SB, IB61, nnd cxti. p. J13S, 1862. fLCatahgac dn Appareili d'Acauitiquf. Paris, 1866. 156 doppler's principle. [298. A modification of this experiment due to Mayer' may also be noticed In this case one fork excites the vibrations of a second in umson with itself, the excitation being made apparent by a small pendulum, whose bob rests against the extremity of one of the prongs. If the exciting fork be at rest, the effect is apparent up to a distance of 60 feet [1830 cm.], but it ceases when the exciting fork is moved rapidly to or fro in the direction of the line joining the two forks. There is some difficulty in treating mathematically the problem of a moving source, arising from the feet that any practical source acts also as an obstacle. Thus in the case of a bell carried through the air, we should require to solve a problem difficult enough without including the vibrations at all. But the solution of such a problem, even if it could be obtained, would throw no particular light on Doppler's law, and we may therefore advan- tageously simplify the question by idealizing the bell into a simple source of sound. In § 147 we considered the problem of a moving source of disturbance in the case of a stretched string. The theory for aerial waves in one dimension is precisely similar, but for the general case of three dimensions some extension is necessary, in order to take account of the possibility of a motion across the direction of the sound rays. From ^ 273, 276 it appears that the effect at any point of a source of sound is the same, whether the source be at rest, or whether it move in any manner on the surface of a sphere described about as centre. If the source move in such a manner as to change its distance (r) from 0, its effect is altered in two ways. Not only is the phase of the disturbance on arrival at affected by the variation of distance, but the amplitads also undergoes a change. The latter complication however may be put out of account, if we limit ourselves to the case in which the source is sufficiently distant. On this understanding we may assert that the effect at of a disturbance generated at time t and at distance r is the same as that of a similar disturbance generated at the time t + Bt and at the distance 7* — aBt In the case of a periodic disturbance a velocity of approach (v) is equivalent to an increase of frequency in the ratio a : a + v, 299. We will now investigate the forced vibrations of the air contained within a rectangular chamber, due to internal souroes 1 PhiK Mag. (4), zuu. p. S78, 1872. RECTANGULAR CHAMBER. found. By § 267 it appears that the result at time ( of an lal condensation confined to the ueighbourhoud of the point ^ = S22taBj,j,cosfca(coa(p — 1 cos f 5 -^1 cos (r — j ...(1), ■ ^y '" {'' t) °°" (« j) '" (' ^l//*-*"'!"''- .(2). , which the effect of an impressed force may be deduced. Sin § 276. The disturbance fff ^fdxdydz communicated at C being denoted by ///*{*') dt'd.rdyda, or ^,{t')dt'. the lltant disturbance at time t is .(,'j)c»('?)-('l)/: ^,(t')coBk-a(t-t')df ...(9). The symmetr)' of this expression with rpspGct to a;, y, z and f, »;, t is an example of the principle of reciprocity (§ 107). In the case of a harmonic force, for which "?>, ((') = A cos mat', we have to consider the value of /> s mat' cos ka (t — 1f)di^ . ..(*)• » Strictly speaking, this integral has no definite value ; but, if wish for the expre^ssion of the forced vibrations only, we must t the integrated function at the lower limit, as may be seen BUpposiug the introduction of very small dissipative forces, thus obtain [' *. (C) COB ka (t -tf)dt = a sin mat ..(5). As might have been predicted, the expressions become infinite in caae of a coincidence between the period of the source and one of the natural periods of the chamber. Any particular normal vibration vrill not be excited, if the source be situated on one nf its loops. The effect of a multiplicity of sources may readily be inferred r ■ompifttiop or integration. 158 UNLIMITED TUBE. [SOO. 300. When sound is excited within a cylindrical pipe, the simplest kind of excitation that we can suppose is by the forced vibration of a piston. In this case the waves are plane from the beginning. But it is important also to inquire what happeiu when the source, instead of being uniformly diffused over the section, is concentrated in one point of it. If we assume (what, however, is not unreservedly true) that at a sufficient distance from the source the waves become plane, the law of reciprocity is sufficient to guide us to the desired information. Let ^ be a simple source in an unlimited tube, B, R two points of the same normal section in the region of plane waves. Ex hypotheaiy the potentials at B and R due to the source A are the same, and accordingly by the law of reciprocity equal sources at B and R would give the same potential at A. From this it follows that the effect of any source is the same at a distance, as if the source were uniformly diffused over the section which passes through it. For example, if B and R were equal sources in opposite phases, the disturbance at A would be nil. The energy emitted by a simple source situated within a tube may now be calculated. If the section of the tube be (x, and the source such that in the open the potential due to it would be A cos t(a^ — r) ,,v * = -i;^- r ^^^' the velocity-potential at a distance within the tube will be the same as if the cause of the disturbance were the motion of a piston at the origin, giving the same total displacement, and the energy emitted will also be the same. Now from (1) 27rr^ -^==\A cos kat ultimately, and therefore if '^ be the velocity-potential of the plane waves in the tube (supposed parallel to z)y we may take <r--^ = ^^co8A?(a< — ^) (2), corresponding to which dA ^ = -2^cosA?(ai-«) -....(8). ENERGY EMITTED. Hence, as in § 245, the energy ( W) the source is given by dW [ , dylr\ _paA-- rf( ■ '(-^f). 1 that in the long run W paA^ 8o- If the tube be stopped by an immovable piston placed close to the source, the whole energy is emitted in one direction ; but this is not alL In consequence of the doubled pressure, twice ..■ much energy as before is developed, and thus in this case W: paA* ,.(5). The narrower the tube, the greater is the energy issuing from •■I given source. It is interesting to compare the efficiency of a source at the stopped end of a cyiindrical tube with that of an equal source situated at the vertex of a cone. From § 280 wt hftve in the latter case. k'aA' , W' = p W: W' = . ; k'o .,(6). ..{7). I !ie energies emitted in the two cases are the same when m = i^tr, :!iiit is, when the section of the cylinder is equal to the area cut off by the cone from a sphere of radius i""'. 301. We have now to examine how far it is true that vibra- tions within a cylindrical tube become approximately plane at a suffident distance from their source. Taking the axis of s parallel lo the generating lines of the cylinder, let us investigate the iimtion, whose potential varies as e**"', on the positive side of a "iiree, situated at z = 0, If be the potential and V stand for ii\dj^+ d^jdy* the equation of the motion is 'it' \ + V" + !•■ U . If ^ be independeut of 2, it represents vibrations wholly -imaverae to the axis of the cylinder. If the potential be then tiiportional to e'P^, it must satisfy (V+p')(f,-0 (2), 160 VIBRATIONB IN UNLIMITED TUBES. [SOLJ as well as the condition that over the boundary of the section In order that these equations may be compatible, p is restricted to certain definite values corresponding to the periods of the natural vibrations. A zero value of p gives ^ = constant, which solution, though it is of no significance in the two dimension pro- blem, we shall presently have to consider. For each admissible value of Pf there is a definite normal function u of ^ and y (§ 9i\ such that a solution is if> = Au^P^ (4). Two functions u, u\ corresponding to different values of p, are conjugate, viz. make fjuu'dxdy = (5), and any function of x and y may be expanded within the contour in the series if>=:AoUo'{'AiUi-¥AiU^ + (6), in which tio» corresponding to p = 0, is constant. In the actual problem (f) may still be expanded in the same series, provided that Aq, Ai, &c. be regarded as functions off. By substitution in (1) we get, having regard to (2), + «,|^ + (A:*-l).')^} + -=0 (7), in which, by virtue of the conjugate property of the normal fiino- tions, each coefficient of u must vanish separately. Thus ^f-' + AM, = 0, ^ + ik,-p>)A^O (8), The solution of the first of these equations is Ao^Ooe^-^fio^-^, giving ^o = aot^^<**-^'»+Awo «*<«*-*» (9). The solution of the general equation in A assumes a difier^ form. accordinflT as jb* — i)* is positive or negative. If the fixoed 01.3 DISCRIMINATION OF CASES. I HI ibration be graver in pitch than the gravest of the purely tiviiiH- erse natural vibrations, every finite value of p' in grf*at(>r thrui Xr\ »r fc» — />* ia always negative. Putting *»-/>> = -/i» (10), we have A—ae^-k- ffe-*^, whence 4> = (ae^^ + ffe^') ii^^ Ml). Now under the circumstances supfKiHo^J, it in f'viiU'ul Ihiti, I,Im» motion does not become infinite with z, w) that all ^M*- rvfinM-ionlM a vanish. For a somewhat different Tf'Jir¥}U i\i" ^suwi im ti Uf of 9,,, as there can be no wave in the negative; tWr^irUnu. W«' umy therefore take an expression which reduces to its fint u-nu *h''/» z U ^uiViiMully great We conclude that in all carje** thh -utk-f^:" »iltj/ri;if'ly hotttitit- plane, if the forced vibration 4« yrnr,^tr M///> tha tfrnvpnl nf llw wUural transterse vibrotionui. In the case of a circular cvi;ndr:r. of rvJii^ /•, *.h' ;^r.i7'-<f '""»'! verse vibration has a wa7e-:»=:r./*r. •^«'; ^aj v, 2^^ •■ / ^ii .-MM; (5339). If then the wav^-l^nrh -t" '^^*'' f'''^'>"'^ /if'.r;.^»''f. """'I 5'tt3r, the waves ultimarriy 'vf^-/.rM^ piar*^ /^ f*"'/ *'"pf"" however that the wav^ ^lirin-.arr:;- '••y^y*ri-.»- p:?ir»', -li'-^'''*!^!' '*•" ^ve-Iength &1I ahon of za^. \r^.r:.. .;.v..r. PV r./;,fr.pl' . '^ "•'» ^rce of vibration be -ivmmerrcai v.r.r, r ■s^v•/■^ v, 'n'- »'•-' "^ "•" ^ube, tf.7. a *implti -"i-iiirnft ^irnar**.": -,r, 'r.*^^ ^i/.w >-''lf '*•' /»'*■"•'*■ ^nutfverse vibration ^:rh -^^hich *•• -ho'i^I n*i'/'- '/* '*''•»' '*' he more than an ■jccav.f hisfht^r 'har» .ji ^h'- '/rutv:%\ '■••■'". ""' the wave-Iensfth jt :he :or?^ v-.r.r^fcfion nvyht. ',m'/" j*-"* '*•'"* '"•" the abt^v*; value. Frorii < 12 .. ':vhen j = ) /: ifiadmiich tf '.^^ //T -f '1*7 sw t.I '*r.i-h I: Aonears ^ircnrrtinelv hAf r.*- .,l?rh<r v;,!,.-^ i.f * fi-»MrifA the .«me as voniri hp iriviur^rj ,t ^ ;Tfi^,| ,,iMf.i.ri •* '"'* ^ 162 REACTION OF AIR [SOL giving the same mean normal velocity as actually exists. Any normal motion of which the negative and positive parts are equal, produces ultimately no effect When there is no restriction on the character of the source, and when some of the transverse natural vibrations are graver than the actual one, some of the values of h^ —p^ are positive, and then terms enter of the form or in real quantities = )Su cos {t a« - V(i» - p») ^} (14), indicating that the peculiarities of the source are propagated to an infinite distance. The problem here considered may be regarded as a generaliza- tion of that of § 268. For the case of a circular cylinder it may be worked out completely with the aid of Bessel's functions, but this must be left to the reader. 302. In § 278 we have fully determined the motion of the air due to the normal periodic motion of a bounding plane plate of infinite extent. If d<f>ldn be the given normal velocity at the element dS, ^-iiifr^ (■) gives the velocity-potential at any point P distant r fix>m dS. The remainder of this chapter is devoted to the examination of the particular case of this problem which arises when the normal velocity has a given constant value over a circular area of radius R, while over the remainder of the plane it is zero. In particular we shall investigate what forces due to the reaction of the air will act on a rigid circular plate, vibrating with a simple harmonic motion in an equal circular aperture cut out of a rigid plane plate extending to infinity. For the whole variation of pressure acting on the plate we have (§ 244) JJSpdS = -. cr Jj ^dS = - tA»cr jT^dflf, 'n-2.J ON A VIBB-ITINO CTRCCrLAK PLATE. H-ri' 17 is the natural dt:nsity, and i^ v-ariea as e'*"'. Thus by (1) jjSpdS-'^^tS'^^dSdS' (2). In the double sum SS* dSdS' (3), iitiich we have uow to evaluate, each pair of elements is to be taken once only, and the product is to be summed after multipli- cation by the factor r~' e~'^, depending on their mutual distance. The best method is that suggested by Prof. Maxwell for the common potential'. The quantity (3) is regarded as the work that would be consumed in the complete dissociation of the matter composing the disc, that is to say, in the removal of every eiement from the influence of every other, on the supposition that the potential of two elements is proportional to r~' e~^. The smount of work required, which depends only on the initial and final states, may be calculatt'd by supposing the operation performed in any way that may be most convenient. For this i'lirpose we suppose that the disc is divided into elementary rings, .i| that each ring is carried away to infinity before any of the . .iLTior rings are disturbed. The first step is the calculation of the potential ( TO at the edge of a disc of radius c. Taking polar co-ordinates {p, Q) with any point of the circumference for pole, we have This quantity must be multiplied by l-jrcdc, and afterwards integrated with respect to c between the limits and fi. But it mil be convenient first to effect a transformation. We have = - i""cos(2jt-c8ine)d^-— f °sin(2tcsiu^)dfl = M2)-iK(z) (4). ivhure r is written for '2Jcc. J a {i) is the Eessel's function of zero ' Theory of Besonance. Phil. Tram. 1870. 164 REACTION 07 AIE [301 order (§ 200), and K(z)iB& function defined by the equation K<2) = -( sin(«8intf)dtf 2( z' «• ^ "l'.S'.6».7»"^ (5). Deferring for the moment tie further consideration of the function K, we have F-^[A-(.)-.-jl-^.Wl] » (6), and thus Now by (6) § 200 and (8) § 204 j\dzMz)=zM^) W; and thus, if A'l be defined by K,{z)={'zdzKiz) (8). 3 may wnte k V ksr )•■ From this the total pressure is derived by introduction of tto fiictor - T^ , BO that TT an The reaction of the air on the disc may thus be divided into two parts, of which the first is proportional to the velocity of the disc, and the second to the acceleration. If | denote the iS» placement of the disc, so that ^ = , , we LaTe^ = ita^=ita^ and therefore in the equation of motion of the disc, the teactitm c the air is represented by a fiictiooal force atr . wit? . ^ ( 1 — -^-W— retarding the motion, and by an accession to the inertia equal 1 When kR is small, we hai :iiS20O, J,(2tK) tR ' k'le ■ t'ram the ascending aeries for .i J^ !^^_ + ,11) .2=. 3''. 4 I.a'.S'.iS.S^"" ^ '• 1.2=. 3^ 1.2=.; 30 that the frictional term is approximately ^aff.TriE'.i'TP.f (12). I From the nature of the case the coefficient of j must be pmtive. otherwise the reaction of the air would tend to augment, instead of to diminish, the motion. That /, (z) is in fact always less than ij may be verified as follows. If d lie between and tt, and I be positive, ain (s sin d) — e sin 8 ia negative, and therefore also il> , {z sin ff)~tsin 0} sin dd h negative. But this integral is Ji {x) — ^z, which is accordingly negative for all positive values of e. When kR is great, ^i {"ikR) tends to vanish, and then the frictional term becomes wimply ac.Tri?.j. This result might have been expected ; for when kR is very targe, the wave motion in the neighbourhood of the disc becomes appi-oximately plane. We have then by (6) and (8) § 245, dp = ap„\, in which p, is the (tensity (o-); so that the retarding force is ■irR'Bp = atr.TrR'.^. We have now to consider the term representing an alteration ■ f inertia, and among other things to prove that this alteration is :i increase, or that ^i (2) is positive. By diruct integration of the .xiuding aeries (5) for K (which is always convergent), A'.(z)- Tr[l'-9 V.S'.s'^VVS'.S'.l (13). When therefore kR is smalt, we tiiay tatte as the expression for the increase of inertia HffR' TrJf. ■ (I*)- This part of the reaction of the air ia therefore represented by iiippoeiiig the vibrating plate to carry with it a mass of air equal 111 that contained in a cylinder whose base is the plate, and whose hfjj;ht is equal to 8fl/3ir ; so that, when the plate is sufficiently ^Tiiall, the mass to be added is independent of the period of ri brat ion. 166 BEACTION OF AIR [309 From the series (5) for K{£), it may be proved immediate! that IU4^^('^-1-^^^^ <^5)' 2 or From the first form (15) it follows that K,{z)^\^ K{z)zdz = ^z^z^p (17)1 By means of this expression for Ki {z) we may readily prove that the function is always positive. For dK{z) ^ rf 2 fi'g^ ^.^ 0)de^- f *'cos {z sin 0) sin Odd.. .(18); dz dz irJa ^ ' ttJo so that irj(^)= — il- I cos (^ sin ^) sin ^d^[ ^—(^sinmzsin0)8m0d0 (19), an integral of which every element is positive. When z is very large, cos {z sin 0) fluctuates with great rapidity, and thus K^ {z) tends to the form K^{z)^^.z (20). When z is great, the ascending series for K and JSTi, though always ultimately convergent, become useless for practical calculation, and it is necessary to resort to other processes. It will be observed that the differential equation (16) satisfied by if is the same as that belonging to the Bessel's function Jo> with the exception o: the term on the right-hand side, viz. 2/7r2:. The function K ij therefore included in the form obtained by adding to the genera solution of BesseFs equation containing two arbitrary constants anj particular solution of (16). Such a particular solution is i7r.ir(^)=-?-^-r-»+l«.8«.^»-l«.8«.5».^^+l«.8«.5«.7«.jr^-...(21), as may be readily verified on substitution. The series on tlM right of (21), notwithstanding its ultimate diwrgency* may b( ui^ Bucceesfully for computation when z is greab. It is in ftd BATING CIRCULAR ] ■CLOalj'tical equivalentof /^'"e-^(j» + /9')-*rf^,and we might take K{z) = — \ !'j"'^+ Complementary Function, termiuiug the two arbitrary coostauts by au examination of the s assumed when z is very great. But it is perhaps simpler to low the method used by Lipschitz ' for Beasel's functions. I By (4) we have 'dS = ..(22). nplex variable o ■• e~'"dv Dsider the integral I — = , where a i form u+iv. Representing, as usual, simultaneous pairs of of u anil V by the co-ordinates of a point, we see that the hie of the integral will be zero, if the integration with respect ; range round the rectangle, whose angular points are respec- rely 0, h,h + i, i, where h is any real positive quantity. Thus from which, if we suppose that A = x , I"' e-^dv ^ _ ^. r (T™ du . r e-"''-*^''du Jo •V'l-l/'^ Vp Vl + U' */ov'rT(M~+"t)'" Replacing m by A "^ ™*y write (23) in the form ;■ e-*"dv __■['_ r>e-^d(iv) _ ..(23). e^d$ e^''^>re-_''0^dd The 6rst term on the right in (24) is entirely imaginary ; it therefore follows by (22) that ^-ttJ^^z) is the real part of the second term. By expanding the binomial under the integral sign, and afterwards integrating by the formula f^''e-'>ff^id0=r(q + ^), s the expansion for J^ (z) in negative powers of z. we obtain COS (s — i t) 8in(z-i7r) (25). Lommi-l, SHulifit llbtT die BfteVachen Functi 168 REACTION OF AIR [302. By stopping the expansion after any desired number of tenn^ and forming the expression for the remainder, it may be pro?ed that the error committed by neglecting the remainder caimot exceed the last term retained (§ 200). In like manner the imaginary part of the right-hand member of (24) is the equivalent of -■^i'rrK{z), so that ir(i:) = -|2ri-^-»H-ia.8«.^-»-l>.8*.5a.^-'+ [ "*" V (^) r ^ ^ '^'- £)' "^ ^ '^' sf/m^ " }sin(«-i7r) (26). "\/(^){i^"iSm-» + }cos(^-i,r) The value of Ki{z) may now be determined by means of (1*^^ We find -- = -- {r-*-8.2r-44-l«.8a.6.2r-«-l«.3«.6«.7.-2r8H- } aZ TT _i_ //'^^«/.o/* 1 \ii _L 8.5.1 8.6.7.9.1.8.6, '^V W^^'^^'"^''M ^^^"^^^^ 1. 2.8.4. (S^)^ "^- /f^\c.;^/^ 1 \J 3 8.6.7.1.8 "\/Ur^'^^^-^^>|lT(8J)- 1.2.8.(S.)» ^8. 6. 7. 9. 11. 1.3. 6. 7 _ ) /o^x "*"~1l.2.3.4.6.(8^)» j ^ ^' The final expression for if j (2:) may be put into the form 2 irj(^)=- {2: + ^i-8.2^^-f l«.82.6.-?-»-l«.3«.6«.7.-?-'+ } TT -^,».(.-i.)|i-i'--'X/ ( l«-4)(3«-4)(5«-4)(7«-4) _ "*■ 1.2.8.4.(82^)* Vf-»n(.-iw)ri-<'-^Ht'>tn }'...W It appears then that Ki does not vanish when z is great, bu^ approximates to 2z/7r, But although the accession to the inertia^ > Ab wm to be ezpeoted, the leries within brackets are the same as fhoM thil ooear in iha axjwirion of the tanrtioii J^V^V ON A VIBRATING CIRCULAR PLATE. H nhich is proportional to A',, becomes infiiiite with R, it vanisheB ultimately when compai-ed with the area of the disc, and with the iitlier term which represents the dissipation. And this agrees i\ii,h what we should anticipate from the theory of plane waves. U, independently of the reaction of the air, the mass of the fii.we be M, and the force of restitution be /if, the equation of iijutioa of the plate when acteii on by an impressed force F, pro- Iportionai to e*", will be ijr by (13), if, as will be usual iu practical applications, kR be .imll, (j/ + '*'f-')f4?^^"£ + ^f--f. (30). Two particular cases of this problem deserve notice. Fii-st let Ji nnd /I vanish, so that the plato, itself devoid of mass, is subject Id no other forct;s than F and those arising from aerial pressures. Since f = ika^, the frictional term is relatively negligible, and we gel when kR is very small, (ztttK'. g^^e = -if (31). Nest let M and ft be sucb that the natural period of the plate, H'hen subject to the reaction of the air, is the same as that imposed upon it Under these circnmstances ' and therefore ta.77rR\^\^'4^F (32). Comparing with (31), we see that the amplitude of vibration is B&ter in the case when the inertia of the air is balanced, iu the do of 16 : S-irkR, shewing a large increase when kR is small. In e first case the phase of the motion is such that comparatively very little work is done by the force F; while iu the second, the inertia of the air is compensated by the spring, and then F, being of the same phase as tho velocity, does the maximum amount of wurk. CHAPTER XVI. THEORY OF RESONATORS. 303. In the pipe closed at one end and open at the other we had an example of a mass of air endowed with the property of vibrating in certain definite periods peculiar to itself in more or less com- plete independence of the external atmosphere. If the air beyond the open end were entirely without mass, the motion within the pipe would have no tendency to escape, and the contained column of air would behave like any other complex system not subject to dissipation. In actual experiment the inertia of the external air cannot, of course, be got rid of, but when the diameter of the pipe is small, the effect produced in the course of a few periods may be insignificant, and then vibrations once excited in the pipe have a certain degree of persistence. The narrower the channel of com- munication between the interior of a vessel and the external medium, the greater does the independence become. Such cavities constitute resonators; in the presence of an external source of sound, the contained air vibrates in unison, and with an amplitude dependent upon the relative magnitudes of the natural and forced periods, rising to great intensity in the case of approxi- mate isochronism. When the original cause of sound ceases, the resonator yields back the vibrations stored up as it were within it, thus becoming itself for a short time a secondary source. The theory of resonators constitutes an important branch of our subject. As an introduction to it we may take the simple case of a stopped cylinder, in which a piston moves without friction. On the further side of the piston the air is supposed to be devoid of inertia, so that the pressure is absolutely constant If now the piston be set into vibration of very long period, it is dear that the contained air will be at any time veiy nearly in the equi- librium condition (of uniioTm den»\i^^ c^ioRtes^tidiiig to the momentary position of the pistoa. If the maas of the piston be very cousitJerable in comparison with that of the included air, the ntural vibrationa resulting from a displacement will occur nearly Bif the air had no inertia: and in deriving the period from the I kinetic and potentnil energies, the foiiner may be calculated with- out allowance for the inertia of the air, and the latter aa if the rarefactioa and condensation were uniform, Under the circum- stJinces contemplated the air acts merely as a spring in virtue of its resistance to compression or dilatation ; the form of the contain- ing vessel ia therefore immatei'ia!. and the period of vibi-ation remains the same, provided the capacity be not varied. When a gas is compi-essed or rarefied, the mechanical value of the resulting displacement is found by multiplying each iufinitesi- mal increment of volume by the corresponding pressure and iutegrating over the range required. In the present case it is of coiitse only the difference of pressure on the two sides of the piston which is really operative, and this for a small change is proportional to the alteration of volume. The whole mechanical i-altie of the small change is the same as if the expansion were opposed throughout by the viean, that is half the final, pressure ; thus corresponding to a change of volume from S to S + SS, since 1) = a'o, r-p.~ SS~ipa-' ..(I)'. ilf* + '^ ^m If A denote the area of the pistoa, M its mass, and o: its linear ^■failacement, S8 = Ax, and the equation of motion is ^Bjical f Lei jjjcating vibrations, whose periodic time is i-aA\ -m. .■(3)- Let us now imagine a vessel containing air, whose interior communicates with the exteinal atmosphere by a narrow aperture or neck. It is not difficult to see that this system is capable of vibrations similar to those just considered, the air in the neigh- bourhood of the aperture supplying the place of the piston, By sufiiciently increasing S, the period of the vibration may be made M l ong as we please, and we obtain finally a state of things in ' Compare (12) g 245. 172 KINETIC ENERGY OF MOTION [303. which the kinetic energy of the motion may be neglected except in the neighbourhood of the aperture, and the potential energy may be calculated as if the density in the interior of the vessel were uniform. In flowing through the aperture under the operation of a difference of pressure on the two sides, or in virtue of its own inertia after such pressure has ceased, the air moves approximately as an incompressible fluid would do under like circumstances, provided that the space through which the kinetic energy is sensible be very small in comparison with the length of the wave. The suppositions on which we are about to proceed are not of course strictly correct as applied to actual resonators such as are used in experiment, but they are near enough to the mark to afford an instructive view of the subject and in many cases a foundation for a suiSciently accurate calculation of the pitch. They become rigorous only in the limit when the wave-length is indefinitely great in comparison with the dimensions of the vessel [On the above principles we may at once calculate the pitch of a resonator of volume S, whose cavity communicates with the external air by a long cylindrical neck of length L and area A. The mass of the aerial piston is pAL; so that (3) gives as the period of vibration =Vff) <*y. or, if X be the length of plane waves of the same pitch, X = aT = 27ry(^) (5). If the cross-section of the neck be a circle of radius R, A = frUr, and we obtain the formula (8) of § 307.] 304. The kinetic energy of the motion of an incompressible fluid through a given channel may be expressed in terms of the density p, and the rate of transfer, or current, Jf , for under the cir- cumstances contemplated the character of the motion is always the same. Since T necessarily varies as p and as X', we may put 2'=ip~ (1). where the constant c, which depends only on the nature of the annel, is a linear quantity, as may be inferred from the fiuait thrt • sions of X are 3 ii s velocity-poUtntial, space aiiil — 1 in time. In fact, if ijy f Green's theorem, where the integration is to be extended over b sor&ce including the whole region through which the moUoQ is Lsible. At a sutGcient distance on either side of the aperture, ^ wmes constant, and if the constant values he denoted by ^ and I the integration be now limited to that half of S towards prhich the fluid flows, we have = J p (*.-*.) //S "iS ' 4p (*. - « -f ■ determined linearly by its surface ^ dS, or X, is proportional to (^ — tp^). If we put = (^ — 00. we get as before T = i^p^'jc. Now, since within n Fig. 58. nature of the constant c will be better understood by coD' idering the electrical problem, whose mditioDS are mathematically identical itb those of that under discussion. et us suppose that the fluid is re- ilaced by uniformly conducting ma- irial, iuid that the boundary of the channel or aperture is replaced by in- lators. We know that if by battery iwer or otherwise, a difference of ilectric potential be maintained on the o sides, a steady euiTent through the ipertnre of proportional magnitude 'ill be generated. The ratio of the stal current to the electromotive force is called the conductivihj { the channel, and thus we see that our constant c represents imply this conductivity, on the supposition that the specific "ucting power of the hypothetical substance is unity. The ime thing may be otherwise expressed by sajHug that c ia the ide of the cube, whose resistance between opposite faces is the une as that of the channel. In the sequel we shall often avail of the electrical analogy. ^ ^^B 174 NATURAL PITCH OF RESONATORS. [304 When c is known, the proper tone of the resonator can be easily deduced. Since F=i/)a'^'. y = ip4^ (2). the equation of motion is Z + ^Z = (3), indicating simple oscillations performed in a time T=2^^y^' (4). If JV be the frequency, or number of complete vibrations executed in the unit time, ^=4Vs » The wave-length X, which is the quantity most closely con- nected with the dimensions of the cavity, is given by -^-vf and varies directly as the linear dimension. The wave-length, it will be observed, is a function of the size and shape of the resonator only, while the frequency depends also upon the nature of the gas ; and it is important to remark that it is on the nature of the gas in and near the channel that the pitch depends and not on that occupying the interior of the vessel, for the inertia of the air in the latter situation does not come into play, while the com- pressibility of all gases is very approximately the same. Thus in the case of a pipe, the substitution of hydrogen for air in the neighbourhood of a node would make but little diflference, but its effect in the neighbourhood of a loop would be considerable. Hitherto we have spoken of the channel of communication as single, but if there be more than one channel, the problem is not essentially altered. The same formula for the frequency is still applicable, if as before we understand by c the whole conduc- tivity between the interior and exterior of the vessel When the channeb are situated sufficiently {slt apart to act independently one of another, the resultant conductivity is the simple sum of those belonging to the separate channels ; otherwise the resultant £s less than that calculated by m^T^ 8ji<dd.t\Qiv. 304.] SUPERIOR AND INFERIOU LIMITS. 175 If there be two precisely similar channels, which do uot interfere, and whose conductivity taken separately is c. we have JV"=^/2> 27r v/| <"• shewing that the nute ia higher than if there were only one cbflimel in the ratio v'2 : 1, or by rather less than a fifth — a law observed by Sondhauss and proveil theoretically by Helmholtz in the case, where the channels of communication consist of simple holes in the infinitely thin sides of the reservoir. 305. The investigation of the conductivity for various kinds of channels is an important part of the theory of resonators ; but in all except a very few cases the accurate solution of the problem is beyond the power of existing mathematics. Some general principles throwing light on the question may however be laid down, and in many cases of interest an approximate solution, suiEcient for practical purposes, may be obtained. We know (^ 79, 242) that the energy of a fluid flowing through a channel cannot be greater than that of any fictitious motion giving the same total current. Hence, if the channel be narrowed in any way, or any obstruction be introduced, the con- ductivity is thereby diminished, because the alteration is of the nature of an additional constraint. Before the change the fluid was free to adopt the distribution of flow finally assumed. In cases where a rigorous solution cannot be obtained we may use the minimum property to estimate an inferior limit to the conductivity; the energy calculated from a hypothetical law of flow can never be less than the truth, and must exceed it unless the hypothetical and the actual notion coincide. Another general principle, which is of frequent use, may be more conveniently slated in electrical language. The quantity with which we are concerned ie the conductivity of a certain con- ductor composed of matter of unit specific conductivity. The jirinciple is that if the conductivity of any part of the conductor ho increased that of the whole is increased, and if the conductivity ■ if any part be diminished that of the whole is diminished, exception being made of certain very particular cases, where no alteration ensues. In its passage through a conductor electricity " "" " '■ 'fjjo that the energy dis^pated is fov a ^\fto, XfiftA 176 SIMPLE APERTURES. [305. current the least possible (§ 82). If now the specific resistance of any part be diminished, the total dissipation would be less than before, even if the distribution of currents remained unchanged. A fortion will this be the case, when the currents redistribute them- selves so as to make the dissi])ation a minimum. If an infinitely thin lamina of matter stretching across the channel be made' perfectly conducting, the resistance of the whole will be diminished, unless the lamina coincide with one of the undisturbed equipoten- tial surfaces. In the excepted case no effect will be produced. 306. Among different kinds of channels an important place must be assigned to those consisting of simple apertures in ud- limited plane walls of infinitesimal thickness. In practical appli- cations it is sufficient that a wall be very thin in proportion to the dimensions of the aperture, and approximately plane within a distance from the aperture large in proportion to the same quantity. On account of the symmetr}' on the two sides of the wall, the motion of the fluid in the plane of the aperture must be normal, and therefore the velocity-potential must be constant ; over the remainder of the plane the motion must be exclusively tiingential, so that to determine <t> on one side of the plane we have the conditions (i) (f) = constant over the aperture, (ii) d^jdn = over the rest of the plane of the wall, (iii) (f) = constant at infinity. Since we are concerned only with the differences of {f> we may suppose that at infinity <f> vanishes. It will be seen that conditions (ii) and (iii) are satisfied by supposing <^ to be the potential of attracting matter distributed over the aperture ; the remainder of the problem consists in determining the distribution of matter so that its potential may be constant over the same area. The problem is mathematically the same as that of determining the distribution of electricity on a charged conducting plate situated in an open space, whose form is that of the aperture under con- sideration, and the conductivity of the aperture may be expressed in terms of the mpucity of the plate of the statical problem. If <^ denote the constant potential in the aperture, the electrical resistance (for one side only) will be the integration extending over the area of the opening. Now 1 1 ^ do- = 27r X (whole quantity of lontter distributed), and thus, if if be the capacity, or charge corresponding to unit- poiential, the total resistance is (wJ/)-'. Accordingly for the con- ductivity, which is the reciprocal of the resistance, c = ■^■^f.. -in So far as I am aware, the ellipse is the only form of aperture fir which c or M can be determined theoretically', in which case the result is included in the known solution of the problem of 'iel^rmining the distribution of charge on an ellipsoidal conductor. From the fact that a shell bounded by two concentric, similar and similarly situated ellipsoids exerts no force on an interna! particle, it IS* easy to see that the superficial density at any point of an ellip- •'M necessary to give a constant potential is proportional to the jwrpeodicular (p) let fall from the centre upon the tangent plane l[ ihe point in question. Thus if p be the density, p = «p ; the whole quantity of matter Q is given by r = \lpdS = K\{pdS-- iTTKohc . ■■(2),= Qp 4vabc " Id the usual notation i-lc'-l-^/a'-y'/f, If we now suppose that c is infinitely small, we obtain the par- I ticular case of an elliptic plate, and if we no longer distinguish 1 the two surfaces, we get -AV^ -(*)■ rbe case of a resonator with an elliptic apertaie was eoneideced b; HelmholU s, Bd. 57, I860), whose reaiUt it equivalent to (8), ' Se bdng for the moment the third principal aiii of the ellipeoid. 178 ELLIPTIC APEBTDBB. [806. We have next to find the value of the constant potential (P). By considering the value of P at the centre of the plate, we see that p.//'-^.//,*^. Integrating first with respect to r, we have jpdr = Q -r 4a V(l - 6"cos*^), e being the eccentricity; and thus ivhere F is the symbol of the complete elliptic function of the fint order. Putting P = 1, we find as the final expression for the capacity of an ellipse, whose semi- major axis is a and eccentricity is e. In the particular case of the circle, « = 0, F(e) = ^tt, and thus for a circle of radius iJ, c = 2i2 (6). If the capacity of the resonator be S, we find from (6) § 304 '2S> X=s7r ^/(f) (n The area of the ellipse (<r) is given by and hence in terms of a l-Wi^'-'-^-"^ <»> When e is small, we obtain by expanding in powers of e pre- vious to integration, i i?'(«) = i^|l+g6» + g^e' + J^|«'+...} (9), 1 whence m :iOG.] COMPABIBON WITH CIRCULAR APERTURE. 171 Neglecting «• and higher powers, we have therefore -ve)-o-B-f*-") ('»'■ From thia result we see that, if its eccentricity be small, the iiinductivity of an elliptic aperture is very nearly the same as that of a circular aperture of equal area. Among various forms iif aperture of given area there must be one which has a minimum cunductivity, aud, though a formal proof might be difficult, it is s^y to recognise that this can be no other than the circle. An inferior limit to the value of c is thus always afforded by the con- ductivity of the circle of equal area, that is i-J^a-JTr), and when the true form is nearly circular, this limit may be taken as a close tppTOximution to the real value. The value of X is then given by X = 2M(r"*S* (U). In order to shew how slightly a moderate eccentricity affects the value of c, I have calculated the following short table with the iid of Legendre's values of F{e). Putting e = sin ^, we have i's>j' as the ratio of axes, and for the conductivity ve :ir) ' 2^/(cos ^) . i" (sin ■^) ' * « = sin 1^. b : a = COS 1^. »^2f(«)(l-«')'. ■00000 rooooo 1-0000 2C ■34-20* ■93969 1^0002 30 ■50000 ■86603 rooi3 40 ■642T9 ■76604 1^0044 5C ■76604 ■64279 roi22 60 ■86603 ■60000 1-0301 70 ■93969 ■34202 10724 30 ■984S1 ■1731)5 11954 90 100000 ■00000 CO The value of the last factor given in the fourth column is the ratio of the conductivity of the ellipse to that of a circle of equal <'reii. It appears that even when the ellipse la so eccea^.Yi.t "Coa-X 180 CALCULATION BASED ON ABEA. [306. the ratio of the axes is 2 : 1, the conductivity is iDcreased by only about 3 per ceut., which would correspond to an alteration of little more ihaa a comma (§ 18) in the pitch of a resonahH'. There seems no reason to suppose that this approximate inde- pendence of shape is a property peculiar to the ellipse, and « may conclude with some confidence that in the case of any mode- rately elongated oval aperHire, the conductivity may be calculated from the area alone with a considerable degree of accuracy. If the area be given, thire is no superior limit to c. For sup- pose the area o- to be distributed over n equal circles sufBcientlj far apart to act indopendently. The area of each circle is (r.'«, and its conductivity is 2(H7r)~*o*. The whole conductivity is b times as great, and therefore increases indefinitely with n. As a general rule, the more the opening is elongated or broken up, the greater will be the conductivity for a given area. To find a superior bmit to the conductivity of a given aperture we may avail ourselves of the principle that any addition to the aperture must be attended by an increase in the value of c. Thiu in the case of a square, we may be sure that c is less than for tht circumscribed circle, and we have already seen that it is greawt than for the circle of equal area. If b be the side of the square, 26 The tones of a resonator with a square aperture calculated from these two limits would differ by about a whole tone ; the graver of them would doubtless be much the nearer to the truth. This example shews that even when analysis fails to give a solution in the mathematical sense, we need uot be altogether in the dark M to the magnitudes of the quantities with which we are dealing. In the case of similar orifices, or systems of orifices, c varies aa the linear dimension. 307. Most resonators used in pmctice have necks of greater or less length, and even when there is nothing that would be called > neck, the thickness of the side of the reservoir cannot alwa^ be neglected. Wo shall therefore examine the conductivity of t channel formed by a cylindrical boring through an obstmctiiig plate bounded by parallel planes, and, though we fail to solve Ut( problem rigorously, we shall obtain information sufficient for bmL ' shall call L. aud Fig. 59. whence (!)■ practical purposes. The thickness of the plate m 'he radiiu of the cylindrical channel R. Whatever the resistance of the channel may be, i; will be lessened by the introduction of infinitely ihin discs of perfect conductivity at A and B. fig. 59. The eflfect of the discs is to produce constant potential Dvor their areas, and the problem thus mollified is susceptible of rigorous solution. Outside A and B the inolioQ is the same as that previously investi- gated, when the obstructiog plate is infinitely thin; between A and B the flow is uniform. The resist- unce is therefore on the whole -L+ '■- 2R wBf " L + i-rR If a denote the correctioQ, which must be added to L on sccouot of an open end, '-i'H (2). This correction is in general under the mark, but, when L is P small in compariaou with R, the assumed motion coincides B and more nearly with the actual motion, and thus the value f 01 in (2) tends to become correct. A superior limit to the resistance may be calculated from a Mthfitical motion of the fluid. For this purpose we will suppose dnjtely thin pistons introduced at A and B, the effect of which 1 be to make the norma! velocity constant at those places, pithio the tube the flow will be uniform as before, but for the lal space we have a new problem to consider : — To determine ! motion of a fluid bounded by an infinite plane, the normal city over a circular area of the plane having a given constant ■lae, and over the remainder of the plane being zero. The potential may still be regarded as due to matter distributed over the disc, but it is no longer constant over the area; the density iif the matter, however, being proportional to d<f>/dn is constant. The kinetic energy of the motion the integration going over the area of the circle. 182 CONDUCTIVITY OP NECKS. [307. The total current through the plane II an an Hence 2 kinetic energy _ jf^da (current)* it^B^ dn If the density of the matter be taken as unity, d^jdn = Stt, and the required ratio is expressed by Pjii^R^ where P denotes the potential on itself of a circular layer of matter of unit density and of radius R, The simplest method of calculating P depends upon the con- sideration that it represents the work required to break up the disc into infinitesimal elements and to remove them from each other 8 influence *. If we take polar co-ordinates (/», 0), the pole being at the edge of the disc whose radius is a, we have for the potential at the pole, V = ffdddp, the limits of p being and 2a cos 6, and those of being — ^ir and + i^r. Thus 7=4a (3> Now let us cut ofif a strip of breadth da from the edge of the disc. The work required to remove this to an infinite distance is iwada . 4a. If we gradually pare the disc down to nothing and carry all the parings to infinity ", we find for the total work by integrating with respect to a from to i2, ^■" 3 • The limit to the resistance (for one side) is thus 8/3w*JB; we conclude that the resistance of the whole channel is less than L 16^ (4). Collecting our results, we see that -R<a<^- R 4 OTT (5). ^ A part of § 802 is repeated here for the sake of those who may wish to ifoii the difficulties of the more complete investigation. * This method of oalcolatiiig P was suggested to the author hj Ckrk MaxweH P] COREECTION TO LENGTH. ViD decimals, a > -785 ill «<-849-R) It must be observed that a here denotes the correction for one nd. The whole reaistance corresponda to a length i + 2a of ube having the section trR'. ..(6). When L is very great in relatio ■n-fl' ^ L bo R, we may take simply m- 1 this case we have from (6) § 304 ''" R ■ ..(8). The correction for an open end (a) is a function of L, coinciding ■ the lower limit, viz. JttJJ, when L vanishes. As L increases. B with it ; but does not, even when L is infinite, attain Biperior limit 8fl/37r, For consider the motion going on in middle piece of the tube. The kinetic energy is greater than inds merely to the length of the piece. If therefore the i be removed, and the free ends brought together, the motion fftherwise continuing as before, the kinetic energy will be dimin- '"lied more than corresponds to the length of the piece subtracted. -■' fortiori will this be true of the real motion which would exist in 'ue shortened tube. That, when i = oo , o does not become 8ii/3'7r 's evident, because the normal velocity at the end, far from being "Mutant, as was assumed in the calculation of this result, must rie from the centre outwards and become infinite at the edge, fiarther approximation to the value of o may be obtained by Vflmming a variable velocity at the plane of the mouth. The c&Iculatioo will be found in Appendix A, It appears that in the B of an infinitely long tube a cannot be so great as '82422 R. 1 value of « is probably not far from '82 R. B^deB the cylinder there are very few forma of lel whose conductivity can be determined mathematically. however the form is approximately cylindrical we may r are useful as allowing us to ealimttte t,^! 184 TUBES OF REVOLUTION. [308.1 eflfect of such departures from mathematical accuracy as miM^' occur in practice. An inferior limit to the resistance of any elongated and approxi- mately straight conductor may be obtained immediately by the imaginary introduction of an infinite number of plane perfectly conducting layers perpendicular to the axis. If a denote the ara of the section at any point x, the resistance between two layen distant dx will be <r~*(ir, and therefore the whole actual resistance is certainly greater than /' -^dx (1). unless indeed the conductor be truly cylindrical In order to find a superior limit we may calculate the kinetic energy of the current on the hypothesis that the velocity parallel to the axis is uniform over each section. The hypothetical motion is that which would follow from the introduction of an infinite number of rigid pistons moving freely, and the calculated result is necessarily in excess of the truth, unless the section be absolutely constant. We shall suppose for the sake of simplicity that the channel is symmetrical about an axis, in which case of course the motion of the fluid is symmetrical also. If U denote the total current, we have ex hypothesi for the axial velocity at any point x u = <r''U (2), from which the radial velocity t; is determined by the equation of continuity (6 § 238), d(ru) ^ d(rv) ^Q dx dr ' Thus rv = const. — J Ur^ -j - , or, since there is no source of fluid on the axis, v=^-kUr^ (8> doD \ 1. V f f boa ] SUPERIOR LIMIT. The kinetic energy may now be calculated by simple intcgra- if If be the r&dius of the channel at the point x, so that a- = Try', Thui i kinetic energy _ (current)' ij?Hm\ ..(*)■ This is the quantity which gives a superior limit to the resist- ance. The first term, which corresponds to the component velocity «, is the same as that previously obtained for the lower limit, as might have been foreseen. The difference between the two, which gives the utmost error involved in taking either of them as the true value, is a^/j-d)'*'- ..(5), In a nearly cylindrical channel dy/dm is a small quantity and so the result found in this manner is closely approximate. It is not necessary that the section should be nearly constant, but only ihat it should vary slowly, The success of the approximation in this and similar cases depends upon the fact that the quantity to be estimated is at a minimum. Any reasonable approximation to the real motion will give a result very near the truth according to the principles of the differential calculus. By means of the properties of the potential and stream functions (§ 288) the present problem admits of actual approxi- mate solution. If ^ and -ijr denote the values of these functions at any point x,r; u, v denote the axial and transverse velocities. _d<^_l rff ~ dr~ r dx" whence by elimination rfy. 1 d0 d'0^ df'^ ■ dr da? d^_lrf^ d^_ rfr" r dr"^ ditf~ ■■(6), ■(7). ..(8). 186 APPROXIMATE CALCULATION. [308. M^ If J^ denote the value of ^ as a function of x when r»0, the la general values of ^ and -^ may be expressed in terms of ^ Ij |i means of (7) and (8) in the series r^F' 7^F'" f^F' f^F^ ' where accents denote differentiation with respect to ^ At the boundary of the channel where r = y, ^ is constant, say -^i. Then "^^^ 2 2«.4"^2«.4>.6 ^ ' is the equation connecting y and ^. In the present problem y is given, and we have to express F by means of it. By successive approximation we obtain from (10) y» 8 (dft" V y« y^ 8 ir« * <iB» V y* /J yL.^l^\ (11) The total stream is given by the integral and therefore the resistance between any two equipotential surfaces is represented by The expression for the resistance admits of considerable simpli- fication by integration by parts in the case when the channel is truly cylindrical in the neighbourhood of the limits of integration. In this way we find for the final result, resistance = /^,jl + i,'«-(^^--i^] (12y. y\ yf' denoting the differential coefficients of y with respect to x. It thus appears that the superior limit of the preceding investigation is in fact the correct result to the second order of EXPERIMENT. 187 ipproximation. If we regard y as a function of ax. where w is a small (juantity, (12) is correct as far as terms containing m*. 309. Our knowledge of the laws on which the pitch of rej»onalorM depends, is due to the labours of several experimentere iiiiti mathematicians. The observation that for a given mouthpiece the pitch of a resonator depends mainly upon the volume S is due to Liacovius, who found that the pitch of a flaak partly filled with water was not altered when the flaak was inclined. This result was con- firmed by Sondhauss'. The latter observer found further, that in the case of resonators without necks, the influence of the aperture depended mainly upon its area, although when the shape was very elongated, a certain rise of pitch ensued. He gave the formula N = 52400 ..(1), Si the unit of length being the millimetre. The theory of this kind of resonator we owe to Helmholtz', whose formula is (2). 2*^*5* applicable to circular apertures. For flasks with long necks, Sondhauss* found JV"= 4670.5- " Li Si '■ corresponding to the theoretical ll N. ..{*). la practice it does not often happen either that the neck o long that the correction for the open ends can be neglected, (*) Bupposes, or, on the other hand, so short that it can iteelf be neglected, as supposed in (2). Wertheim' was the fiist ' Ueber dea BrmnmltreiBel uuil das SobwiagQngBgeietE der oubiaoheii Pfeifen. ~ tf. Ann. Lixii. pp. 2S5, ail. 1B50. • Cwlle. Bd. Lvu. 1—73. 1800. ' Vtbta die Schnllftchwingnngen der Laft ia ethitstea OlagrBbren nod in godook- uigieiohei Weite. Poflff. Jnn. nin. p. 1. 1860. IT les vibrations aonoreg do I'air. Ann. d. Chim. (S) ixii. p, 385. 188 UELMHOLTZ'S INVESTIGATION. [309. to shew that the effect of an open end could be represented by an addition (a) to the length, independent, or nearly so, of L and X. The approximate theoretical determination of a is due to Helmholtz, who gave ^irR as the correction for an open end fitted with an infinite flange. His method consisted in inventing forms of tube for which the problem was soluble, and selecting that one which agreed most nearly with a cylinder. The cor- rection ^ttR is rigorously applicable to a tube whose radius at the open end and at a great distance from it is 12, but which in the neighbourhood of the open end bulges slightly. From the fact that the true cylinder may be derived by in- troducing an obstruction, we may infer that the result thus obtained is too small. It is curious that the process followed in this work, which was first given in the memoir on resonance, leads to exactly the same result, though it would be difiicult to conceive two methods more unlike each other. The correction to the length will depend to some extent upon whether the flow of air from the open end is obstructed, or not. When the neck projects into open space, there will be less ob- struction than when a backward flow is prevented by a flange as supposed in our approximate calculations. However, the un- certainty introduced in this way is not very important, and we may generally take a = ^7ri2 as a suflicient approximation. In practice, when the necks are short, the hypothesis of the flange agrees pretty well with fact, and when the necks are long, the correction is itself of subordinate importance. The general formula will then run ^"27rV Sl/i + iV(^^)} ^^^' where a is the area of the section of the neck, or in numbers 6¥832 fif V(li + -8863 V<^) ^ '' A formula not differing much from this was given, as the em- bodiment of the results of his measurements, by Sondhaoss^ who ^ Pogg, AnuL ozl. pp. 58, 810. 1870. at the same time expressed a conviction that it was no mere etnpiriciil formula of interpolation, but the expression of a natural law. The theory of resonators with necks was given about the same time' in a memoir 'on Resonance' published in the Pkilo- scpliical Transactions for 1871, from which most of the last few pages is derived. 310. The simple method of calculating the pitch of resonators with which we have been occupied is applicable to the graveet mode of vibration only, the character of which is quite distinct, The overtones of resonators with contracted necks are relatively very high, and the corresponding modes of vibration are by no means independent of the inertia of the air in the interior of the reservoir. The character of these modes will be more evident, when we come to consider the vibrations of air within a com- pletely closed vessel, such as a sphere, but it will rarely happen that the pitch can be calculated theoretically. There are, however, cases of multiple resonance to which our theory is applicable. These occur when two or more vessels com- municate by channels with each other and with the external air ; and are readily treated by Lagrange's method, provided of course that the wave-length of the vibration is sufficiently large in com- parison with the dimensions of the vessels. Suppose that there are two reservoirs, S, S', comraimicating _mtb each other and with the externa! air by narrow passages or Fig. 60. necks. If we were to consider SS' as a single reservoir and apply our previous formula, we should be led to an erroneous result ; for thai formula is founded on tbe assumption that within the reaervoii- ihe inertia of the air may be left out of account, whereas it is evident that the energy of the motion through the connecting S^ may be as great as through the two others. However, an ' Froetediagt of the Royal Society, Nov. 34, 1870. 190 DOUBLE BESONATOB. [810. investigation on the same general plan as before meets the cue perfectly. Denoting by Xi, X,, X^ the total tranafers of fluid through the three passages, we have as in (2) § 304 for the kinetic energy the expression C e. in and for the potential energy, ,(2). An application of Lagrange's method gives as the diflferential equations of motion, — + a» - -V- " = Ci S By addition and integration, Ci Cs Cs (4). (5). Hence on elimination of Xs, Assuming X, = ^e^*, X, = 5eP*, we obtain on substitution and determination of A : B, p* + p»a«(^J^ + ^|+^|c,c, + c,(c, + c,)| = 0...(6), as the equation to determine the natural tones. If ^ be the frequency of vibration, N* — -p'/^TT^, the two values of p* being of course real and negative. The formula simplifies considerably if Ct»Ci> iSf'=»jSf; but it will be more instructive to work out this case from the beginning. Let Ci » Og^mct^me. \ :0.] DOUBLE KESONATOR. The differential equations take the fonn .1=0 ■ m. I-X, ..(8). while from (4) X, Hence The whole motion may be divided into two parts. For the first of these X, + X. = (9). which requires that A', = 0. The motion is therefore the same as might take place were the communication between 5 and 8' cut off, and has its frequency given by iV' = ..(10). The density of the air is the same in both reservoirs. For the other component part, ^, — X, = 0, ao that ^.-?'i J^-="^?^° (U). The vibrations are thus opposed in phase. The ratio of frequencies is given by N'' :N^ = m + 2:m, shewing that the second mode has the shorter period. In this mode of vibration the connecting passage acte in some measure as a second opening to both vessels, and thus raises the pitch. If the passage be contracted, the interval of pitch between the two notes is small. A particular case of the general formula worthy of notice is obtained by putting c, = 0, which amounts to suppressing one of fe^fhe comnjuni cations with the external air. We thus obtain ^^^fhe communi ^-■^(t + I^W^o ('^>' 192 PARTICULAIt 0A8B. [31A.| or, if S«=iS', Ci^mCi^mc, p4 + a»pfV_.^_Z +__«o (13). whence a'c JV» = g^{m + 2±V(m» + 4)} (M) If we further suppose m = 1, or c, = Ci, If JV^' be the frequency for a simple resonator (S, c), i^'»= «'^ and thus iV? : N'' = ^^^ = 2618, iV'«:iVV= 0-^ = 2-618. It appears that the interval from Ni to N' is the same as from JV'' to iV^2, namely, ^(2*618) = 1*618, or rather more than a fifth. It will be found that whatever the value of m may be, the interval between the two tones cannot be less than 2*414, which is about an octave and a minor third. The corresponding value of m is 2. A similar method is applicable to any combination, however complicated, of reservoirs and connecting passages under the single restriction as to the comparative magnitudes of the reser- voirs and wave-lengths; but the example just given is sufficient to illustrate the theory of multiple resonance. A few measure- ments of the pitch of double resonatore are detailed in my memoir on resonance, already referred to. 311. The equations which we have employed hitherto take no account of the escape of energy from a resonator. If there were really no transfer of energy between a resonator and the external atmosphere, the motion would be isolated and of little practical interest : nevertheless the characteristic of a resonator consists in its vibrations being in great measure independent Vibrations, once excited, will continue for a considerable number of periods without much loss of energy, and their frequency will be almost entirely independent of the rate of dissipation. The rate of dissipation is, however, an important feature in the character I..] COMMUNICATION OF ENERGY. F I of a resonator, on which its behaviour under certain circumstances I materially depends. It will be understood that the dissipation lii're spoken of means only the escape of energy fi'om the vessel md its neighbourhood, and its diffusion in the surrounding medium, and not the transformation of ordinary energy into beat, or such transformation our equations take no account, unless special terms be introduced for the purpose of representing the effects of viscosity, and of the conduction and radiation of heat. [The influence of the conduction of heat has been considered byKol^k'.] In a previous chapter (§ 278) we saw how to express the motion JD the right of the infinite flange (Fig. 61), in terms of the normal ivWity of the fluid over the disc A. We found, § 278 (3), ^ 2ttJJ dii r where <t> is proportional to e*"'. If r be the distance between any two points of the disc, ^ is a mati quantity, and e~**' = 1 — ikr approximately. ^- ^^'-imT-'^isfj')- ..(1). The first terra depends upon the distribution of the current. If ■ I* suppose that d^jdn is constant, we obtain ultimately a term r- presenting an increase of inertia, or a correction to the length, H|Ual to 8ii/37r. This we have already considered, under the supposition, of a piston at A. The second term, on which the dissipation depends, is independent of the distribution of current, 1 iVIrd. Ann. 1. 12, p. 353, 1881, 194 RATE OF DISSIPATION. [311. being a function of the total current (X) only. Confining our attention to this term, we have *-t5 ® Assuming now that if> oc e^^, we have for the part of the varia- tion of pressure at A, on which dissipation depends. The corresponding work done during a transfer of fluid SX is a ^^ 9 ^^^ since, as in § 304, the expressions for the potential and kinetic energies are F = ipa'|-'. T=y^ (4), the equation of motion (§ 80) is ^*&^*T^"> (')■• in place of (3) § 304. In the valuation of c an allowance must be included for the inertia of the fluid on the right-hand side of A, corresponding to the term omitted in the expression for 8p. Equation (5) is of the standard form for the free vibrations of dissipative systems of one degree of freedom (§ 45). The amplitude varies as e~^*^^*^, being diminished in the ratio e : 1 after a time equal to ^ira/n^c. If the pitch (determined by n) be given, the vibrations have the greatest persistence when c is smallest, that is, when the neck is most contracted. If S be given, we have on substituting for c its value in terms of S and n, 47ra _ 4nra* ,^. shewing that under these circumstances the duration of the motion increases rapidly as n diminishes. In the case of similar resonators c x nr^, and then 47ra 1 n*c n ' ^ Equation (5) is only approximate, inaamuoh as the dissipative loroe is oakn* lated on the sapposilion that the vibration is permanent ; bat this will kad to no material enor when the dissipation is smalL 195 I irbional Iohs of ' 311.] NUMEEICAL EXAMPLE. which shews that In this case the same proportional Iobs of amplitude always occiirs after the lapse of the same number of periods. This result may be obtained by the method of di- mensions, as a consequence of the principle of dynamical similarity. As an example of (5), I may refer to the case of a globe with a neck, intended for burning phosphorus in oxygen gas, whose capacity is '251 cubic feet [7100 c.c.]. It was found by experiment that the note of maximum resonance marie 120 vibrations per secoiid, so that n = 1 20 x Z-rr. Taking the velocity of sound (o) at 1120 feet [34200 cent,] per second, we find from these data ^ „ = ; of a second nearly. Judging from the sound produced when the globe is struck, 1 think that this estimate must be too low; but it should be observed that the absence of the infinite Hange assumed in the iheury must influence very materially the rate of dissipation. We will now examine the forced vibrations due to a source of sound external to the resonator. If the pressure &p at the mouth of the resonator due to the source, i.e. calculated on the supposition that the mouth is closed, be F^', the equation of tnotioQ coiTeaponding to (5), but applicable to the forced vibration I n + '^'H + e^X-Fe^' (7). If Z = X.e'""", where X, a real. F^ [i_^\' (J^]' p'a'X,'~\'S o) '^Xiirl ■ The inaximum varifttion of pressure ((?) iuside the resonator ta connected with jr„ by the equation « = ^-' W. ItDoe X, -^ S is the maximum condensation. Thus which agrees with the equation obtained by Helmholtz for the coiie where the commuDicstton with the external air is by a )$tture (§ 306). The present problem is nearly, but not 13—2 ■tell 196 FORCED VIBRATIONS. [311. quite, a case of that treated in § 46, the difference depending upon the fiict that the coefficient of dissipation in (7) is itself a function of the period, and not an absolutely constant quantity. If the period, determined by k, and iS be given, (9) shews that the internal variation of pressure (G) is a maximum when c=l[^8, that is, when the natural note of the resonator (calculated without allowance for dissipation) is the same as that of the generatiog sound. The maximum vibration, when the coincidence of periods is perfect, varies inversely as S; but, if iSf be small, a very slight inequality in the periods is sufficient to cause a marked fidling off in the intensity of the resonance (§ 49). In the practical use of resonators it is not advantageous to carry the reduction of S and c very far, probably because the arrangements necessaiy for connecting the interior with the ear or other sensitive ap- paratus involve a departure fix>m the suppositions on which the calculations are founded, which becomes more and more important as the dimensions are reduced. When the sensitive apparatus is not in connection with the interior, as in the experiment of reinforcing the sound of a tuning-fork by means of a resonator, other elements enter into the question, and a distinct investigation is necessary (§ 319). In virtue of the principle of reciprocity the investigation of the preceding paragraph may be applied to calculate the effect of a source of sound situated in the interior of a resonator. 312. We now pass on to the iurther discussion of the problem of the open pipe. We shall suppose that the open end of the pipe is provided with an infinite flange, and that its diameter is small in comparison with the wave-length of the vibration under consideration. As an introduction to the question, we will further supp<^ that the mouth of the pipe is fitted with a freely moving piston without thickness and mass. The preceding problems, boxn which the present differs in reality but little, have already given us reason to think that the presence of the piston will cause no important modification. Within the tube we suppose (§ 255) that the velocity-potential is ^ = (ilcosAw?-h5sin&F)c*«< (1), where, as usual, k « 2ir/\ = n/a. At the mouth, where « = 0, ilation between J,{2kR))_ -K,{2kR) (3), >eiDg the radius of the pipe. From this the solution of the ibleni may be obtained without any restriction as to the I of kR: since, however, it is only when kR is small the presence of the piston would Dot materially modify I question, we may as well have the benefit of the simplification mce by taking aa in (1) § 311 1*-" - Q.-'T-f <*'■ [ Now, since the piston occupies no space, the values of (d^jdx\ t be the same on both sides of it ; and since there is no mass, I like moat be true of the values of j70„rftr. Thus ^Bd '1 2 }■■ Substituting in (1), we find on rejecting t putting for brevity 5=1, *= (5). imaginary part, 1 containing s -.(6). In this expression the term containing sin nt depends upon the diBsipation, and is the same as if there were no piston, while that involving SA'A/Stt represents the effect of the inertia of the external ■ in the neighbourhood of the mouth. In order to compare with rious results, let a be such that 8A:R , . ,^ COB fcr = Bin A: (a; — o) ; sinfcc- 37r ■■(7). I, the 8<]uarea of small ciuantities being neglected, 8R "Si = sini{,c — a) vasnt — \l^R^ cos kx &\ant (8), s formulsB shew that, if the dissipation be left out of account, city-potential is the same as if the tube were lengthened 198 THEORY OF OPEN END& [312. by 8/3^ of the radius, and the open end then behaved as a loopi The amount of the correction agrees with what previous investi- gations would have led us to expect as the result of the intro- duction of the piston. We have seen reason to know that the true value of a lies between ^ttjS and SR/Sir, and that the presence of the piston does not affect the term representing the dissipation. But, before discussing our results, it will be advantageous to in- vestigate them afresh by a rather different method, which besides being of somewhat greater generality, will help to throw light on the mechanics of the question. 313. For this purpose it will be convenient to shift the origin in the negative direction to such a distance from the mouth that the waves are there approximately plane, a displacement which according to our suppositions need not amount to more than a small fraction of the wave-length. The difficulty of the question consists in finding the connection between the waves in the pipe, which at a sufficient distance from the mouth are plane, and the diverging waves outside, which at a moderate distance may be treated as spherical. If the transition take place within a space small compared with the wave-length, which it must evidently do, if the diameter be small enough, the problem admits of solution, whatever may be the form of the pipe in the neighbourhood of the mouth. Fig. 62. 0^ f At a point P, whose distance from A is moderate, the velocity potential is (§ 279) ^ = — 6-^6*'^ (1), whence efo.= - — V 0+^^) (*)• Let us consider the behaviour of the mass of air included be- tween the plane section at and a hemispherical surfiu^ whoae THEORY OF OPEN ENDS. 199 313.] centre is A, and radius r, r being large in comparison with the diuneter of the pipe, but small in comparison with the wave- len^h. Within this apace the air must move approximately as an incompressible fluid would do. Now the current acrosH the lierai- -■ipherical surface = 27rr'"- -STT^'e'"' -2TrA'il+ikr)e''" if the square of A»- be neglected. If, as before, we take for the velocity-potential i <t) = {A cos kx-t- B sin fer) e'"' , we have for the current across the section at 0, ..(3), ithin the pipe w. 'O. -nkBtf". ..(5); and thus <TkB = -2TA' (6). This is the firet condition ; the second is to be found from the wnaideration that the total current (whose two values have just been equated) is proportional to the difference of potential at the terminals. Thus, if c denote the conductivity of the passage be- tween the terminal surfaces, .rtS A' .^ , An On substituting for A' its value from (6), v : have 1 "I this expression the second term is negligible in comparison with ^he first, for c is at most a quantity of the same order as the radius ^r the tube, and when the mouth is much contracted it is smaller *till. Thus we may take ^-"(-^S <«»• substituting this in (4), we have for the imaginary expression of t velocity-potential within the tube, if S be put equal to unity, akx + ck {-\-t) cos fcj; \ e*"*, 200 CORRECTION TO LENGTH. [313. or, if only the real part be retained, 6^ \smkx cos kx cosn^ — ^r- co8ibrsmiit....(9). Following Helmholtz, we may simplify our results by introdudng a quantity a defined by the equation tanAa = — (10> c Thus , sinA(a:-a) . k^a . - ^ /nx ^ as i_ _^coQ^« C08A:;2r8mn< (11), cos KCL 2tir and the corresponding potential outside the mouth is yfr^-^^cosint-kr) (12> If jB be the radius of the tube, we may replace a by wiP. When the tube is a simple cylinder, and the origin lies at a distance Ai from the mouth, we know that crc""* = AL + /a-B, where /i is a number rather greater than ^tt. In such a case (the origin being taken sufficiently near the mouth) ka is a small quantity, and therefore from (10) a = - = AZ+/Ai2 (13). At the same time cos ka may be identified with unity. The principal term in ^, involving cosn^, may then be calcu- lated, as if the tube were prolonged, and there were a loop at a point situated at a distance ^R beyond the actual position of the mouth, in accordance with what we found before. These results, approximate for ordinary tubes, become rigorous when the diameter is reduced without limit, friction being neglected. If there be no flange at A, the value of c is slightly modified by the removal of what acts as an obstruction, but the principal effect is on the term representing the dissipation. If we suppose as an approximation that the waves diverging from A are spherical, we must take for the current ^Trr^d-^ldr instead of 2irr^d'^ldr, The ultimate effect of the alteration will be to halve the expression for the velocity-potential outside the mouth, as well as the corresponding second term in ^ (involving sinn^). The amount of dissipation is thus seen to depend materially on the degree in which the waves are free to diverge, and our anaiytioal expressions must not be regarded as more than rough estimalefli AND CLOSED PIPES. The correct theory of the open organ-pipe, ioctuding equations 11) and (12), was discovered by Helmholtz', whose method, however, differs considerably from that here adopted. The earliest solutions of the problem by Lagrange, D. Bemouili, and Elder, were founded on the assumption that at an open end The pressure could not vary from that of the surrounding atmo- tphere, a principle which may perhaps even now be considered applicable to an end whose openness is ideally perfect. The fact I that in all ordinary cases energy escapes ia a proof that there is L not anywhere in the pipe an absolute loop, and it might have been k^ expected that the inertia of the air just outside the mouth would ^|llBve the effect of an increase in the length. The positions of the ^FBodea in a sounding pipe were investigated experimentally by r Savurt'and Hopkins", with the result that the interval between the mouth and the nearest node is always less than the half of that separating consecutive nodes. [The correction necessary for an open end is the origin of a (iepavture from the simple law of octaves, which according to elementary theory would connect the notes of closed and open pipes of the same length. Thus in the application to an organ-pipe let afl denote the correction for the upper end when open, and I the length of the pipe including the correction for the mouth at the lower end. The whole effective length of the open pipe is then i + ttfi, while the effective length of the pipe if closed at the upper end ia I simply. The open pipe is practically the longer, and the interval between the notes is less than the octave of the simple theory*. It may be worthy of remark that the correction, assumed to be independent of wave-length, does not disturb the harmonic rela- l^iuns between the partial tones, whether a pipe be open or closed.] 314. Experimental determinations of the correction for an open end have generally been made without the use of a flange, and it therefore becomes important to form at any rate a rough estimate of its effect. No theoretical solution of the problem of an unflanged open end has hitherto been given, but it is easy to ' Crelle. Bd. 57, p. 1. 1860. * B«ohereliM eur les vtbratioDR ie I'tit. Ann. d, Chlm. t. nxiv, 1B23. ' Aeiial vibrationa in ojlindrioal ttibea. Cambridge Trantaetiont, Vol. v. p. aSl. < Bosauquet, Phil. Mag. \i. p. 63, 1678. INFLUENCE OF FLANGE. see (^ 79, 307) that the removal of the flange will reduce 1 correction materially below the value '82 B (Appendix A). In t absence of theorj- I have attempted to determine the influence of a flange experimentally*. Two organ-pipes nearly enough in unison with one another to give countable beats were blown from an organ bellows ; the effect of the flange waa deduced from the difference in the frequencies of the beats according as one of the pipes was flanged or not. The coirection due to the flange was about •2R. A (probably more trustworthy) repetition of this experiment by Mr Bosancjuet gave -ZoR. If we subtract SZS from 82^, we obtain 'GR, which may be regarded as about the probable value of the correction for an unflanged open end, on the supposition that the wave-length is great in comparison with the diameter of the pipe. . Attempts ti> determine the correction entirely from experiment 1 have not led hitherto to very precise results. Measurements by 1 Wertheim' on doubly open pipes gave as a mean (for each end) ■663 R, while for pipes open at one end only the mean result was 7i6R. In two careful experiments by Bosanquet' on doubly open pipes the correction for one end was 635 R, when \ = 12R, and '543 R, when X = 30R. Bosanquet lays it down as a general rule that the correction (expressed ae a fraction of R) increases with the ratio of diameter to wave-length ; part of this increase may however be due to the mutual reaction of the ends, which causes the plane of symmetry to behave like a rigid wall. When the pipe is only moderately long in proportion to its diameter, a state of things is approached which may be more nearly repre- sented by tho presence than by the absence of a flange. The comparison of theory and observation on this subject is a matter of some difficulty, because when the correction is small, its value, as calculated from observation, is affected by uncertainties as to^ absolute pitch and the velocity of sound, while for the case, whoM the correction is relatively larger, which experiment is more cotoH potent to deal with, there is at present no theory. Probably a more" accurate value of the con-ection could be obtained from a resonator of the kind considered in § 306, where the communication with ' Phil. Hag. (3) in. 156. 1877. [The earliest experiments of the kind ue thoaa of Uripon (.^nn. d. Cliim. m. p. 384, 1874) wlio shewed that the effect of > l&rge flaoga is proportional to the diameter of the pipe.] ' ^nn. d. Chim. (3) t. iwu. p. 394, 1851. » Phil. Hag. (5) iv. p. 319. 1877. EXPERIMENTAL METHODS. the outside air is by a Bimple aperture ; the " length " is in that B zero, and the "correction" is everything. Some measurements of this kind, in which, however, no great accuracy was attempted, will be found in my memoir on resonance'. [Careful experimental determinations of the correction for an unflanged open end have been made by Blaikley', who employed a vertical tube of thin brass 208 inches (53 cm.) in diameter. The tower part of the tube was immersed in water, the surface of which defined the " closed end," and the experiment consisted in varj-ing the degree of immersion until the resonance to & fork of known pitch was a maximum. If the two shortest distances of the water stirface from the open end thus found be I, and I3, (^ — ^1) represents the half wave-leugth, and the "correction for the open end" is ^(li — li)~l,. The following are the results obtained by Blaikley, expressed as a fraction of the radius. They relate to the same tube resounding to forks of various pitch. I c 253-68 -565 I e 317-46 -595 »g' 3S081 -oGi V?' 444-72 -587 c" 507-45 -568 The mean correction is thus '576 Jt.] Various methods have been used to determine the pitch of resonators experimentally. Most frequently, perhaps, the resonators have been made to speak after the manner of organ-pipes by a stream of air blown obliquely across their mouths. Although good results have been obtained in this way, our ignorance as to the mode of action of the wind renders the method unsatisfactory. In Bosanquet'a experiments the pipes were not actually made to speak, but short discontinuous jets of air were blown across the open end, the pitch being estimated from the free vibrations as the sound died away. A method, similarin principle, that I have sometimes employed with advantage consists in exciting free vibra- tions by means of a blow. In order to obtain as well defined a note as possible, it is of importance to accommodate the hardness of the substance with which the resonator comes into contact to the pitch, ' Phil. Tram. 1971. 8ea alw SondhatiBB, Pogg. Ann. 1. 140, B3, 319 (1870), uid Bome remarks thereupon b; mjeelf {Phil. Mag,, Sept. 1870). ' PMl. Mag. vol. 7. p. 389, 1879. 204 DISCUSSION OF MOTION [314 a low pitch requiring a soft blow. Thus the pitch of a test-tube may be determiued in a moment by striking it against the hfsA knee. In using this method we ought not entirely to overlook the &ct that the natural pitch of a vibrating body is altered by a term depending upon the square of the dissipation. With the notation of § 45, the frequency is diminished from n to n(l — iifc'n"'), or if a? be the number of vibrations executed while the amplitude falls in the ratio e : 1, from n to ?i V Stt^x') ' The correction, however, would rarely be worth taking into account. The measurements given in my memoir on resonance were conducted upon a diflFerent principle by estimating the note of maximum resonance. The ear was placed in communication with the interior of the cavity, while the chromatic scale was sounded. In this way it was found possible with a little practice to estimate the pitch of a good resonator to about a quarter of a semitone. In the case of small flasks with long necks, to which the above method would not be applicable, it was found sufficient merely to hold the flask near the vibrating wires of a pianoforte. The resonant note announced itself by a quivering of the body of the flask, easily per- ceptible by the fingers. In using this method it is important that the mind should be free from bias in subdividing the interval between two consecutive semitones. When the theoretical result is known, it is almost impossible to arrive at an independent opinion by experiment. 316. We will now, following Helmholtz, examine more closely the nature of the motion within the pipe, represented by the formula (11) § 313. We have <l>^Lcos(nt-0) ...(1), where 1>= V, — -^ ^ ,cos*fci? (2), ^ X^a-cmkacoskx ,^^ tan(?«— ^ — ""iL? \ W 2^8mA;(« — a) ^ :n5.] ORIGINATING WITHIN AN OPEN PIPE. 205 In the expression for L' the second tenn ia very small, and therefore the maximum values of tf> occur very nearly when k (a; - a) = (- m + ^) tt, I -j: = ^„i\-^\-a (4), ..here m i.s a positive integer. The distance between consecutive maxima is thus JX, and the value of the maximum ifi sec^ ka. The minimum values of L' occur iipproximatcly when A- (iC — a) = — vtir, ! —x = ^m\ — a (5), .iiid their magnitude is given by IL' = -r- , cos' ka: = - — , cos' ^'h (6). ■iTT- 471-' ' In like manner, ^J = J"co8(n(-x) (V). where ^' = jf --, '+-r-,B\nH'x (8), cos' Ka iir i^iT cos ka sill ka: H ""'t- a^coaM^-. ) ""• ^^H The maximum values of lA' occur when ^P -fl: = J,nX-a (10), ^Knd the minimum values, when P -x = ^in\~l\-a (11). The approximate magnitude of the maximum is f sec' kci, and that of the minimum li^a^ cos' ia -- 4w'. It appears that the maxioia uf velocity occur in the same parts of the tube as the iiiiuima of condensation (and rarefaction), and the minima of \ clocity in the same places as the maxima of condensation. The htries of loops and nodes are arranged as if the first loop were at a distance a beyond the mouth. With regard to the iihases. we see that both $ and x "■^^ ^ general small ; and therefure with the exception of the places where L' and J' are near their minima the whole motion is Bjucbrooous, as if there were no dissipation. Hitherto we have considered the problem of the passage of e waves along the pipe and their gradual diffusion from the ^uigin of the plane waves them- 206 MOTION DUE TO [315. selves. All that we have assumed is that the origin of the motia w- is somewhere within the pipe. We will now suppose that 4e|»^ motion is due to the known vibration of a piston, situated at a:= — i, the origin of co-ordinates being at the moutL Thua, when a; = — i, ^^Gcosnt (12), ax and this must be made to correspond with the expression for the plane waves, generalized by the introduction of arbitrary amplitude and phase. We may take g = BJco8(n<-€-x) (13). where J and x ^^^^ ^^® values given in (8), (9), while B and 6 are arbitrary. Comparing (12) and (13) we conclude that ^'= 2^coskil + aj ^^*^' G. = £.;fc.l«2^1^ + ^8in.Jfc4 (15). I cos'Ara 4nr* J ^ by which B and € are determined. In accordance with (12) § 313, the corresponding divergent wave is represented by -Jr = — 5 — co8(n^ — € — At) (16). If Cr be given, B is greatest, when cos fc (Z + a) = 0, that is when the piston is situated at an approximate node. In that case shewing that the magnitude of the resulting vibration is veiy great, though not infinite, since cos Axe cannot vanish. When the mouth is much contracted, cos ka may become small, but in this case it is necessary that the adjustment of periods be very exact in order that the first term of (15) may be negligible in comparison with the second. In ordinary pipes cosA;a is nearly equal to unity. The minimum of vibration o<^' I is such tb< '■SXTERNAL SOURCES. tii9 1 (i + a) = + 1 , that is, wheL the piston is situated at a loop. Iti (liat case n G cos ka ..(18). The vibration outside the tube is then, acconling to the value of 1. (iqual to or smaller than the vibration which there would be if there were no tube and the vibrating plate were made part of \\\6ijz plane. 316- Our equations may also be applied to the investigation ijf the motion excited in a tube by external sources of sound. Let us suppose in the first place that the mouth of the tube is cli.>se<i by a fixed plate forming part of the yz plane, and that the potential due to the external sources (approximately constant over the plate) is under these circumstances yfr = H cosnt (1), where ^ is composed of the potential due to each source and its imnge in the i/t plane, as explained in § 278. Inside the tube let the potential be j (^ = if cos tiC cos (it (2), ' to that ij) and its differential coefficient are continuous across the hairier. The physical meaning of this is simple. We imagine within the tube such a motion as is determined by the conditions tbat the velocity at the mouth is zero, and that the condensation lit the mouth is the same as that due to the soui-cea of sound when ihe mouth is closed. It is obvious that under these circumstances the closing plate may be removed without any alteration in the motion. Now, however, there is in general a finite velocity at x= — [, and therefore we cannot suppose the pipe to be there stopped. But when there happens to be a node &t x = — I, that is tu say when t is such that [sin kl] = 0, all the conditions are jiitisfied, and the actual motion within the pipe is that expressed by (2)', This motion is evidently the same as might obtain if the pipe were closed at both ends; and in external space the potential is the same as if the mouth of the pipe were closed with the rigid plate. In the genei-al case in order to reduce the air at a; = — i to rest, we must superpose on the motion represented by (2) another of ' (An error, pointed out by Dr Burton, in here corrected.] 208 ENLARGEMENT AT A CLOSED END. [316, |^' the kind investigated in § 313, so determined as to give at « = -l 1^ a velocity equal and opposite to that of the first. Thus, if the second motion be given by cUf>/dx = BJ 0O8 (nt — « — x), we have e + x = 0, and ^ fcos"A:(i + a) h<r* • .,1) tt. - mtt /o\ ( cos'Ara 4f/r* J ^ When sin kl = 0, we have, as above explained, B^O. The maxi- mum value of B occurs when cos A; (2 + a) » 0, and then *-w <*)•• It appears, as might have been expected, that the resonance is greatest when the reduced length is an odd multiple of |X. 317. From the principle that in the neighbourhood of a node the inertia of the air does not come much into play, we see that in such places the form of a tube is of little consequence^ and that only the capacity need be attended to. This consideration allows us to calculate the pitch of a pipe which is cylindrical through most of its length (l), but near the closed end expands into a bulb of small capacity (8). The reduced length is then evi- dently / + a+fif<r-' (1), where a is the correction for the open end, and <r is the area of the transverse section of the cylindrical part This formula is often useful, and may be applied also when the deviation from the cylindrical form does not take the shape of an enlargement. When the enlargement represented by /Sf is too large to allow of the above treatment, we may proceed as follows. The dissipa- tion being neglected, the velocity-potential in the tube may be taken to be = sin A; (a? — a) cos nt, the origin being at the mouth, while a = \'rrR approximately. At a? = — i, we have ^ = n sin A: (Z + a) sin nt, and ^ = i:co8A:(Z-Ha)co8?i^. ^ HelmholU, CreUe, Bd. 67, 1860. ' be assumed that the condensation within iS is sensibly Thus '^ sin k (1 + 0.)= <rk cos k (I + a), tan t- (( + «) = .,(3) kS the equation determining the pitch. Numerical examples of ipplication of (3) are given in my memoir on resonance ...v. Trans. 1871. p. 117). Similar reasoning proves that in any case of stationary vibra- ions, for which the wave-length is several times aa great as the iameter of the bulb, the end of the tube adjoining the bulb lehaves approximately as an open end if hS be much greater ban ff, and as a stopped end if kS be much less than a. 318. The action of a resonator when under the influence of a ource of sound in unison with itself is a point of considerable lelicacy and importance, and one on which there has been a [Ood deal of confusion among acoustical writera, the author not ixcepted. There are cases where a resonator absorbs sound, as it were attracting the vibrations to itself and so diverting them from 'egions where otherwise they would be felt. For example, lUppose that there is a simple source of sound B situated in a larrow tube at a distance i\ (or any odd multiple thereof) from a itosed end, and not too near the mouth : then at any distant iZteroal point A, its effect is nil. This is an immediate conse- [OADce of the principle of reciprocity, because if A were the lource, there could be no variation of potential at B. The -rtiction, precluding too great a proximity to the mouth, may dispensed with, if we suppose the source B to he diffused T .roily over the cross section, instead of concentrated in It. Then, whatever may be the size and shape of the section, I •.- ip absolutely no disturbance on the further side. This ptn the theory of vibrations in one dimension; 14 nay i - , ion, I rhi. I the J aad th> reqaira^ at ibe Kmice, lor iffl^U^bnt dai the taij Hilr tk |Aa« of the M»d In the eembiaaAmB ct fifet up C LWJ ted in ¥ig. 63, the coten frcclf ■! ^ ; ac AitfiMfartaetf at tbetnoatbofarewiii of (Mtdi ideatieai vitb ifei ova. Vader these dTcunutM it is alMarlwtL and tktrt it bo nhntim pnpagatod along J It is clear that tbe i^bainai tmbe BC loajr be replaced by cAhei rtanBrtcr at the aame fatcfa (7), without prejudice to actioo of the qifwiataa. The onfinar; expUnatioD by interfert (ao called) of dtiect sad leAected waves is then leas applicable* Tm-». (•) M OS) TheHe caAes where the source is at the mouth of a reson muHt not be confused with others where the source lienor. U B be a, source at the bottom of a stopped tube v\ ' PogB- Ann I, vn. \Mi&. RESONATOR CLOSE TO SOURCE. 2lJ reduced length ia ^\, the intensity at an external point A majf be vastly greater than if there had been no tube. In fact the potential at A due to the source at B is the same as it would be at B were the source at A. 319. For a closer examination of the mechanics of resonance, irt shall obtain the problem in a form disembarrassed of unne- itMary difficulties by supposing the resonator to consist of a ^mall circular plate, backed by a spring, and imbedded in an ijj'lefinite rigid plane. It was proved in a previous chapter, (30) ^ 302, that if M be the mass of the plate, f its displacement, nl the force of restitution, R the radius, and a- the density of the air, the equation of vibration is (A,+ S^-)f + «J!^f+^f,f (!)• vhere F and f are proportional to e'*"'. If the natural period of vibration (the reaction of external air incloded) coincide with that imposed, the equation reduces to ^aa-Trk'R'^-^F (2). Let us now suppose that F is due to an external source of BDQiid, giving when the plate is at rest a potential ^g. which will be nearly constant over the area of the plate. Thus F=-Sp.irIi' = ikaff.TrE^.yfr, (3); sothat Tri?f = .? = 2»Vfc-'-f, = iX-f, (4), and the potential i^ due to the motion of the plate at a distance r will be * — 2.— — IT — -'''•15? «• independent, it should be observed, of the area of the plate. Leaving for the present the case of perfect isochronism, let us suppose that -(M+^-^y^'a' + f.=0 (6). •"■■I that Stt/^' is the wave-length of the natural note of the resunator. If M' be written for M + ^a-R?, the equation corre- sponding to (5) takes the form 212 REINFORCEMENT OF SOUND [319. from which we may infer as before that if k' = k the efficiency of the resonator as a source is independent of jR. When the adjust- ment is imperfect, the law of falling o£f depends upon U'Br*. Thus if M' be great and 22 small, although the maximum efficieiM^ of the resonator is no less, a greater accuracy of adjustment ii required in order to approach the maximum (§ 49). In the ciae of resonators with simple apertures M'^^aR^, so that M'Br* varies as Ilr\ Accordingly resonators with small apertures re- quire the greatest precision of tuning, but the difference is m^ important. From a comparison of the present investigation with that of § 311 it appears that the conditions of efficiency aie different according as internal or external effects are considered. We will now return to the case of isochronism and suppose further that the external source of sound to which the resonator A responds, is the motion of a similar plate B, whose distance c from ^ is a quantity large in comparison with the dimensions of the plates. The intensity of B may be supposed to be such that its potential is t = >- (8^ Accordingly -^o = cr^ er^, and therefore by (5) shewing that at equal distances from their sources 4> : ylr^e-^ : ike (10). The relation of phases may be represented by regarding the induced vibration if> as proceeding fix)m B by way of A, and as being subject to an additional retardation of ^X, so that the whole retardation between B and -4 is c + iX. In respect of amplitude is greater than -^ in the ratio of 1 : kc. Thus when kc is small, the induced vibration is much the greater, and the total sound is much louder than if A were not permitted to operate. In this case the phase is retarded by a quarter of a period. It is important to have a clear idea of the cause of this augmentation of sound. In a previous chapter (§ 280) we saw that, when A is fixed, B gives out much less sound than might at first have been expected from the pressure developed The explanation was that the phase of the pressure was unfiskVooraUe; BY RESONATORS. 213 Parger part of it is concerned only in overcomiDg the inertia fXe surrounding air, and is ineffective towards the performance jrk. Now the pressure which sets A in motion is the whole U and not merely the insiguificaDt part that would of itself irk. The motion of ^ is determined by the condition that I componeut of the whole pressure upon it, which has the phase Be velocity, shall vanish. But of the pressure that is due to motion of .4 , the lai^er part has the phase of the acceleration ; L therefore the prescribed condition reijuires an equality reen the small component of the preas>jre due to A'e motion, I a pressure comparable with the lafge component of the nire due to ffs motion. The result is that A becomes a b more powerful source than B. Of cotirse no work is done ! piston A : its effect is to augment the work done at B, ^y modifying the otherwise unfavourable relation between the (ihases of the pressure and of th*; velocity. The infinite plane in the preceding discussion is only required in order that we may find room behind it for our machinery of springs. If we are content with still more highly idealized - "Tirces and resonators, we may dispense with it. To each piston , ist be added a duplicate, vibrating in a similar manner, but in ' opposite direction, the effect of which will be to make the imal velocity of the fluid vanish over the plane AB. Under I > se circumstances the plane is without influence and may be iitoTcd. If the size of the plates be reduced without limit they become ultimately equivalent to simple aoorces of fluid ; and we conclude that a .simple soaree B will become more efficient than before in the ratio of 1 : Jtc, when at a small distance c from it there is allowed to operate a simple reao&ator (a» we may call it) of like pitch, that ia. a eonrce to which the inertia of the tnunedistely BorroutKling fluid is compensated by some adequate biaerf, aod which is set in mniian hf estemal catuea only. 1 the present state of our koowledge ot the mecfaaotea of log fluids, while th^ difficulties trf" dedtutioD are tar Ae i part still to be overcome, any omptificatioii of conditioaN Hi kllow* progreM to be made, witboot wboUy destroyii^ Ae character of the queHtion, may be a iCep of great £>uch, for example, waa the intiodocticn hj Hefaa- % of tbe idea of a soaroe eooceutraled in one potn^ mwiMiiiiil dly by the riolation at that point of the eqnatioa ai ; AND DOUBLK SOUBCE. oootinuity. Ferhape in like maiuter the idea of a simple reso- nabn- may be naefut, altboogli the thing would be atill mor»i imposBibU to coostnict thao a simple somce. 3ML We bare weB that there is a great augmeotation at aaumd, ■ fa» a aohablj toned resonator is close to a simple ■MMBfc. MmA mtKK is this the ease, vben the aosrce of sound u t^iftmti, Tfe pnfaatial due to a doable tmorce is (§§ 294. 324) '(•-i>- ..(1). r be at a snail distaooe c. e to tbe resonator at a distance r' ii ■r ■ witboot effect i but when /i,= il, r &M ^ tbe axis of the double source, e its potential is ..(3). ♦-K- ..(4). doe to the resonaUR in tbe ratio jfc* : 1, kind of motion to the d at its centre ; but tha t k> B «Blf pennissible when thfl I with c: otherwise ■i&B Aa acbon of the resonatoB I how powerfa ■^■1 ete aaluui gf » OMMBtar ■ whea placed in a suitabl fe otiMs «• « iiiiimiiiinrf ««■ if amd. whoee characbe I ;:fiac it wodd of tieelf prndnee bat Itttle effect at TWO OR MORE BOdb of the beet examples of this use of a resonator U affurdod I vibrating bar of glass, or metal, held at the nodes. A strip late glass abont a foot [30 cm.] long and an inch [2'5 cm,] id, of medium ihJctness (say ^ inch [32 cm.]), Hupportod at iDt 3 inches [76 cm.] from the ends hy meana of Hiring Iwiitod it, answera the purpose very well. When utruck by a r it gires but little souud e.xcept overtonos ; and even thuw r mlmost be got rid of by choosing a hammer of HuitubluKoftnon. I de6ciency of sound is a coni^equence of the small dinieniiionB ihe bar in comparison with the wave-length, which allowH of tho r transference of air from one side to the other. If now the mth of a resonator of the right pitch' be held ovor on<! of l,ho ends, a sound of considerable force and purity miiy bii ained by a well-managed blow. In this way an iniprovod [ntmicoD may be constrncted, with tones much lower thiiii aid be practicable without resonators. In tho ordinary inntru- 3it the wave-lengths are sufficiently short to permit thi' Imr to tDmunicate vibrations to the air independently. ' The reinforcement of the sound of a boll in a wi<ll-knowu riment due to Savart' is an example of tho wmm inndti uf but perhaps the moat striking inntancu Im in tho ht- ingement adopted by Helmholtz in his exporinientH r("iuiring s tones, which are obtained by holding tuning-forkn tivnr th" toutbs of resonators. When two simple resonators A,, A^, Mipiirutuly in tunc ith the source, are close together, the effect in Iohn than if tlitin* B only one. If the potentials due respectively t<i A,, A, ho 1, ^, we may take 4>i = A^—^ , <t>t = At-^ . , B represent the distance A,At, and ^,, ^„ tho potonliali tat would exist at A„ A,, if there were no roaonatorH; thou thn toditiona to determine A,. A, are by (5) § 310 f^+AJR^ + ilcA,' ^, + A,IIt = + HcA, > To get the best effect, the moath of the lenonaXot oujjht lu bu prolt; olote to d then the piteh is decidadl; lower than it would ha in the up«u, Tlw tl •djuaUnent may be made b; vKrfing the amount □( obatruolion. Thia naa ot ra ia o( great antiquity. • jMh i. Chim. (. ntv. 1833. ■■(1). 210 FORMATION OF JETS [321. By hypothesis -^j and y^^ are nearly equal, and therefore ^« = ^'=rT|^t (2X Since ikR is small, the effect is much less than if there were only one resonator. It must be observed however that the diminished effectiveness is due to the resonators putting one another out of tune, and if this tendency be compensated by an alteration in the spring, any number of resonators near together have just the effect of one. This point is illustrated by § 302, where it will be seen (32) that though the resonance does not depend upon the size of the plate, still the inertia of the air, which has to be compensated by a spring, does depend upon it. 322. It will be proper to say a few words in this place on an objection, which has been brought forward by Bosanquet' as possibly invalidating the usual calculations of the pitch of re- sonators and of the correction to the length of organ-pipes. When fluid flows in a steady stream through a hole in a thin plate, the motion on the low pressure side is by no means of the character investigated in § 306. Instead of diverging after passing the hole so as to follow the surface of the plate, the fluid shapes itself into an approximately cylindrical jet, whose form for the case of two dimensions can be calculated' from formulae given by Kirchho£ On the high pressure side the motion does not deviate so widely from that determined by the electrical law. In like manner fluid passing outwards from a pipe continues to move in a cylindrical stream. If the external pressure be the greater, the character of the motion is different. In this case the stream lines converge from all directions to the mouth of the pipe, afterwards gathering themselves into a parallel bundle, whose section is considerably less than that of the pipe. It is clear that, if the formation of jets took place to any considerable extent during the passage of air through the mouths of resonators, our calculations of pitch would have to be seriously modified. The precise conditions under which jets are formed is a subject of great delicacy. It may even be doubted whether they would occur at all in frictionless fluid moving with velocities so small that the corresponding pressures, which are proportional to the squares of 1 PhiL Mag. Vol. it. p. 125, 1877. > PkiL Mag. YoL n. p. Ul, 1876. I 322.] DURING SONOROUS MOTION. 217 the velocities, are incoDsidemble. But with air, as we actually have it, moviog under the actioD of the pressures to be found in resonators, it must be admitted that jets may sometimes occur. While experimenting nbout two years ago with one of Ktinig'e brass resonators of pitch c', I noticed that when the corresponding fork, strongly excited, was held to the mouth, a wind of consider- able force issued from the nipple at the opposite side, This effect txiHy rise to such intensity as to blow out a candle upon whose n-iek the stream is directed. It does not dt'pend upon any peculiar mntion of the air near the ends of the fork, as is proved by mounting the fork upon its resonauce-box and presenting the open ftid of the box, instead of the fork itself, to the mouth of the resonator, when the effect is obtained with but slightly diminished intensity. A similar result was obtaiued with a fork and re- sonator, of pitch an octave lower (c). Closer examination revealed ihe fact that at the sides of the nipple the outward flowing stream was replaced by one in the opposite direction, so that a tongue of tiame from a suitably placed candle appeared to enter the nipple at the same time that another candle situated immediately in front was blown away. The two effects are of course in reality alternating, and only appear to be simultaneous in conse'iuence of the inability of the eye to follow auch rapid changes. The formation of jets must make a serious ilraft on the energy of the motion, and this i.-i no doubt the reason why it is necessary to close the nipple in order to obtain a powerful sound from a resonator of this form, when a suitably tuned fork is presented to it. , At the same time it does not appear probable that jet forma- tion occurs to any appreciable extent at the mouths of resonators as ordinarily used. The near agreement between the observed and the calculated pitch is almost a sufGcient proof uf this. Another argument tending to the same conclusion may be drawn from the {lersistence of the fi^e vibrations of resonators (§311), whose dura- tion seems to exclude any imixirtant cause of dissipation beyond ^the communication of motion to the surrounding air. HH In tbe case of organ-pipes, where the vibrations are very ' powerful, tbese arguments are less cogent, but I see no reason for thinking that the motion at the upper open end differs greatly fr(>m that supposed in Belmhollz's calculation. No conclusion to f can, 1 think, safely he drawn from the phenomena of MAINTENA^'CE OF SOUNB [322. stead; motion. Id the opposite extreme eaae of impulsive motica jets certainly cannot be formed, as follows from Thomson's prini cipie of least energy (§ 79), and it is doubtful to which extrent^ the case of periodic motion may with greatest plausibility I assimilated. Observation by the method of intermittent ilium nation (§ 42) might lead to further information upon this subject 1 322a. Aa has already been mentioned, the free vibrations of thft' body of air contained in a resonator may be excited by a suitabli blow delivered to the latter. The gas does not at first partake of th sudden movement imposed upon the walls, and the relative motio thus initiated is the origin of free vibrations of the kind considere in preceding sections. When corks are drawn from partial! empty bottles, or when the lids are suddenly removed tubular pasteboard pencil-cases, free vibrations of the resonatio air columns are initiated in like manner. If the vibrations are to be maintained with a view to th embsion of a continued sound, the vibrating body must be i communication with a source of energy (§ 68 a), and the reactioi between the two must be rightly accommodated with respect i phase. The question whether the source of energy or the resona tor is to be regarded as the origin of the sound is of no particuls significance and will be variously answered according to the poin of view of the moment. In the organ the pipe, rather than t compressed air within the bellows or even the escaping wind, ; regarded as the parent of the sound, but when a similar pip is maintained'- in action by a flame the credit of the joint perfo) mance is usually given to the latter. Up to this point the explanation of maintained vibrations i simple enough ; but the complete theory in any particular < demands such an investigation of the reaction as will determii the phase relation. On this depends the whole question wbetbt the reaction is favourable or unfavourable to the continuance i the vibrations, and the determination is often a matter of d culty. Before proceeding to discuss the action of the blast it will 1 desirable to say something further upon the organ-pipe consider) I Mmply as a resooator. We have seen {§ 314) how to take aocou] f an upper open end, but according to tho rule of Cavailld-C he whole addition which must be madt- to the measured lei a.] IN FLUTE OKUAN-PIPE8. 219 at an open pipe in order to bring about agreeinent with the Bimple formula (8) § 255 amounts to as much as 3^ 72, ay much greater than the correction (12K) neceasary for a mple tube of circular section open at both ends. This dis- repancy is sometimes attributed to the blast. But it must be membered that the lower end is very much less open than the pper end, and that if a sensible correction on account of deficient >enness is required for the latter, a much more iinporlant correc- poD will probably be necessary for the former. Observations by e author' have shewn that this is the case. A pipe fitted with r alidiDg prolongation was tuued to maximum resonance with a iren (250) fork as in Blaikley's experiment (§ 314). It was then lown from a well-regulated bellows with measured pressures of rind, and the pitch of the sounds so obtained was referred to that f the fork by the method of beats (§ 30). The results shewed tliat at practical pressures the pitch of the pipe as sounded by VrtDd was higher than its natural note of maximum resonance ; so Uiat the considerable correction to the length found by Cavaill6- Coll is not attributable to the blast, but to the contracted ^aracter of the lower end treated as open in the elementary l^eory. In order to estimate the natural note an even larger ''correction to the length" would be required. The rise of pitch due to the wind increases with prosaure. Thus in the case referred to above the pipe under a pressure of l"06 inches (2'7 cm.) of water gave a note about 2 vibrations per BDCond sharper than that of the fork, but when the wind pressure Was raised to 42 inches (107 cm.) the excess was as much as 11 vibrations per second. When the pressure was raised much fiirther, the pipe was "over blown" and gave the octave of itB proper pitch, This, of course, corresponda to another mode of vibration of the aerial column. It remains to consider the maintaining action of the blast. The vibrittiona of a ctiiumn of air may be encouraged either by the introduction of fluid at a place where the density varies and At n moment of condensation (and by the similar abstraction of fluid at a moment of rarefaction), or by a suitable acceleration of ' parts of the column situated near a loop. Since the blast of organ acta at an open end of the pipe, it is clear that here we I Phil. Mag. m. p. 162, 1877 ; iiii. p. 340. 1883. FLUTE 0EGAN-PIPE8. have to do with the latter alternative. The sheet of wind directed across the lip of the pipe is easily deflected. When during the vibration the external air tends to enter the pipe, it carries the jet with it more or lesa completely. Half a period later when the natural flow is outwajxla, the jet is deflected in the corresponding direction. In either case the jet enoourageB the prevailing motion, and thus renders possible the maintenance of the vibration. For ready speech it ia necessary that the sheet of wind be accurately adjusted. But Schneebeli' has shewn that when the vibration is once started there is more latitude. In an experi- mental arrangement the jet was so adjusted as to piiss entirely outside the pipe. Under these circumstances there was failure to speak until by a temporary strong blast directed upon it from outside the jet was bent inwards to the proper position. The pipe then spoke and continued in action until by a pressure in the reverse direction the jet was beat back. The motion of the jet may be made apparent with the aid of smoke or by means of- a piece of tissue paper held so as to vibrate with it. Botb Schneebeli and H. Smith' insist upon a comparison between the jet and the tongue of a reed organ-pipe, but the modes of action appear to be essentially diflferent. The above view of the matter, which is that adopted by V. Helmholtz in the fourth edition of his gi-eat work, appears to be: satisfactory as a general explanation of the maintenance of fr continued vibration, but it cannot be regarded as complete. In- mattera of this kind practice is usually in advance of theory and many generations of practical men have brought the organ, pipe to a high degree of excellence. Another view that has been favourably entertained by many good authorities I'Cgards the pipe as merely reinforcing by ii resonance a sound primarily due to the friction of the jet playing^ against the lip, and there seems to be no doubt that sounds may thus originate". Perhaps after all there is less ditfercnce than might at first appear between the two views, and the latter may be especially appropriate when the initiation of the sound ratheC than its maintenance is under consideration. A detailed discussion 1 Pogg. Aim. Bd. 15a, p. 301. 1874. ' Nal«re, 1873, 1874, 1875. mple Melile'a Jfciunt:, p, '253; SondhauBs Pusiy. ^Ppei OVERTONES. of the question will be found in an essay by Van Schaik'. For a fuller esplanatiiin we must probably await a better knowledge of Uie mechanics of jets. 322 b. The character of the eonnd emitted from a pipe mds upon the presence or absence of the various overtones, a matter which requires further consideration. When a system vibrates freely, the overtoues may be harmonic or inharmonic according to the nature of the system, and the composition of the sound depends upon the initial circumstances. But in the case of a maintained vibration like that now befoi-e us the motion strictly periodic, and the overtones must be harmonic if present alL The freijuency of the whole vibration will correspond proximately with that natural to the pipe in its gravest mode', the agreement between the pitch of an audible overtone and bt of any free vibration may be much less close. The strength of overtone thus depends upon two things : first upon the extent I vhich the maintaining forces possess a component of the right I, imd secondly upon the degree of approximation between the me and some natural tone of the vibrating body. In organ- the sharpness of the upper lip and the comparative thinness sheet of wind are favourable to the production of overtones ; ^ that in narrow open pipes v. Uelmholtz was able to hear plainly Irst six partial tones. In wiiler open pipes, on the other hand. agreement between the overtones and the natural tones is less . In consequence, pipes of this class, especially if of wood, a 8i)fter quality of sound, in which besides the fundamental the octave and twelfth are to be detected*. When a bottle (§ 26), or a spherical resonator, is blown by bid after the manner of an organ-pipe, there are no natural i in the neighbourhood of the harmonics, and the resulting tnd is almost free from overtones. When two organ-pipes of the same pitch stand side f side, complications ensue which not unfrequently give trouble i practice. In extreme cases the pipes may almost reduce one other to silence. Even when the mutual influence is more tderate, it may still go so far as to cause the pipes to speak > Veber die Tonerrrgut^ in Labialpfei/en. BotterduD. 1B91. > We are aot now speaking of " over blowing." * Tuiieniiifindviigin. Fourth edition, p. 155, 1877. 222 MUTUAL INFLUENCE [322 in absolute unison, io spite of inevitable small natural differeaci The simplest case that can be considered is that of a pipe, aloE the mediau plane of which a tbin resisting wall is supposed to t introduced. If this wall occupy the whole place, the origi pipe is divided into two, indt^pendent of, and perfectly simili to one another. And the pitch of these segments is the san as that of the original pipe, fluid friction being neglected, a during the vibrations of the latter there is no motion aero the median plane of symmetry. But the case is altered if tl wall be limited to the part of the plane included within t] pipe, for then the two vibrating columns are free to react up< one another. The system as a whole has two degrees of freedom- we are not now regarding overtones — and free vibrations are j formed in two distinct periods. The first of these is charactei ised by synchronisra of phase between the vibrations of the con ponent columns, and the pitch is accordingly the same as befoi the separation into two parts. But in the second mode tl phases of vibration of the component columns are opposed, . that the air which escapes from one open end is absorbed b the contiguous open end of the other part. In consequence I "correction for the open ends" is much diminished in amoun and the pitch in this mode is correspondingly raised. So Ion as the motion is free, temporary vibrations in both modes i co-exist, and would give rise to beats ; but it does not follo' that both can be maintained by the blast. This would indee seem improbable beforehand, and experiment shews that afta the first moment the vibrations are confined to the sect mode. The contiguous open ends act as opposed sources, e but little sound escapes, although within the pipes, and in U outside in the immediate neighbourhood of their mouths, th vigour of the vibrations is unimpaired. Effects of the sa kind are produced when two distinct but similar pipes i mounted side by side, and under the influence of the blast th compound system may vibrate in one mode only, in spite i small differences of pitch between the notes of the pipes vrhen sounding separately'. 322 d. Direct observation of the state of things wittuo L vibrating air column ia of course a matter of great difficulty, bi ^^^^^B ' Praceediivit of the .1/riii<-<i<' \-~~i». OF TWO OROAN-PIPEa. interesting results have been obuined by Topler and Boltzniann', calling to their aid the method of optical interference to meet the difficulty arising out of the invisibility of air and the method of stroboscopic vision to meet that arising out of the rapidity of the changes. The upper end of an organ-pipe, closed by a thin plate of metal, was provided with sides of worked glass projecting above beyond the metal plate, and by suitable optical arrangements ioter- ference waa produced between light which passed above and below. The space above being occupied by air at normal density and that below by air in a state of increased or diminished density according to the phase at the moment, the interference bands undergo displacements synchronous vrith the aerial vibration. Observed directly these displacements would escape the eye ; but by the aid of a fork electrically maintained and provided with suitable slits (§ 42) the light may he rendered intermittent in a period nearly coincident with that of the vibration, and then the sequence of changes becomes apparent From the observed raove- mvut of the bands it is possible to infer not merely the total change of density from maximum to minimum, but the law of the variation of density as a function of time. When a pipe of large section was but moderately blown, the change of density at the node amounted to 009 of an atmosphere, and the law was very nearly simple harmonic. Under a greater pressure of wind the simple harmonic law was widely departed from, the bands shifting themselves almost suddenly from one extreme position to the other. In this case the amplitude of the first overtone (the twelfth) was about one quarter of that of the fundamental tone. The whole variation of density was -019 atmosphere. 12 e. In some experimental investigations a form of pipe completely symmetrical with respect to the axis has been mployed'. The lip is constituted by the entire circular edge of the pipe as defined by a plane perpendicular to the axis, and upon this au annular sheet of wind is bnrnght to bear. A similar arrangement is adopted in the ordinary steam whistle. Another way of applying wind to evoke the speech of small pipes has been experimented upon by Sondhauss', and the rationale > Paag. Ann. ciu. p. 331, 1870. ' Qripoa, Ann. d. Chtmie, iii. p. 884. 1874. * Pogs. Ann, ici. p. 126, 1661, ^gtore I 1W II w i II '* «f *ftniMB am- be takm as eridence 1 tfc« NMdjr iow vf atr Aii%h ifce piaapt in qnobMi » onrta OKrtnrj lo what netma m Ac ray-fipe aitd in seaattve flame 1^ defawMtini «f the je« swdd aeeB here t» be of the B^m trial aorl. Time ia petbapa a tendenc y altenateiy to foUcnr u to Afrfmrt frvta the eoane maAed oat I7 tbe walh. 322/ All importaiit part of ottr present subject relates t ihtt rnnirit/iuaiuMi of ribntioDfl by meana of heat, and it will I (xmoible to fpve at leaat a g<eoeral accoaot of the masDer in wbicb lhi< I'ffcct takftit place. In almost all caae^ where he&t is com' iniliiicatcj lo a Ixidy expansion ensues, and this expansion may b tUwU* to ilo mi!':hiinioa) work. If the phases of the forces thui ^VBKtlVK Ih! favoiirubh-, a vibration may be maintained. fAn /iwtnictivK i!xanip\p i« aSo'ciiwi b^ TtevelYan'a rocker, coa trevelyan's rocker. 225 listing of a mass of iron or copper, so shaped that during vibration the weight is alternately carried on one or other of two adjacent and parallel ridges. When the instrument is heated and placed upon a block of cold lead, the vibrations persist so long as the beat remains sufficient. "Sir John Leslie first suggested that the cause of these vibrations is to be found in the expansion of the cold block by the heat which flows into it from the hot metal at the points of contact. Faraday ', Seebeck ', and Tyndall ' have adopted this explanation ; and they have shewn that most of the facts that they and others have ascertained respecting these vibrations are easily explained upon this view of their cause, supposing only that the expansion is sufficiently great to produce any sensible effect. Forbes '. on the other hand, after an extensive series of experiments, was led to reject Sir John Leslie's ex- planation, one of his principal reasons for doing so being the impossibility, as it appeared to him, that the expansion occasioned ■■by so slow a process as the conduction of heat could produce any insible mechanical effect." Davis, from whom" the above sentences are quoted, has Kamined the question mathematically, and has shewn that the :planation is adequate. It is evidently important that the lower body should possess a high rate of expansibility with temperature. In this respect lead stands high among the metals, and rock salt, which Tyndall found to answer well, is even more expansible. The objection taken by Forbes may be met by the reply that ^e conduction of heat is not a slow process when small distances md masses are in question ; and the special repulsion invoked by the basis of an alternative explanation would be of isuitable character in respect of phase. It is essential that the "phase of the force should be in arrear of the phase of the negative displacement. In an experiment due to Page* the vibrations are made independent of an initial difference of temperature, the local heating at the points of contact being obtained with the aid of an ' Prop. 0} Ron. Iml. vol. ii, p. 119, 1831. ' Pogg. Aim. vol. u. p. 1. !M0. ■ FhU. Mag. vol. vm. p. 1, 1864. * Phil, Mag. vol. n. pp. 16, 183. 19U. • Phil. Man. vol, iLV. p. 296. 1873. ' Sillitnan'e Journal, vol, ii. p. 106, 1860. t. u. \'b 226 MAINTENANCB BY HEAT [322/ electric current caused to pass from one body to the other. In this arrangement there is no contraction in the upper body to be deducted from the expansion in the lower. On a similar principle Gore^ has contrived a continuous motion of a copper ball which travels upon circular rails themselves connected with a powerful battery. 322^* But the most interesting examples of vibrations maintained by heat are those which occur when the resonating body is gaseous. *' If heat be periodically communicated to, and abstracted from, a mass of air vibrating (for example) in a cylinder bounded by a piston, the effect produced will depend upon the phase of the vibration at which the transfer of heat takes place. If heat be given to the air at the moment of greatest condensation, or be taken from it at the moment of greatest rarefaction, the vibration is encouraged. On the other hand, if heat be given at the moment of greatest rarefaction, or abstracted at the moment of greatest condensation, the vibratioo is discouraged. The latter effect takes place of itself (§ 247) when the rapidity of alternation is neither very great nor very small in consequence of radiation; for when air is condensed it becomes hotter, and communicates heat to surrounding bodies. The two extreme cases are exceptional, though for different reasons. In the first, which corresponds to the suppositions of Laplace's theory of the propagation of sound, there is not sufficient time for a sensible transfer to be effected. In the second, the temperature remains nearly constant, and the loss of heat occurs during the process of condensation, and not when the condensation is effected. This case corresponds to Newton's theory of the velocity of sound. When the transfer of heat takes place at the moment of greatest condensation or of greatest rarefaction, the pitch is not affected. If the air be at its normal density at the moment when the transfer of heat takes place, the vibration is neither encouraged nor discouraged, but the pitch is altered. Thus the pitch is raised if heat be communicated to the air a quarter period before the phase of greatest condensation ; and the pitch is lowered if the heat be communicated a quarter period after the phase of greatest condensation. ^ PML Mag. toL xt. p. 619, 1858 ; toL xmi. p. 94, 1869. OF AERIAL VIBRATIONS. In general both kinds of effects are produced by a periodic transfer of heat. The pitch is altered, and the vibrations are either encouraged or discouraged. But there is no effect of the second kind if the air concerned be at a loop, i.e. a place where the density does not vary, nor if the communication of heat be the same at any stage of rare&ction as at the corresponding stage of condousatioD '." Thus in any problem which may present itself of the main- tenance of a vibration by heat, the principal question to be considered is the phase of the communication of heat relatively to that of the vibration. 322 A. The sounds emitted by a jet of hydrogen burning in a pipe open at both ends, were noticed soon after the discovery of the gas, and have been the subject of several elaborate inquiries. The fact that the notes are subatantially the same as those which may be elicited in other ways, e.g. by blowing, was announced by Chladni. Faraday' proved that other gases were competent to take the place of hydrogen, though not without disadvantage. But it is to Sondhauss' that we owe the most detailed examina- tion of the circumstances under which the sound is produced. His experiments |M\)ve the importance of the part taken by the column of gas iii the tube which supplies the jet. For example, lund cannot be got with a supply tube which ia plugged with iton in the neighbourhood of the jet, although no difference can detected by the eye between the flame thus obtained and others which are competent to excite sound. When the supply tube is unobstructed, the sounds obtainable by varying the resonator are limited as to pitch, often dividing themi^elves into distinct groups. In the intervals between the groups no coaxing will induce a maintained sound ; and it may be added that, for a part of the interval at any rate, the influence of the flame is inimical, so that a vibration started by a blow is damped more rapidly than if the jet were not ignited. Forms of resonator other than the open pipe may be employed, sometimes with advantage. Very low notes can be got from lerical resonators, such as the large globes employed for demon- Frae. Hwj. Iiul. toI. vm. p. 536. 1878; N-tCure. vol. itiii, p. 319, 1878. Quart. Jimm. Sci. vol. t, p. 274, 1818. 1. vol. c:x. pp. I. ue, laso. Kmu Bee 228 SINGING FLAMES. [322 h. strating the combustion of phosphorus in oxygen gas. A globe of this kind gave in its natural condition a deep and pure tone of 64 vibrations per second. When it was fitted with a longer and narrower neck formed from a pasteboard tube, the calculated fi:^uency fell to 25, and the vibrations, though vigorous enough to extinguish the liame, were hardly audible. When it is desired to excite very deep sounds, the supply tube should be made of considerable length, and the orifice must not be much con- tracted. Singing flames may sometimes replace electrically maintained tuning-forks for the production of pure tones, when absolute constancy of pitch is not insisted upon. In order to avoid progressive deterioration of the air, it is advisable to use a resonator open above as well as below. A bulbous chimney, such as are often used with paraffin lamps, meets this require- ment, and at the same time emits a pure tone. Or an otherwise cylindrical pipe may be blocked in the middle by a loosely fitting plug'. As Wheatstone shewed, the intermittence of a singing flame is easily made manifest by an oscillating, or a revolving, mirror. A more minute examination is best effected by the stroboscopic method, § 42. Drawings of the transformations thus observed have been given by Topler', from which it appears that at one phase the flame may withdraw itself entirely within the supply tube. Vibrations capable of being maintained are not always self- starting. The initial impulse may be given by a blow ad- ministered to the resonator, or by a gentle blast directed across the mouth. In the striking experiments of Schaffgotsch and Tyndall' a flame, previously silent, responds to a sound in unison with its own. In some cases the vibrations thus initiated rise to such intensity as to extinguish the flame. The experiments of Sondhauss shew that a relationship of proportionality subsists between the lengths of the supply tubes and of the sounding columns. When the nature of the gas is varied, the same supply tube being retained, the mean lengths of 1 PMl. Mag. vol. vn. p. 149, 1879. * Pogg* Ann, vol. oxxvin. p. 126, 1866. * Sound, 8rd edition, p. 824, 1876. ^|b2 A,] LENGTH OF SUPPLV-PtPE. 239 the speaking columns are approximately as the s(|iiare roots of the density of the gas, A connection is thus established between the natural note of a supply tube and the notes which can be sonuded with its itid. Partly in consequence of the peculiar and ill understood be- haviour of flames, anil partly for other reasons, the thorough explanation of the phenomena now under consideration is a matter of some difficulty ; but there can be no doubt that they fall under the head of vibrations maintained by heat, the heat being com- municated periodically to the mass of air confined in the sounding tube at a place where, in the course of a vibration, the pressure varies. Although some authors have shewn a tendency to lay stress upon the effects of the draught of air through the pipe, the sounds, as we have seen, can be readily produced, not only when there is no through draught, but even when the flame is so situated that there is no aenaible periodic motion of the air in its □eighbourhaod. aun In consequence of the variable pressure within the resonator, issue of gas, and therefore the development of heat, varies uring the vibration. The question is xinder what circumstances the variation is of the kind necessary for the maintenance of the vibration. If we were to suppose, as we might at first be inclined to do. that the issue of gas is greatest when the pressiire in the resonator is least, and that the phase of greatest development of heat coincides with that of the greatest issue of gaa, we should have the condition of things the most unfavourable of all to the persistence of the vibration. It is not difficult, however, to see that both suppositions are incorrect. In the supply tube (sup- [>o9ed to be unplugged, and of not too small bore) stationary, or approximately stationary, vibrations are excited, whose phase is either the same or the opposite of that of the vibration in the inator. If the length of the supply tube from the burner to le open end in the gaa-generatiug flask be less than a quarter of the wave-length in hydrogen of the actual vibration, the greatest issue of gas precedes by a quarter period the phase of greatest condensation; so that, if the development of heat is retarded somewhat in comparison with the isaue of gas, a state of things its /amuraUe to the maintenance of the sound. Some such irdation is inevitable, because a jet of inflammable gas can outside; but in many cases a still more potent 230 SINGING FLAMES. [322 h. cause may be found in the fact that during the retreat of the gas in the supply tube small quantities of air may enter firora the interior of the resonator, whose expulsion must be effected before the inflammable gas can again begin to escape. If the length of the supply tube amounts to exactly ooe quarter of the wave-length, the stationary vibration within it will be of such a character that a node is formed at the burner, the variable part of the pressure just inside the burner being the same as in the interior of the resonator. Under these circumstances there is nothing to make the flow of gas, or the development of heat, variable, and therefore the vibration cannot be maintained. This particular case is free from some of the difficulties which attach themselves to the general problem, and the conclusion is in accordance with Sondhauss' observations. When the supply tube is somewhat longer than a quarter of the wave, the motion of the gas is materially different from that first described. Instead of preceding, the greatest outward flow of gas follows at a quarter period interval the phase of greatest condensation, and therefore if the development of heat be some- what retarded, the whole effect is unfavourable. This state of things continues to prevail, as the supply tube is lengthened, until the length of half a wave is reached, after which the motion again changes sign, so as to restore the possibility of maintenance. Although the size of the flame and its position in the tube (or neck of resonator) are not without influence, this sketch of the theory is sufficient to explain the fact, formulated by Sondhauss, that the principal element in the question is the length of the supply tube. 322 1. Another phenomenon of the class now under considera- tion occasionally obtrudes itself upon the notice of glass-blowers. When a bulb about 2 cm. in diameter is blown at the end of a somewhat narrow tube, 12 or 15 cm. long, a sound is sometimes heard proceeding from the heated glass. For experimental pur- poses it is well to use hard glass, which can afterwards be reheated at pleasure without losing its shape. As was found by De la Rive, the production of sound is facilitated by the presence of vapour, especially of alcohol and ether. It was proved by Sondhauss^ that a vibration of the glass ^ Pogg. Atm. vol. uoxz. p. 1, 1860. 322 J.] SONDHAUSs' INVESTIGATIONS. 231 itself ia no essential part of the phenonienon, and the same indefatigable observer was very successful in discovering the con- uectioii (§§ 303, 309) between the pitch of the note and the dimensions of the apparatus. But no adequate explanation of the pi\)duction of sound was given. For the sake of simplicity, a simple tube, hot at the closed end and getting gradually cooler towards the open end, may be con- sidered. At a quarter of a period before the phase of greatest condensation (which occurs almost simultaneously at all parts of the column) the air is moving inwards, i.e. towards the closed end, and therefore is passing from colder to hotter parts of the tube ; bat the heat received at this moment (of normal density) has no ■flffect either in encouraging or discouraging the vibration. The Btne would be true of the entire operation of the heat, if the idjustment of temperature were instantaneous, so that there was ■sever any sensible difference between the temperatures of the air and of the ueighboiinng parts of the tube. But in fact the adjustment of temperature takes (me, and thus the temperature of the air deviates from that of the neighbouring parts of the tube, incliuing towards the temperature of that part of the tube fi'mti which the air has just come. From this it follows that at the phase of greatest coudensation heat is received by the air, and at the phase of greatest rarefaction heat is given up from it, and thus there is a tendency to maintain the vibrations. It must not be forgotten, however, that apart from transfer of heat altt^ether, the condensed air is hotter than the rarefied air, and that in order that the whole effect of heat may be on the side of encourage- ment, it is necessary that previous to condensation the air should pass not merely towards a hotter part of the tube, but towards a part of the tube which is hotter than the air will be when it arrives thei-e. On this account a great range of temperature is necessary for the maintenance of vibratiim, and even with a great range the influence of the transfer of heat is necessarily unlavour- able at the closed end, wliere the motion is very small. This is probably the reason of the advantage of a bulb. It is obvious that if the open end of the tube were heated, the effect of the transfer of heat would be even more unfavourable than in the case of a temperature uniform throughout. ;j. The last example of the production of sound by heat shall here consider is a very striking phenomenon ^L letnperatun 232 SOUNDS DISCOVERED [322 j\ discovered by Rijke^ When a piece of fine metallic gauze, stretching across the lower part of a tube open at both ends and held vertically, is heated by a gas flame placed under it, a sound of considerable power and lasting for several seconds is observed almost immediately after the removal of the flame. DifiFering in this respect from the case of sonorous flames (§ 322), the genera- tion of sound was found by Rijke to be closely connected with the formation of a through draught impinging upon the heated gauze. In this form of the experiment the heat is soon abstracted, and then the sound ceases; but by keeping the gauze hot by the current from a powerful galvanic battery Rijke was able to obtain the prolongation of the sound for an indefinite period. These notes may be obtained upon a large scale and form a very effective lecture experiment. For this purpose a cast iron pipe 5 feet (152 cm.) long and 4f inches (12 cm.) in diameter .may be employed. The gauze (iron wire) is of about 32 meshes to the linear inch (2*54 cm.), and may advantageously be used in two thicknesses. It may be moulded with a hammer on a circular wooden block of somewhat smaller diameter than that of the pii)e, and will then retain its position in the pipe by friction. When it is desired to produce the sound, the gauze caps are pushed up the pipe to a distance of about a foot (30"5 cm.), and a gas flame from a large rose burner is adjusted underneath, at such a level as to heat the gauze to bright redness. For this purpose the ver- tical tube of the lamp should be prolonged, if necessarj', by an additional length of brass tubing. When a g6od red heat is attained, the flame is suddenly removed, either by withdrawing the lamp or by stopping the supply of gas. In about a second the sound begins, and presently rises to such intensity as to shake the room, after which it gradually dies away. The whole duration of the sound may be about 10 seconds*-*. In discussing the question of maintenance in accordance with the views already explained, we have to examine the character of the variable communication of heat from the gauze to the air. So far as the communication is affected directly by variations of pressure or density, the influence is unfavourable, inasmuch as the air will receive less heat from the gauze when its own tem- perature is raised by condensation. The maintenance depends ^ Fogg. Amu vol. cvii. p. 889, 1869 ; F)^, Mag. vol xvii. p. 419, 1859. s PhiU Mag. yoL vii. p. 165, 1879. '•i22j.'] BY BUKE AND B03SCBA. 233 u[K>n the variable tiHnsfer of heat due to the varj-iDg morioiw of the air through the gauze, this motion being compounded of a uniforni iiiotioQ upwards with a, motion, alternately upwards and downwards, due to the vibration. In the tuwer half of the tube these motions conspire a quarter period be/ore the phase of greatest L-oudetisation, and oppo§e one another a quarter period after that phase. The rate of transfer of heat will depend mainly upon the temperature of the air in contact with the gauze, being greatest when that temperature is lowest. Perhaps the easiest way to trace the mode of action is to begin with the case of a simple vibration without a steady current. Under these circumstances le whole of the air which comes in contact with the metal, in course of a complete period, becomes heated ; and after this istote of things is established, there is comparatively little further transfer of heat. The effect of superposing a small steady up- wards current is now easily recognized. At the limit of the inwards motion, i.e. at the phase of greatest condensation, a small quantitv of air comee into contact with the metal, which has not done 60 before, and is accordingly cool -, and the heat communicated to this quantity of air acts in the most favourable manner for the maintenance of the vibration, A quite ditferent result ensues if the gauze be placed in the ujtper half of the tube. In this case the fresh air will come into the field at the moment of greatest rarefaction, when the commu- nication of heat has an unfavourable instead of a favourable effect. The principal note of the tube therefore cannot be sounded. A complementary phenomenon discovered by Bosscha' and Rioss* may be explained upon the same principles. If a current of hot air impinge upon cold gauze, sound is produced; but in order to obtain the principal note of the tube the gauze must be ill the upper, and not as before in the lower, half of the tube. In an experiment due to Ries» the sound is maintained indeHnitely. The up|)er part of a brass tube is kept cool by water contained in a Hurrounding vessel, through the bottom of which the tube passes. In this way the gauze remains comparatively cool, although • '^posed to the heat of a gas flame situated an inch or two below ii. The experiment sometimes succeeds better when the draught ^^^^^H ' Pogg. Ann. vol. cm. p. aiS, 1659. ^^^^^^h* Pogg. Ann. voU cvm. p. 663, 18S9: cii. p. US, 181(0. 234 MAINTENANCE OF SOUND [322 J. is checked by a plate of wood placed somewhat closely over the top of the tube. Both in Rijke's and Riess' experiments the variable transfer of heat depends upon the motion of vibration, while the effect of the transfer depends upon the variation of pressure. The gauze must therefore be placed where both effects are sensible, i,e. neither near a node nor near a loop. About a quarter of the length of the tube, from the lower or upper end, as the case may be, appears to be the most favourable position \ 322 k. It remains to consider briefly another class of main- tained aerial vibrations where the maintenance is provided for by the actual mechanical introduction of fluid, taking effect at a node and near the phase of maximum condensation. Well-known examples are afforded by such reed instruments as the clarinette, and by the various wind instruments actuated directly by the lips. The notes obtained are determined in the main by the aerial columns, modified, it may be, to some extent by the maintaining appliances. The reeds of the harmonium and organ come under a different head. The pitch is there determined almost entirely by the tongues themselves vibrating under their own elasticity* resonating air columns being either altogether absent or playing but a subordinate part. In the instruments now under discussion the air colunm and the wind-pipe are connected by a narrow aperture, which is opened and closed periodically by a vibrating tongue. Tongues are distinguished by v. Helmholtz as in-beating and out-beating. In the first case the passage is opened when the tongue moves inwards, i.e. against the wnnd, as happens in the clarinette. Lip instruments, such as the trombone, belong to the second class, the passage being open when the lips are moved outwards or with the wind. Let us consider the case of a cylindrical pipe, open at the further end, in which the air vibrates at such a pitch as to make the quarter wave-length equal to the length of the pipe. The end of the column where the tongue is situated thus coincides with an approximate node, and the aerial vibration can be maintained if the passage is open at the moment of greatest condenaatjon, ' ^ Froe. Boy. In»U vol. vm. p. Wft, lBn^\ N«lwt,^oVTrau^«a, 322 k.] IN REED INSTRUMENTS. 235 that air from the wind-pipe is then forcibly injected. The tongue is maiotained in motion by the variable pressure within the pipe, and the phase of its motion will depend upon whether it is in- beating or out-beating. In the latter case its phase is nearly the opposite to that of the forces operative upon it, being open when the pressure tending to close it is greatest. This is the state of things in lip instruments, the lips being heavy in relation to the rapidity of the vibrations actually perffrmed, § 46. When the tongue is light and stiff, it must be made in-beating, as in the clarinettf. and its phase is then in approximate agreement with the phase of the forces. A slight departure in the proper direction from precise opposition or precise E^renpient of phase, as the case may be, will allow of the communication of sufficient energy to maintain the motion in spite of dissipative influences, A more complete analytical statement of the circumstances has been given by v. Helraholtz', to whom the whole theory is due. The character of the sounds from the various wind instru- ments used in music diffeni greatly. Strongly contrasted qualities are obtained from the trombone and the euphonion, the former brilliant and piercing, and the latter mellow. Blaikley' has analysed the sounds from a number of instruments, and has called attention to various circumstances, such as the size of the bell- mouth, and the shape of the cup applied to the lips, upon which the differences probably depend. The pi-essures used in practice, rising to 40 inches (102 cm.) of water in the case of the euphonion, have been measured by Stone'. ' ToneiapJinduniieH, 4ch edition. api>eQdii ril. CHAPTER XVII. APPLICATIONS OF LAPLACE's FUNCTIONS. 323. The general equation of a velocity potential, when referred to polar co-ordinates, takes the form (§ 241) If k vanish, we have the equation of the ordinary potential, which, as we know, is satisfied, if -^^r^/Sn, where 8n denotes the spherical surface harmonic' of order n. On substitution it appears that the equation satisfied by Sn is 1 d f . ^ dSf\ 1 d*Sn , t\ CI /v /rt\ sm 6 dO \ da J sin* dool* ^ Now, whatever the form of -i/r may be, it can be expanded in a series of spherical harmonics ^^ = -^0 + ^1 + ^^,4- 4-V^n4- (3), where -i/r^ will satisfy an equation such as (2), Comparing (1) and (2) we see that to determine -^n as a function of r, we have r«'^+2r^-ii(n + l)^n + A'r«tn = 0; or, as it may also be written, W-^wr'^-'*'*"'' <♦> ^ On the theoxy of these fanotionB the Utest Engliih worki we Todhiuiltr% nFunetiam ofLapUi/oe^ Lami^ and Beucl^ttaxai'Bwtenk^ Sl^Ketioal K m r m m lm * - 323.] SOLUTION IN Laplace's functions. 237 Id order to solve this equation, we may observe that when i- is very great, the middle term is relatively negligible, and that then the solution is i~>{r„= Ae^ -i- Be~"^ (5). The same form may be assumed to hold good for the complete equation (4), if we look upon A and B no longer as constants, but as functions of r. whose nature is to be determined. Substituting in (4), we find for B. het ua assume a-B. + B,(a-r)-' + B,(>jT)-+... + i).(iir)- + ...(7), aod substitute in (6). Equating to zero the coefficient of (I'ir)-"-*, t obtain Thus A = i»(« + 1)A, SO that fl _ B Jl . " (" + 1) , ("-l)-( '^ + 2) . ( n-2)...f» + 3 ) "'^'Y'^^Yrikr '^ 2.i.(ikrY + ~2.i,6.{ih-y 1.2 .3...2 n I "•" - "^ 274". 6 ... 2n.(ikry\ *^'- Denoting with Prof. Stokes' the series within brackets by /n (if^)f we have B=B„/„{ikr) (10). In like manner by changing I he sign of i, we get _ A=A„M-ih-) (11). ^B The symbola .^o and B„, though independent of r, are functions ^fcf the angular co-ordinates: in the most general case, tbey are ^Oay two spherical surface harmonics of order n. Equation (5) may ^Hberefore be written ■^ rf „ = Snfl-*-/, (iitr) + S„' «*'*'/„ C- iAt) (12). > On Uie ComiDuiiicBtion ot VitiratioiiB frum n Vibrating Body to a Barrouniling qti. iW(.gw«.iBw. EXPHESSION FOR RADIAL VELOCITY. By differentiation of (12) dr " where ■"'J„(aT)-^e*"'.F.(-ilT)....(13), K (•*>■) - (1 + ifc-)/, (ilT) - t7,T/.' (ijT) . I The foi-ms of the functions F, aa fai- as n = 7, are exhibited in the accompanying table : F,iy) = y^ 2+ Br' F,{[/) = i/+ 4+ Oi/-'+ 9r' F,(j)=y + J6 + 135s-'+ 735;,-^+ 2625;;-'+ 6670^-'+ 5670y-' F,(!/J = j + 22 + 262i/-i4-1890j-»+ 97fiEp-'+ 8W20y-'+ 73765 j- ' + 72705^ F,(S() = i/ + 39 + 434y-" + 42B4if-' + S9995(f-'+l*Be96|;-*+-609356i/^' + 10810BOv-* + 10B1080J-' In order to find the leading terms in F„ (Her) when ikr is small, we have on reversing the seriee in (9) /„(iib-) = 1.3.5...(2n-l){a-r)-"jl+i;lT-l .(ikryn whence by (14) we find F„{ikr)= 1 . 3 .5 ...(2« - l)(n + l)(ikr)- 324. An important caae of our general fonnulEe occurs whi ■<fr represents a disturbance which is propagated wholly outwat At a great distance from the origia, /„ (ikr) =/„{■- ikr) thus, if we restore the time factor (c*"), we have wanJi 1, ant ryji-„ = 8„ e'^'^-^> + 8„' e*"^'*^ . ■•(1). of which the second part represents a disturbance travelling inwards. Under the circumstances contemplated we are there fore to take S„' = 0, and thus ■■(2). = S„/„(i7-r)e"«"-" which represents in the most general manner the n"" harmoni( component of a disturbance of the given period diffusing itsel outwards into infinite space. 324.] DIVEHGENT WAVES. 239 The origin of the diatnrbance may be in a prescribed normal motion of the surface of a sphere of radius c. Let us suppose that at any point on the sphere the outward velocity ia repre- Bented by Ue'^'. V being in general a function of the position of the point considered. If U be expanded in tlio spherical harmonic series U= U.+ U,+ U,+ ... + U„+ (3), we must have by (13) §323 ^/■-.('X'c) (4). The complete value of ^ Is thus U„ Jn{<h-).. ..(5), where the summation is to be extended, to all (integral) values of M, The real part of this equation will ^ve the velocity potential due to the normal velocity i/cosiaC at the surface of the rere r = c. Prof. Stokes has applied this solution to the explanation of a remarkable experiment by Leslie, according to which it appeared that the sound of a boll vibrating in a partially exhausted receiver is diminished by the introduction of hydrogen. This paradoxical phenomenon has its origin in the augmented wave-length due to the addition of hydrogen, in consequence of which the bell loses ite hold (so to speak) on the sniTounding gas. The general expla- S^Bation cannot be better given than in the words of Prof Stokes: I " Suppose a person to move his hand to and fro through a small 'flpoce. The motion which is occasioned in the air is almost exactly the same aa it would have been if the air had been an incompres- fiibie fluid. There is a mere local reciprocating motion, in which the air immediately in front is pushed forward, and that imme- diately behind impelled after the moving body, while in the anterior space generally the air recedes from the encroachment of the moving body, and in the posterior space generally flows in from all sides to supply the vacuum which lends to be created ; so that in lateral directions the flow of the fluid is backwards, a ' TliB MsumpUon of ' vdocily Id ba ' ilJDg the iiunnik] To ioclude the i 240 FORMATION OF SONOROUS WAVES. [324. portion of the excess of fluid in front going to supply the de- ficiency behind. Now conceive the periodic time of the motion to be continually diminished. Gradually the alternation of move- ment becomes too rapid to permit of the full establishment of the merely local reciprocating flow ; the air is sensibly compressed and rarefied, and a sensible sound wave (or wave of the same nature, in case the periodic time be beyond the limits suitable to hearing) is propagated to a distance. The same takes place in any gas; and the more rapid be the propagation of condensations and rare- factions in the gas, the more nearly will it approach, in relation to the motions we have under consideration, to the condition of an incompressible fluid ; the more nearly will the conditions of the displacement of the gas at the surface of the solid be satisfied by a merely local reciprocating flow." In discussing the solution (5), Prof. Stokes goes on to say, " At a great distance from the sphere the function f^ {ikrY be- comes ultimately equal to 1, and we have ^ = -?.%tt,a<-r+c,2_^ (6). " It appears (from the value of dylr/dr) that the component of the velocity along the radius rector Is of the order r~^ and that in any direction perpendicular to the radius vector of the order r^, so that the lateral motion may be disregarded except in the neighbourhood of the sphere. " In order to examine the influence of the lateral motion in the neighbourhood of the sphere, let us compare the actual disturb- ance at a great distance with what it would have been if all lateral motion had been prevented, suppose by infinitely thin conical partitions dividing the fluid into elementary canals, each bounded by a conical surface having its vertex at the centre. " On this supposition the motion in any canal would evidently be the same as it would be in all directions if the sphere vibrated by contraction and expansion of the sur&ce, the same all roundi and such that the normal velocity of the surface was the same it is at the particular point at which the canal in question al on the surface. Now if CTwere constant the expansion o( U^ * Ihaveiiu^icnaM«iiQ)Q.tdiKi^SMmBc<if. SlolBBi^^ EFFECT OF LATERAL MOTION. 241 ldac«d to ite first term U„, and seeing thai f^{ikr)= 1, wo I have fojm (5), f— [expression will apply to any particular canal if wi^ take Us to B the normal velocity at the sphere's surface for that pai-ticular ; and therefore to obtain an expression applicable at once s canals, we have merely to mite U for U,. To facilitate taparisoQ with (5) and (6), I shall, however, write S(/„ for U. i&ve then, Jr = - - e«(«-r+w ._^"_ (7) ^ r^ F,(ikc) ^ ' It must be remerabei-ed that this is merely an expression appli- »ble at once to alt the canals, the motion in each of which takes place wholly along the radius vector, and accordingly the expres- non is not to be differentiated with rt;spect to or u with the d^ of finding the transvei-se velocities, ^VOn comparing (7) with the expression for the function ^ in ^factual motion at a great distance from the sphere (ti), we see that the two are identical with the exception that Un i» divided by two different constants, namely i^o('^'c) in the former case and F» {ike) in the latter. The same will be true of the leading terms (or those of the order r~') in the expressions for the condensation and velocity. Hence if the mode of vibration of the sphere be nich that the normal velocity of its surface is expressed by a I^place's function of any one order, the disturbance at a great disiance from the sphere will vary from one direction to another tecording to the same law as if lateral motions had been pre- ^■bd, the amplitude of excursion at a given distance from the ^Hre vai^'ing in both cases as the amplitude of excursion, in a ^■rnal direction, of the surface of the sphere itself. The only '"' fence is that expressed by the symbuHc ratio F^iikc) : Fi,{ikc). ive suppose Fn(ikc) reduced to the form /*„ (cos a„ + i sin a„), ^iunplitude of vibration in the actual case will b'.' tjj that in the e as fia to f*H, 'ind the phages in tin- Iwn iMs.-, nill r the normal velocity of the surface of tho eible by a single Laplace's FuQCblOl auch fiiPctiQM 242 EFFECT OF LATERAL MOTION. [324. great distance from the centre will no longer vary from one direc- tion to another according to the same law as the normal velocity of the surface of the sphere, since the modulus /in a^d likewise the amplitude On of the imaginary quantity Fn (ike) vary with the order of the function. '' Let us now suppose the disturbance expressed by a Laplace's function of some one order, and seek the numerical value of the alteration of intensity at a distance, produced by the lateral motion which actually exists. "The intensity will be measured by the vis viva produced in a given time, and consequently will vary as the density multiplied by the velocity of propagation multiplied by the square of the amplitude of vibration. It is the last factor alone that is difTerent from what it would have been if there had been no lateral motion. The amplitude is altered in the proportion of /Xq to /i^i so that if yLt^*:^« =/„,/„ is the quantity by which the intensity that would have existed if the fluid had been hindered from lateral motion has to be divided. " If X be the length of the sound-wave corresponding to the period of the vibration, k = 27r/X., so that kc is the ratio of the circumference of the sphere to the length of a wave. If we sup- pose the gas to be air and \ to be 2 feet, which would correspond to about 550 vibrations in a second, and the circumference 27rc to be 1 foot (a size and pitch which would correspond with the case of a common house-bell), we shall have kc = ^, The following table gives the values of the squares of the modulus and of the kc n=0 n = l n = 2 n=3 n = 4 o o "ft % EL i o 4 2 1 0-5 0-25 17 5 2 1-25 1-0625 16-25 5 5 16-25 64062 14-879 9-3125 89 13S0-2 20878 13*848 80 3965 236191 14837899 20177 1495-8 800137 72086371 18160 X 10« 4 2 1 0-6 0-25 1 1 1 1 1 0*95588 1 2-5 IS 60-294 0-87523 1-8625 44-5 1064-2 19650 0-81459 16 1982-5 188953 13965 X 10» 1-1869 299-16 150068 57669097 17092 X 10« ratio In for the functions F^ (ike) of the first five orders, for eadi ^ the values 4, 2, 1, \, aad \ of fcc. It will presently appear why K' -'.■_'4.] stokes' investigation. 243 the tabln has beun extended further in the direction of values g^reater than J than it has in the opposite directiou. Five signi- ficaut figures at least are retaiued. " When Ar = X we get from tlic analytical expressions /„ = 1. We see from the table that when Arc is somewhat large /„ is liable to be a ttUle less than I, and consequent!]' the sound to be n little more intense than if lateral motion had been prevented. The pos.sibility of that is explained by considering that the waves of condensation spreading from those compartments of the sphere which at a given moment ai'e vibrating positively, i.e. outwards, after the lapse of a half period may have spread over the neigh- bouring compartments, which are now in their turn vibrating positively, so that these latter compartments in their outward motion work against a somewhat greater pressure than if such comportment had opposite to it only the vibration of the gas which it had itself occasioned ; and the same explanation applies mutatis mutandis to the waves of rarefaction. However, the in- crease of sound thus occasioned by the existence of lateral motion is but small in any case, whereas when kc is somewhat small 7„ increases enormously, and the soimd becomes a mere nothing compared with what it would have been had lateral motion been prevented. "The higher be the order of the function, the greater will be the number of compartments, alternately positive and negative as to their mode of vibration at a given moment, into which the surface of the sphere will be divided. We see from the table that for a given periodic time as well as radius the value of /„ becomes con- siderable when " is somewhat high. However practically vibra- tions of this kind are produced when the elastic sphere executes, not its principal, but one of its subordinate vibrations, the pitch corresponding to which rises with the order of vibration, so that k increases with that order. It was for this reason that the table was extended from kc = 0-5 further in the direction of high pitch than low pitch, namely, to three octaves higher and only one octave .lower. " When the epheie vibrates symmetrically about the centre, ie, that any two opposite points of the surface are at a given moment moving with equal velocities in opposite directions, or generally when the mode of vibration is such that there is no chang e of position of the centre of gravity of the volume, there 244 Leslie's experiment. [324. is no term of order 1. For a sphere vibrating in the manner of a bell the principal vibration is that expressed by a term of the order 2, to which I shall now more particularly attend. " Putting, for shortness, i*c* = q, we have _ g» - 2g» 4- 9g -f 81 " The minimum value of /, is determined by 9* - 6g« - 84^ - 54 = 0, giving approximately, q = 12-859, kc = 3-586, /io« = 13-859, /Lt,' = 12049, /, = '86941 ; so that the utmost increase of sound produced by lateral motion amounts to about 15 per cent. "I now come more particularly to Leslie's experiments. Nothing is stated as to the form, size, or pitch of his bell; and even if these had been accurately described, there would have been a good deal of guess-work in fixing on the size of the sphere which should be considered the best representative of the bell. Hence all we can do is to choose such values for k and c as are comparable with the probable conditions of the experiment. "I possess a bell, belonging to an old bell-in-air apparatus, which may probably be somewhat similar to that used by Leslie. It is nearly hemispherical, the diameter is 196 inch, and the pitch an octave above the middle c of a piano. Taking the number of vibrations 1056 per second, and the velocity of sound in air 1100 feet per second, we have X = 125 inches. To represent the bell by a sphere of the same radius would be velry greatly to underrate the influence of local circulation, since near the mouth the gas has but a little way to get round from the outside to the inside or the reverse. To represent it by a sphere of half the radius would still apparently be to underrate the effect. Nevertheless for the sake of rather under-estimating than exaggerating the influence of the cause here investigated, I will make these two suppositions suo- cessively, giving respectively c = *98 and c = '49, ho » '4926» m V » '2468 for air. 324.] NUMERICAL RESULTS. 245 " If it were not for lateral motion the intensity would vary from gas to gas in the proportion of the density into the velocity of propagation, and therefore as the pressure into the square root of the density under a standard pressure, if we take the factor de- pending on the development of heat as sensibly the same for the gases and gaseous mixtures with which we have to deal. In the following Table the first column gives the gas, the second the II o» 1-H o eo 2 1-1 GO 8 $ ? S ip ^ ^ o QP op 00 C9 00 CO s 9> W5 S 2 s o» w4 3 s C9 ? <? S • s • lid • • iH • S s s © CO CO s S II »j» iH t* 1-4 ^p w^ 1-4 eo H iH ^^ »H ^5 iH iH -^ u s c« •^ 2 ^ s; CO 04 ^ s; S^ • 3 • 1-4 9> • o • • s ?* s; S 00 Oi C9 »o eo s 8 s o 9 Ud 00 »o 3 3 £. •8 :5 I so d O § 246 DEFICIENCY OF TERM OF ZERO ORDER. [324. pressure jo, in atmospheres, the third the density D under the pressure p, referred to the density of the air at the atmospheric pressure as unity, the fourth, Q^, what would have been the inten- sity had the motion been wholly radial, referred to the intensitT in air at atmospheric pressure as unity, or, in other words, a quantity varying as p x (the density at pressure 1)*. Then follow the values of g, /,, and Q, the last being the actual intensity referred to air as before. "An inspection of the numbers contained in the columns headed Q will shew that the cause here investigated is amply sufficient to account for the facts mentioned by Leslie." The importance of the subject, and the masterly manner in which it has been treated by Prof. Stokes, will probably be thought sufficient to justify this long quotation. The simplicity of the true explanation contrasts remarkably with conjectures that had pre- viously been advanced. Sir J. Herschel, for example, thought that the mixture of two gases tending to propagate sound with different velocities might produce a confusion resulting in a rapid stifling of the sound. [The subject now under consideration may be still more simply illustrated by the problems of §§ 268, 301. The former, for in- stance, may be regarded as the extreme case of the present, in which the spherical surface is reduced to a plane vibrating in rectangular segments. If we suppose the size of these segments, determined by p and q, to be given, and trace the effect of gradu- ally increasing frequency, we see that it is only when the frequency attains a certain value that sensible vibrations are propagated to infinity, the law of diminution with distance being exponential in its form. On the other hand vibrations whose frequency exceeds the critical value are propagated without loss, escaping the attenuation to which spherical waves must of necessity submit.] 325. The term of zero order ^. = fe*'-'-" (1> where iSo is a complex constant, corresponds to the potential of a simple source of arbitrary intensity and phase, situated at the mtre of the sphere (§ 279). If, a& oft^u V^yS^"^ ^ ^^radaoep th^ 325.] REACTION ON RIGID VIBRATIXG SPHERE. 247 source of sound be a solid body vibrating without much change of volume, this terra is relatively deficient. In the case of a rigid sphere vibrating about a position of equilibrium, the deficiency is absolute', inasmuch as the whole motion will then be represented by a term of order 1 ; and whenever the body is very small in comparison with the wave-length, the term of zero order must be iiLsignificant. For if we integrate the equation of motion, V'yjr + i^^fr = 0, over the small volume included between the body and a sphere closely surrounding it, we see that the whole quan- tity of fluid which enters and leaves this space is small, and that therefore there is but little total flow across the surface of the sphere. Patting n = 1, we get for the term of the first order rf, = S,e*i'"-^|l + ^l (2). and iS, is proportional to the cosine of the angle between the direction considered and some fixed axis. This expression is of the same form as the potential of a double source (§ 294), situated at the centre, and composed of two eqna! and opposite simple sources Ij-ing on the axis in question, whose distance apart is iofinitely small, and intensities such that the product of the Intensities and distance is finite. For, if tr- be the axis, and the wine of the angle between x and r be fi, it is evident that the tential of the double source is proportional to It appeal^ then that the disturbance due to the vibration of a ! as a rigid body is the same as that corresponding to a aouble source at the centre whose axis coincides with the line of the sphere's vibration. The reaction of the air on a small sphere vibrating as a rigid body with a harmonic motion, may be readily calculated from preceding fonnulse. If | denote the velocity of the sphere at time t, U.i^^^tj. (3), anrl therefore for the value of ifr at the surface of the sphere, ' have from (5) § 324. ., f Mike) ' The oentre of the sphere being the origlB d 248 INCREASE OF EFFECTIVE INERTIA. [325. The force B due to aerial pressures accelerating the moti<»i is given by If we write then H = — p.fTrpc*.^— qka . J tt/x^ .f.. (6), inasmuch as f = ika ^. The operation of the air is therefore to increase the effective inertia of the sphere by p times the inertia of the air displaced, and to retard the motion by a force proportional to the velocity, and equal to | tt/oc* . qkd^, these effects being in general functions of the frequency of vibration. By introduction of the values of /i and Fi we find F,{ikc) 4 + ifc*c* ^^' «^*'^^*' ^ = 4TF^' ?=4T:fcv ^^)- When kc is small, we have approximately p==^, ? ~ i ^^' Hence the effective inertia of a small sphere is increased by one- half of that of the air displaced — a quantity independent of the frequency and the same as if the fluid were incompressible. The dissipative term, which corresponds to the energy emitted, is of high order in kc, and therefore (the effects of viscosity being disregarded) the vibrations of a small sphere are but slowly damped. The motion of an ellipsoid through an incompressible fluid has been investigated by Green*, and his result is applicable to the calculation of the increase of effective inertia due to a compressible fluid, provided the dimensions of the body be small in comparison with the wave-length of the vibration. For a small circular disc vibrating at right angles to its plane, the increase of effective inertia is to the mass of a sphere of fluid, whose radius is equal to ^ Edinburgh Transaetiom, Deo. 16, 1888. Also Green's MathemaHeal Pi^fen^ Bdited by Feiren. Macmillan & Co., 1811. 325.1 MULTIPLE SOURCES. 249 that of the disc, as 2 to w. The result for the case of a sphere given above was obtained by Poisson', a short time before the publication of Green's paper. It has been proved by Maxwell' that the various tenns of the harmonic expansion of the common potential may be regarded as due to multiple points of corresponding degrees of complexity. Thus Vi is proportional to jr-rr ht- (-). where there are » differentiations of ?'"' with respect to the axes h,, A,, &c., any number of which may in particular cases coincide. It might perhaps have been expected that a similar law would hold for the velocity potential with the substitution of j-'e"'*' for r-'. This however is not the case ; it may be shewn that the potential of a d' e~''^' quadruple source, denoted by ji j. ■ — . corresponds in general g-itr not to the term of the second order simply, viz., ij — — ftiih"), it to a combintition of this with a term of isero order. The Httlal< il poin' ^Smagii logy therefore holds only in the single instance of the double point or source, though of course the function r^ig-**^ after any number of differentiations continues to satisfy the fundamental equation It is perhaps worth notice that the disturbance outside any nary sphere which completely encloses the origin of sound may be represented as due to the normal motion of the surface of any smaller concentric sphere, or, as a particular case when the radius of the sphere is infinitely small, as due to a source concen- trated in one point at the centre. This source will in general be composed of a combination of multiple sources of all orders of complexity, 326. When the origin of the disturbance is the vibration of a rigid body parallel to its axis of revolution, the various spheiical harmonics S„ reduce to simple multiples of the zonal harmonic Pn ifi), which may be defined as the coefficient of e" in the expan- sion of {1 — 2e/i-l-fl']*^ in rising powers of e. [For the forms of B functions see § 334.] And whenever the solid, besides being Mimoiru dt VAeadfatie da ScifHco. Tom. ». p. 531. llAcell's KUetriciltj and Marinetiim, Cb. IX. 250 CASES OF SYMMETRY. [326. S3niimetrical about an axis, is also symmetrical with respect to an equatorial plane (whose intersection with the axis is taken as origin of co-ordinates), the expansion of the resulting disturbance in spherical harmonics will contain terms of odd order only. For example, if the vibrating body were a circular disc moving perpen- dicularly to its plane, the expansion of -^ would contain terms proportional to Pi (/i), Pj (^i), Pg (ji\ &c. In the case of the sphere, as we have seen, the series reduces absolutely to its first term, and this term will generally be preponderant. On the other hand we may have a vibrating system symmetri- cal about an axis and with respect to an equatorial plane, but in such a manner that the motions of the parts on the two sides of the plane are opposed. Under this head comes the ideal tuning- fork, composed of equal spheres or parallel circular discs, whose distance apart varies periodically. Symmetry shews that the velocity-potential, being the same at any point and at its image in the plane of symmetry, must be an even function of /a, and there- fore expressible by a series containing only the even functions Po(/i), Pa(/i), &c. The second function Pa (/a) would usually preponderate, though in particular cases, as for example if the body were composed of two discs very close together in comparison with their diameter, the symmetrical term of zero order might become important. A comparison with the known solution for the sphere whose surface vibrates according to any law, will in most cases furnish material for an estimate as to the relative importance of the various terms. [The accompanying table, p. 251, giving Pn as a function of 0y or cos~*/i, is abbreviated from that of Perry*.] 327. The total emission of energy by a vibrating sphere is found by multiplying the variable part of the pressure (proportional to yjr) by the normal velocity and integrating over the surface (§ 245). In virtue of the conjugate property the various spherical harmonic terms may be taken separately without loss of generality. We have (§ 323) S e^<«<-'^ \ yfrn = ika ^^—- /„ {ikr) \ (1), dr " r» ^'^^^^^ ) 1 FhiX, Mag. "voL xnn., ^. S16> 1891. ■ ■ ■ ^^^H 1 ■ ■ 327 •] TABLE OF ZONAL HARMONICS. 251 I Table of Zonal Spherical Harmonics. ! e p, F, P, p. p. P, Fj 1-0000 1-0000 1-0000 1-0000 10000 1-0000 l^OOOO 2 -9994 ■9988 -9963 ■9939 -9900 9873 ■9839 i ■9976 ■9927 ■9854 -9768 -9638 9496 9399 B -9945 -9886 ■9B74 ■9469 ■9194 8881 -8638 e -9903 ■9709 -!1423 -9048 -8689 8063 ■7448 10 -9848 ■9648 -9106 ■8583 ■7840 7046 ■6164 12 -9781 ■93G2 '8724 ■7920 ■6966 5892 •4733 -9703 -9122 ■8283 ■7224 ■6990 4635 ■8219 16 ■9613 ■8860 ■7787 ■6454 ■4937 3323 -1699 18 ■9SU ■8368 ■7240 -5824 ■3836 3002 -0889 20 ■9397 ■8246 -0649 -4760 ■3716 0719 -■1073 22 ■9272 -7895 -8019 ■3845 '1602 0481 -■2201 24 ■0136 '7618 ■5357 ■29-26 '0525 1659 -■3095 20 ■8088 ■7117 ■4670 -i007 --0489 2478 - '3717 SH ■8829 ■6694 ■3961 ■1105 --U15 3211 - '4052 30 ■8660 ■6250 ■3248 ■0234 -■2233 3740 - ^4101 H2 ■8480 ■6788 ■2527 -0691 -■2923 4002 - '3876 34 ■8290 -S310 -1809 - -1357 -■3473 4148 -'3409 3tl ■8090 -4818 ■1102 --2052 - ■3871 4031 --2738 88 ■7880 ■4314 ■0413 -■3868 -■4112 3719 --1918 40 ■7880 ■3802 -■0352 -■3190 - 4197 3234 --1003 42 -7431 ■3284 --0887 - -3616 -■4128 3011 -•0065 44 •7193 -2762 - -1486 --3940 - 3914 1878 -0846 4B ■6947 ■2338 --204O - -4158 -■3368 1079 -1666 48 ■6691 ■1716 --2647 -■4370 - -3103 0251 ■3349 50 ■6428 -1198 -•3003 - -4276 --2545 ■I- 0363 ■2854 52 -6137 -0086 - -3401 - ^41 78 - -1910 + 132C -3133 54 ■5878 -0183 - -3740 --3984 -1223 + 2002 -3334 SO ■5592 -■0310 - ■4016 -■3698 -■0510 + 2359 '309G 58 '5290 --0788 -■4229 -■3331 ■0206 3976 ■2753 60 '6000 -■1260 -■4376 -■2891 ■0898 3332 -3231 ■4695 - -1694 -■4463 - -2390 -1543 3321 -1671 64 ■4384 -■2117 - -4470 --1841 -2133 3240 -0818 66 ■4067 --2318 -■4419 - -12S6 ■3616 + 2996 ■0021 68 ■3746 -■2896 -•4305 -■0660 ■3006 3605 --0763 70 ■B430 -•8213 ^-4190 -■0038 ■3381 2089 -■I486 72 ■3080 -■3568 -■3898 -0568 ■3434 1473 -■2099 74 ■27S6 -'3860 -■3611 ■1153 '3461 0796 -■2569 76 ■2419 - -4113 -■3275 ■1705 ■3362 0076 -■2848 7H ■9079 -■4358 -■2894 ■2211 '3143 0644 - -2943 80 ■1786 -■4548 -■3474 ■2639 SSIO 1321 --3835 82 ■1392 --4709 -■3020 ■3040 ■2378 1936 --2636 M ■1046 --4886 -■1539 -3346 ■18B1 3431 --2067 86 •0G9B -■4927 -■1038 -1278 2811 -1460 88 -0949 •'4983 -■0633 •3704 ■0651 3045 -■0736 00 ■0000 --6000 -■0000 ■3760 ■0000 -'31SE 0000 1 i M ■ ■ 1 252 ENERGY EMITTED [327. or on rejecting the imaginary part •^„ = {^cosk {at — r) + a' sin k (at — r)} .(2). ^•' = - §f a cos A (a< - r)-/9 sin A (of -r)l ar T^ where i^=a + i/3, /=a' + i^ (3). " T^f/^n'^^ l*'^ cos» A: (a« - r) - o'/S sin* A: (a« - r) + (oa' — fiff) sin A: (erf — r) cos i (at — r)]. When this is integrated over a long range of time, the periodic terms may be omitted, and thus j.jjirJ^^^dS.dt^^(a^-a'^)ffSn*do- (4). Now, since there can be on the whole no accumulation of energy in the space included between two concentric spherical surfaces, the rates of transmission of energy across these surfaces must be the same, that is to say r~* (a'^ - ^a) must be independent of r. In order to determine the constant value, we may take the particular case of r indefinitely great, when Fn (ikr) = ih\ a = 0, /8 = At, /n(tAT) = l, a' = l, /8' = 0. Thus a'/8 - /8'a = At, identically (5). It may be observed that the left-hand member of (5) when multiplied by i is the imaginary part of (a + 1/8) (a' — %I3') or of Fn (ikr)fn (— ikr), so that our result may be expressed by saying that the imaginary part o{ Fn{ikr) fn{— ikr) is tAr, or Fn(ikr)M^ikr)- Fn("ikr)Mikr)=^2ikr (6). In this form we shall have occasion presently to make use of it. The same conclusion may be arrived at somewhat more directly by an application of Helmholtz's theorem (§ 294), i.e, that if two functions u and t; satisfy through a closed space S the equation (V« + A;*) u =: 0, then •S-S)-^-"-- ■<'^ //( 327.] FROM A VIBRATING SPHERICAL SURFACE. 253 If we take for S the space between two concentric spheres, making u , v- - , we find that r~*{^n(^)/n(-**'')-"-'^n(— i^)/n(**^)} niust be independent of r. We have therefore f I U^ ^dS.dt^^ ik^atjjSn'da ; so that the expression for the energy emitted in time t is (since W=^k'patjjSn*da' (8). It will be more instructive to exhibit TT as a function of the normal motion at the surface of a sphere of radius c. From (2) -Z^ ="--^ [cos kat (a cos kc-¥fi sin kc) + sin kat (a sinkc — fi cos Arc)], so that, if the amplitude of d'^njdr be Un, we have as the relation between Sn and Un C*trn» = (a» + ^)Sn» (9). This formula may be verified for the particular cases n = and n= 1, treated in §§ 280, 325 respectively. 328. If the source of disturbance be a normal motion of a small part of the surface of the sphere {r — c) in the immediate neighbourhood of the point /a = 1, we must take in the general solution applicable to divergent waves, viz. ^ = _^,a,.-...2^^^^/„W (1), U„ = i(2n + 1) P„ 0*) .J^l frP„0*) d,, = i(2n + l)P»0*)/^V<iM = ^1^ Pn(f^)jjUdS (2) ; 254 SOURCE SITUATED [328, for where U is sensible, Pn(M') = 1- Thus ^'.//™.S(2n + l)P„(M)^> (3> In this formula 1 1 UdS measures the intensity of the source. If ike be very small, /o(iAT) /(tfer) / 1\ . so that ultimately ^=-W//^'^^ (*)• and the waves diverge as from a simple source of equal magnitude. We will now examine the problem when kc is not very small, taking for simplicity the case where yjr is required at a great distance only, so that /n(ikr) = 1. The factor on which the rela- tive intensities in various directions depend is - (2n + 1) P„(m) ... 2 Fnitkc) ^ ^' and a complete solution of the question would involve a discussion of this series as a function of /i and kc. Thus, if ^ = - 2^ jjUdS . {F' + (?»}» . e«(««-r+«)+» (7), where tan^ = G : F (8). The intensity of the vibrations in the various directions is thus measured by F* + 0^. If, as before, Fn = a + i^, 2 a» + /S« The following table gives the means of calculating F and for any value of /i, when kc = ^, 1, or 2. In the last case it is necessary to go as far as n = 7 to get a tolerably accurate result, and * larger values of fcc the calculation would soon beoome ' (9). ON THE SURFACE OF A SPHERE. 255 klxmous. In all problems of this sort the harmonic analyais seems lose its power when the waves are very small in cooaparisoQ nth the dimensions of bodies. ic = 4. " 2a 3^ (n + i)«-=-(«»+,S^ (K + iJ^-ta'+p-) + a + 1 + ■4 + -2 + * 7 + 1846168 - -8230768 64 36 - -0601391 --0328886 3 - m + 863 - -0034527 + 0063201 4- 1*902 + 8141 + -0004653 + ■0008642 + 175593 - 321419 + -0000144 - -0000264 kc = 1. • » p .(n+i)»+(«»+ffl (» + i);9+(.»+^ + 1 + 1 + •25 + •25 + 2 + ■« 3 5 8 - -140449 224719 63 + 34 - -040784 330013 + 296 + 461 + ■004438 306912 + 1951 - B179 + ■000787 300505 ~ 40613 ~ 63331 - -000047 300073 -936340 + 601317 -■000000 + -000004 he = 2. ■ ■ p (» + i)«-5-(«»+^, (» + Dp-(a» + p») + 1 + 2 + -1 + -a + 175 - 2 5 ■46980 B7IU - 8 36 176 - IB -1875 + 35 ■125 Q4870 10567 + 186-625 + 85'4375 1)2436 01115 + 638-eO -1177-3 ooao'j - -00466 -8621-7 -8946-8 -00078 -■00033 he most interesting question on which this analysis informs the inflaeuce which a rigid sphere, situated ckwu to the NWTce, has on the intensity of sound in different directions. Bj" tJie principle of reciprocity (§ 2U-i) the source and the place of " " I may be interchanged. When therefore we know the i 256 NUMERICAL RESULTS. [328. relative intensities at two distant points B, B\ due to a source A on the surface of the sphere, we have also the relative intensities (measured by potential) at the point A, due to distant sources at B and B. On this account the problem has a double interest As a numerical example I have calculated the values o( F+%0 and F^-\- CP for the above values of kc, when /a = 1, /i = — 1, /a=sO, that is, looking from the centre of the sphere, in the direction of the source, in the opposite direction, and laterally. When kc is zero, the value of F^ + G^ is '25, which therefore represents on the same scale as in the table the intensity due to an unobstructed source of equal magnitude. We may interpret kc as the ratio of the circumference of the sphere to the wave-length of the sound. kc 1 F+iO F^+0* i 1 -1 •521508 + •149417t '159U9 - '4841491* '480244 - -2165891 '294291 '259729 '231999 1 1 -1 •667938+ •238869i - -440055 - '3026091 + '321903 - '3649741 '502961 '285220 '236828 2 1 -1 1 '79683 +'23421t '24954 + '505861 -15381 -'57662i '6898 '3182 '3562 In looking at these figures the first point which attracts attention is the comparatively slight deviation from uniformity in the intensities in different directions. Even when the circum- ference of the sphere amounts to twice the wave-length, there is scarcely anything to be called a sound shadow. But what is perhaps still more unexpected is that in the first two cases the intensity behind the sphere exceeds that in a transverse direction. This result depends mainly on the preponderance of the term of the first order, which vanishes with /i. The order of the more important terms increases with kc\ when kc is 2, the principal term is that of the second order. Up to a certain point the augmentation of the 8|Aere wiD increaae the total energy emiUed, b^sause a simple soaxoa en 328.] NTTMERICAL RESULTS. 257 twice as much energy when close to a rigid plane as when entirely in the open. Within the limits of the table this effect masks the obstruction due to an increasing sphere, so that when /* = — !, the intensity is greater when the circumference ia twice the wave- length than when it is half the wave-length, the source itself remaining constant. If the source be not simple harmonic with respect to time, the relative proportions of the various constituents will vary to some fxtent both with the size of the sphere and with the direction of the point of observation, illustrating the fundamental character ijf the analysis into simple harmonics. When kc is decidedly less than one-half, the calculation may be conducted with sufficient approximation algebraically. The result is >+term8 in jfc* (10). It appears that so far as the term in k'(^, the intensity is an even function of ft, viz. the same at any two points diametrically opposed. For the principal directions ^= ± 1, or 0, the numerical calculation of the coefficient of f c* is easy on account of the simple values then assumed by the functions P. Thus {^=1). F'+G' = \ + -^k^c' + -177ahk'c*+ (>* = -!). /" + ff'=i-l-^i-=c' + -02755 A-<c*+ (/i = 0). F^ + CP = i- i k-d' + -W5Z4-k*c'+ When jfc* can be neglected, the intensity is less in a lateral direction than immediately in front of or behind the sphere. Or, by the reciprocal property, a source at a distance will give a greater intensity on the surface of a small sphere at the point furthest from the source than in a lateral position. If WB apply these formulae to the case of Ac = ^, we get »(fi=l), /^ + G' = -3073, (^ = -1). F'+O' = -2Q0i, (fi. = <>). F'+G' = -2Sii. which agree pretty closely with the results of the more complete calouUtio o. EFFECT OF SMALL SPHERE. For Other values of ft, the coefficieut of fc* in (10) might \ calculated with the aid of tables of Legendre'e functions, or fronfl the following algebraic expression in terms of ^', 1 + I/. + g^i', + If P,' - sVmP. + T^^. = -781 38 + 1 5 /* + -85938 fi' - -03056 /**. The difference of intensities in the directions ^ = ^ = — 1 may be very simply expressed. Thus (/" + ff'V.i - (f + G\= _j = J l*<^. li kc=^, Ji''c' = -014S. If Ac=|, Jfc*c* = -0029. If A,-c=J, |/:*c* = -0002. At the same time the total value of f + C approximates tol "25, when kc is small. These numbers have an interesting bearing on the explanation I of the part played by the two ears in the perception of the quarter J from which a sound proceeds. It should be observed that the variations of intensity in differen directions about which we have been speaking are due to th presence of the sphere as an obstacle, and not to the fact tha the source is on the circumference of the sphere iustead of i the centre. At a great distance a small displacement source of sound will affect the phase but not the intensity in anj direction. In order to find the alteration of phase we have for a smai sphere f=i + i-.c'(-i + f^-^P,), (? = Ac(-i + f/-). tan^ = G:i''=A-c(-l+f^), or = kci-\+%^L) nearly. Thus in (7) eain(-r+fi+« = e*(«-r+Wi, from which we may infer that the phase at a distance is the sai as if the source bad been situated at the point /i = l, r = | (instead of r = c), and there had been no obstacle. 329. The functional symbols / and F may be expressed terms of P. It ia known' that P r„w,_" ?L+I l-A, «(i-l) ( n + l)(« + 2) ( l-^)- "^'^' 1* 1 " 2 1.2 1.2 2* ~' ' For Ibe fonuB of the fonctioQB P, gee % 334. '' Thomson and TaiVa Nat. Phil, %181 (abated from Murphy), 29.] ANALYTICAL EXPRESSIONS. 259 r, on chaDging /a into 1 -* /a, ^ n-.x^-l-!? !Lhi M , n(n-l) (n + l)(n + 2) /i»_ *VA /a;-i 1' 1-21. 2' 1.2 ^^ "^ ^' Consider now the symbolic operator Pnfl — t")» *"^d ^^^ i^ perate on ff\ "V dy/y ^* 1.2 *^^^ 2.4 ^* A comparison with (9) § 323 now shews that /.(y)=yP.(l-|).l (2X om which we deduce by a known fonnula, t'/.(,)=^p.(i-l)i-(-i)-p.(l).^ w n like manner. If we now identify y with titr, we see that the general solutiim, L2) § 323, may be written "om whidi the seccHid term is to be omitted, if no part of the Lstorbanoe be jffopagated inwards. Again bam (14) § 323 we see that y* ~V dy)- y ' •hence ^.0'> = i^^-(l -|) (l -|)-^ <''>' - '-^-^-^OryY <^^ «^. >k(:^=_P.(^)^A- ,, Yl— 1 260 ANALYTICAL EXPRESSIONS. [329. Using these expressions in (13) § 323, we get dr -^ ^ ^^""^""[ddkr) ddkr' ikr 330. We have already considered in some detail the form assumed by our general expressions when there is no source at infinity. An equally important class of cases is defined by the condition that there be no source at the origin. We shall now investigate what restriction is thereby imposed on our general expressions. Reversing the series for/», we have +(-l)»S„'e+*-(l-tifcr +...)}, shewing that, as r diminishes without limit, r^„ approximates to In order therefore that -^n may be finite at the origin, fifn4-(-l)~Sn' = (1) is a necessary condition ; that it is sufficient we shall see later. Accordingly (12) § 323 becomes rtn = S«{6-*'/n(tfcr)-(-.l)»e+*-/n(-ifcr)} (2). If, separating the real and imaginary parts of /n, we write (as before) /„ = a' + t/8' (3>, (2) may be put into the form nfr^ = - 2t"+^ Sn [a' sin (kr + ^nir) -/8' cos (At + i nir)] (4). Another form may be derived from (4) § 329. We have ^, = -2*(-l)-S,i>.(j^).'-!!^^ -2»(-ir*.p.(j^).5^ ^.> 330.] MOTION CONTINUOUS THROUOH POLE. 261 Since the fuDctiou P„ is either wholly odd or wholly even, the expression for ^^„ ia wholly real or wholly imaginary. In order to prove that the value of ^„ in (5) remains finite when r vanishes, we begin by observing that 2 sin It sinAr f''"' .^ , T?-=J-,^"''' '[d.ihrl kr l^:"[d.ikrl' "'' = j P.{f.)if^'dii ..(6). ■■(?)■ as is obvious when it is considered that the effect of differentiating e**"" any number of times with respect to ikr is to multiply it by the corresponding power of fi. It remains to expand the expres- sion on the right in aHcending powers of r. We have ,M'' + .. Now any positive integral power of /i, such as /^p, can be expanded in a terminating series of the functions P. the function of highest order being Pp. It follows that, if ^ < n, ■ j^\>'P„(,x)d,i = 0. liy known properties of these functions ; so that the lowest power of ih- in I Pn ifi) e** d/t is (Her)". Retaining only the leading Bt6nn, we may write I j"p.<,i.)^d^.^i'^J*\'P,Md„. ^H From the expression for Pn (m) in terms of fi. viz. Wp ,, 1.3.5...(2»-1 )(. .(»-!) 'p.w-- vfe see that I.S.3.. !.3...n r 2(2»-l)'- n(« -l)(.- 2)(,.-3) 2.1.(2n-i)(2»-3l '^ \Pi. (/*) + terma in ft of lower order than ft 262 ANALYTICAL EXPRESSIONS [330. and therefore 1 . 2 . 3 ... 71 2 rQv ~1.3.5...(2n-l) ■ 2^+1 ^ '' Accordingly, by (5) and (7) ^»'-^^(-^)"^ M.3.5?2n + l) ^ (10). which shews that y^^ vanishes with r, except when n = 0. The complete series for y^nt when there is no source at the pole, is more conveniently obtained by the aid of the theory of Bessel's fdnctions. The differential equations (4) § 200, satisfied by these functions, viz. may also be written in the form -it- + (^— ^-)y^=<^ <^2). It is known (§ 200) that the solution of (11) subject to the condition of finiteness when z = Ofisy=^AJm (z), where ^ f z^ ^^ ^'^ '^ 2"»r(m+T) 1^ ~ 2. (2m + 2) "* T4)--} •<^3>' is the Bessers function of order m. When m is integral, F (m + 1) = 1 . 2 . 3 . . . m; but here we have to do with m fractional and of the form n-\-]i, n being integral. In this case _ , -V 1 . 3. 6 ...(2n4-l) / /ij\ r(m + l)=. 2iH5 -V'^ (I*)- Referring now to (12), we see that the solution of g,(:.^)..„ (,.; under the same condition of finiteness when <; 3= 0, is 6-A«IJ»(«^ ..«(!" ■^2.4.(2TO + 2)(2m 330.] FOR VELOCITY-POTENTIAL. 263 Now the function i^„, with which we are at present concerned, satisfies (4) § 323, viz. d»(n^«) P(nK) . /, n (n + 1) \ _ „ which is of the same forai as (15), if m = n 4- ^ ; so that the solu- tion is yfrn^'A (kr)-^ Jn+^ {kr) = A {krY V2 1.3...(2n + l) V-w ( 2.(2n + 3) ( 2.(2n + ■*"2:4.(2n + 3)(2w + 5)""-J ^^^^• Determining the constant by a comparison with (10), we find Vr„ = - 2 (- 1)» t»« kS^ (£)*«^«+i (*^) (tifcr)' A'r* 2tA;(-l)"S,j g g ^g^^j^jl 2(2n + 3) "*'2.4.(2n + 3)(2w + 5) 2.4.6.(2n-|-3)(2n + 6)(2n + 7)'^ •") (19). as the complete expression for -^n ^^ rising powers of r. Comparing the different expressions (5) and (19) for i^^, we obtain ^■(5^)-^-'(^)'^".<'') « If -P = a + 1/8, the corresponding expressions for d^ltn/dr, are 2i»+i 5 = --^{asm(kr + ^nir)'-/3coa{kr + ^nir)] 2n(-l)»&'g,(ifcr)»- ( n + 2 ) , . 1.3.5...(2» + 1) f 2n(2n + 8r*^^-\ ^^^'' 264 PARTICULAR CASES. [330. It will be convenient to write down for reference the fonns of y^ and dyfr/dr for the first three orders. n = ^ dr sin At 2%kSo f sin At ^0 = - 2ikS, n = l At cos At — — cos At sin At] [dr = — -^ ]2cosAT-f (*^ — ji) sin At}-. n = 2 i , . , sin At 4- T- cos At r i\ J(^J" "" ' At •1- d^2 . dr = ^*f'{(4-j^)8inAT-(AT-l,)cosAr}. 331. One of the most interesting applications of these results is to the investigation of the motion of a gas within a rigid spherical envelope. To determine the free periods we have only to suppose that dy^/dr vanishes, when r is equal to the radius of the envelope. Thus in the case of the symmetrical vibrations, we have to determine A, tan A?i* = At (1), an equation which we have already considered in the chapter on membranes, § 207. The first finite root (At = 1*4303 tt) corre- sponds to the symmetrical vibration of lowest pitch. In the case of a higher root, the vibration in question has spherical nodes, whose radii correspond to the inferior roots. Any cone, whose vertex is at the origin, may be made rigid without affecting the conditions of the question. The loops, or places of no pressure variation, are given by (At)""^ sin At s= 0, or AT = m7r, where m is any integer, except zero. The case of n = l, when the vibrations may be called dia- metral, is perhaps the most interesting. 8i, being a harmonic of order 1, is proportional to cos where is the angle between r id some Sxed direction of lefeieiiDLQe. €»i3CL<^ ch|\/<M vaniaheB a p.1.] DIAMETRAL VIBRATIONS. 265 at the poles, there are no conical nodes' with vertex at the ceiitn>. Any meridianal plane, however, is nodal, and may be siippiweii rigid. Along any specified radius vectoi'. ^, and 'i^Jd$ vaiiiah, and change sign, with cos kr — (kr)~' sin kr, viz. when tiuihr = kT. To find the spherical nodes, we have 2kr l-fi^i^ tan kr = ..(21. The first root is At = 0. Calculating from Trigonometrical Tables by trial and error, I find for the next root, which oor- responda to the vibration of moat importance within a sphere, kr = 119-26 X 7r/180 ; so that r:\ = -3313. The air sways from side to side in much the same manner rh in a doubly closed pipe. Without analysis we might anticipate that the pitch would be higher for the sphere than for a closed pipe of equal length, because the sphere may be derived from the cylinder with closed ends, by filling up part of the latter with obstructing material, the effect of which must be to sharpen the spring, while the mass to be moved remains but little changed. In fact, for a closed pipe of length 2r, r:\ = -25. The sphere is thus higher in pitch than the cylinder by about a Fourth. The vibration now under consideration is the gravest of which the sphere is capable; it is more than an octave graver than the gravest radial vibration. The next vibration of this type is such that kr = 340-35 w/180, or r:X=i-94.H, and is therefore higher than the first radial. When kr Ls great, the rooti of (2) may be conveniently calcu- lated by means of a series. If ytr - o-w — ij, [where a is an integer,] then 2(<rir- y) t8ny = . from which we find ^H^ Anodaia m A noda b a mrfu* wbMi mii^t \m Mip|>owd rifti ,.(»). 26G VIBRATIONS OF SECOND ORDER. [331. When n = 2, the general expression for Sn is S^ = Ao (cos'tf " i) + (-^1 cos (o + Bi sin oo) sin cos tf -f (-4a cos 2(0 + -Bj sin 2w) sin*^. . ..(4), from which we may select for special consideration the following notable cases: (a) the zonal harmonic, S, = ilo (cos«tf - i) (4a). Here d^^dd is proportional to sin 20, and therefore vanishes when 6 = ^7r. This shews that the equatorial plane is a nodal surface, so that the same motion might take place within a closed hemisphere. Also since S^ does not involve (o, any meridianal plane may be regarded as rigid. (^) the sectorial harmonic S2 = iljCOs2(»sin«tf (5). Here again dyjrJdO varies as sin 20, and the equatorial plane is nodal. But dyjrjdo) varies as sin 2a>, and therefore does not vanish independently of 0, except when sin 2(d = 0. It appears accordingly that two, and but two, meridianal planes are nodal, and that these are at right angles to one another. (7) the tesseral harmonic, S2—A1CO8 ©sin dcos (6). In this case d'^f/d0 vanishes independently of o) with cos 20, that is, when tf = ^tt, or f tt, which gives a nodal cone of revolution whose vertical angle is a right angle, d^lt^do) varies as sin q>, and thus there is one meridianal nodal plane, and but one \ The spherical nodes are given by tanAT=> ^^^^g (7), of which the first finite solution is kr = 3-3422, giving a tone graver than any of the radial group. In the case of the general harmonic, the equation ^ p. owe to Prof. Ij«mb the remark that the difietanee ^ ly in nUiiUm to the axes ot x^tocsiM.'V . « tones possible within a sphere of radius r may he written (21) §330 Un(kr + inw) = $:ii (8), or agaJD, 2 irJ'.,^i(ir) = y„^i (/,■/■) (10). [For the roots of ^(r4y,(.))-0 (11), eqviivalent to (10), Prof. M°Mahon gives' ,«-g "^ + ^ 4(7m'+ 154m + 95) 32(8G.«' + 3535m" + 3561». + ei33) ,,„, iTyW <^'' ■where m = ii^, and /3' = l(2i/ + 4s + l) (13). If K = 1, 80 that f = §, „i = 9, ^ = s+l, and (12) gives a result in harmony with (3),] Table A shews the values of X for a sphere of radius unity, corresponding to the more important modes of vibration. In B ia exhibited the frequency of the various vibrations referred to the gravest of the whole system. The Table is extended far enough to include two octaves. Tablb a, CHving the values of \ lot b sphere of unit radios. Order of Harmonic. 1-8983 1 » 8 i 6 6 8-0186 1-8800 1-392 1-llS ■9300 ■eooa II 1 2 ■81884 -67622 1-0677 ■68251 ■M195 ■59308 ■7320 ■5218 ■68eB 3 4 ■J«70 ■aM8B -40380 -46380 ■ k 6 ■gnm- ■ 268 WAVE-LENGTHS OF VIBRATIONS. Table B. [331. Pitch of each tone, referred to gravest. Order of Harmonio. Number of internal spherical nodes. Pitch of each tone, referred to gravest. Order of Harmonio. Number ofinterntl spherifld Dodei. 1-0000 1-6056 21588 2*169 2-712 1 2 8 4 2 8540 8-2458 8-5021 8-7114 8-772 1 5 2 6 1 1 1 332. If we drop unnecessary constants, the particular sola- tion for the vibrations of gas within a spherical case of radius unity is represented by irn = Sn(kr)'iJn+^(kr)cos(k(U-0) (1), where A; is a root of 2kJ'nMk) = Jn^i(k) (2). In generalising this, we must remember that Sn may be com- posed of several terms, corresponding to each of which there may exist a vibration of arbitrary amplitude and phase. Further, each term in Sn may be associated with any, or all, of the values of fc, determined by (2). For example, under the head of n = 2, we might have -^2 = A (cos'd - 1) (A,r)-* Jn+i (Air) cos {k^at 4- 0i) -f B cos 2tt) sin*d (Ajr)"* Jn^^ (k^r) cos (k^cU + 0^), ki and k^ being different roots of 2kJ'^(k) = J^(ky Any two of the constituents of ^jt are conjugate, %,e, will vanish when multiplied together and integrated over the volume of the sphere. This follows from the property of the spherical harmonieSy wherever the two terms considered correspond to different values^ n, or to two different constituents of Sn. The only case remamiB|g for consideration requires us to shew that ff^dr . (fc,r)-* Jn^ (kr) . (A^r)-* /«+* (M - . Jo UNIFORM INITIAL VELOCITY. where k, and ij are different roots of ikJ'„^i(k) = J.,^iik) (4), and this is an immediate consequence of a fundamental property of these functions (§ 203), There is therefore no difficulty in adapting the general solution to prescribed initial circ urn stances. In order to illustrate this subject we will take the case where initially the gas is in its position of equilibrium but is moving with constant velocity parallel to x. This condition of things would be approximately realised, if the case, having been pre- viously in uniform motion, were suddenly stopped. Since there is no initial condensation or rarefaction, all the quantities $n vanish. If d-^jdx be initially unity, we have ■^ = x = r(i, which shews that the solution contains only terms of the first order in spheiical harmonics. The solution is therefore of the form 1^ = -d, (yfc,r)-* J ^{kjr) ficoa k,at + A,(k,r)-iJf{k,r)ftcosk,at+ (5). where &i, k„ &c. are roots of 24 J,' (*■)./, (4) (6). To determine the coefficients, we have initially for values of r from to 1, r = Ai(k,r)-iJ,ik,r) + A,{k,r)-iJi{k,r)+ (7). Multiplying by r*Jf{kr) and integrating with respect to r from to 1, we find Prij,(kr)dr = Ak-^(^[J^{kr)yrdr (8), the other terms on the right vanishing in virtue of the conjugate property. Now by (16), § 203, L 2 _[ [/, {kr)yrdr = [/,' (k)]' + ( 1 - ^,) [J, (k)]' ' =(l-|)[^!(i-)]' (3), by (6). The evaluation of / r''J,{k~r)dr may be effected by the aid of 270 CASE OF UNIFORM [331 a general theorem relating to these functions. By the fundamental differential equation whence by integration by parts we obtain, k' r r^^Jn(kr)dr = n7^Jn(h')-r^^'^^^^^^ (10), or, if we make r = 1, A;»f r»+Vn(AT)dr = nJ„(A)-fc/n'(*) (11). Thus in the case, with which we are here concerned, ifc» [' r* Jj (At) dr = f /|(A;) - kJ^\k) = J^{k) by (6). Equation (8) therefore takes the form '*~(A?-2)J,(ib) ^^'*^' and the final solution is ^'^W^J^"^^"^ <13>' where the summation is to be extended to all the admissible values of k. When ^ = 0, and r = 1, we must have V^ = /a, and accordingly ^¥^2 = ^ <!*)• It will be remembered that the higher values of k are approxi- mately, (3) § 331, k=^(nr (15). air ^ ' The first value of A? is 2*0815, and the second 5*9402, whence ;^ = -85742, j^= 06009, shewing that the first term in the series for '^ is by far the most important 332.] INITIAL VELOCITY. 271 It may be well to recall here that Elquation (14) may be verified thus : the quantities k are the roots of or, if =s2r^Ji{z\ the roots of if> = 0, where <f> satisfies *"+j*'+(i-|)*=^ <i^)- Now, since the leading term in the expansion of if> in ascending powers of 2: is independent of z, we may write <^' = const.{l-^,}(l-g whence, by taking the logarithms and differentiating. If we now put ^ = 2, we get by (17), + ... 2 = --4, (^'=2)=1. )fc»-2 ZKf) 333. In a similar manner we may treat the problem of the vibrations of air included between rigid concentric spherical surfieu^es, whose radii are Vi and r,. For by (13) § 323, if d^njdr vanish for these values of r, i-n(- ikr ,) _ .^^ Fnj- ikr,) ^ Fn{-¥ihr,r Fn{+ikr,y whence tanA,(,. r,) - J ^ ^^^^^^^^^^ (1). where as before Fn(+ikr) = a'^i^ (2). When the difference between rj and ?•, is very small compared with either, the problem identifies itself with that of the vibration of a spherical sheet of air, and is best solved independently. In (1) 272 SPHERICAL SHELL. [333. § 323, if ^ be independent of r, as it is evident that it most approximately be in the case supposed, we have whose solution is simply fn^Sn (4), while the admissible values of k* are given by A-«r«=n(ri + l) (5). The interval between the gravest tone (n = 1) and the next is such that two of them would make a twelfth (octave 4- fifth). The problem of the spherical sheet of gas will be further considered in the following chapter. [For a derivation of (5) fix)m the funda- mental determinant, equivalent to (1), the reader may be referred to a short paper* by Mr Chree.] 334. The next application that we shall make of the spherical harmonic analysis is to investigate the disturbance which ensues when plane waves of sound impinge on an obstructing sphere. Taking the centre of the sphere as origin of polar co-ordinates, and the direction from which the waves come as the axis of ^t, let ^ be the potential of the unobstructed plane waves. Then, leaving out an unnecessary complex coefficient, we have </» = e^ («*+«! ^e^^.^f' (1), and the solution of the problem requires the expansion of e*^ in spherical harmonics. On account of the symmetry the harmonics reduce themselves to Legendre's functions Pn (/i)> so that we may take 6'*^ = ilo4-iliPi+...4-ilnPn+ (2), where Ao».. are functions of r, but not of fi. From what has been already proved we may anticipate that An, considered as a function of r, must vary as but the same result may easily be obtained directly, Multiplyiiig ^ Meaatger of UathmaHc$^ voL xv« p. SO, 1886. PLANE WAVES. 273 1 2) by P^ifi), and integrating ft = + l. we find p'ith respect to /i fi-oni /i = — I to ■-dM = §330. (P,.)'rf/.- W . .iJ ..(3); sin ir so that finally 2» + l f d \ sin^_ .^ / 2*1- ■/.♦iW (4). In the problem in hand the wKole motion outside the sphere may be divided into two parts ; the first, that represented by aiid corresponding to undisturbed plane waves, and the second a disturbance due to the presence of the sphere, and radiating outwards from it. If the poteutial of the latter part be ifr. we have (2) § 324 on replacing the general hannonic S^ by a,./'„(;i), r^,. = ««i^«(/*).B-*-/;(»AT) ..(5). The velocity-potential of the whole motion is found by addition of ^ and ^, the constants a„ being determined by the boundary conditions, whose form depends upon the character of the obstruc- tion presented by the sphere. The simplest case is that of a rigid and fixed sphere, and then the condition to be satisfied when r = c is that t^t'" <«>■ a relation which must of oonrse hold good for each harmonic element separately. For the element of order n, we get ic'e* p i d \ d nakc ^ Fu(ikcy Corresponding to the pUoe wave* ^ — e** '"**', the disturbance due to the presence of the sphere is expreased by eiemenv tHjpaiBHiiy. jror but; eiuuicui. ui uruer r», wt t,, kt^tfl" „ { d \ d nakc -(7). , £n+ 1 .P-(;*)./-iaT>...(8). 274 DISTURBANCE DUE TO [334. At a sufficient distance from the source of disturbance we may take /n (^) = 1- III order to pass to the solution of a real problem, we may separate the real and imaginary parts, and throw away the latter. On this supposition the plane waves are represented by [<!>] = cosk(at + x) (9). Confining ourselves for simplicity's sake to parts of space at a great distance from the sphere, where /n(t*T) = l, we proceed to extract the real part of (8). Since the functions P are wholly even or wholly odd, P f ^ \ d sinfe ? * \d . ike) d,kc* kc is wholly real or wholly imaginary, so that this factor presents no difficulty. {Fn{%kc)]''\ however, is complex, and since Fn{ikc)=^ai-xfi, (r.(ae)l-i^-;^,, where tan 7 = — jS/ou [If the positive value of V(a' 4- j8*) be taken in all cases, 7 must be so chosen that cos 7 has the same sign as a.] Thus ^ = S (2ri +1) 'rl g<[*(a<-r+e,+yl '<i"^^i-'^"(s^)5i-^-p-<^>- w When therefore n is even, kd^ [^3 = (2n + 1) — cos {k{at^r + c) + y} while, if n be odd, Arc* [yjr] = (2n + 1) — i sin {A: (at - r + c) + 7} As examples we may write down the terms in volving harmonics of orders 0, 1, 2. The following ^ functiouB Pn (/i) will be xjyaefaV 334.] A RIGID SPHERICAL OBSTACLE. We have, k'c- rf' sin tc 2fc [l[/;(«(~r ,..(M); [fJ- tJtC I B 81 r faCfc*- The solution of the problem here obtained, though analytically <]Qite geoeral, is hardly of practical use except when /ire is a small quantity. In this case we may advantageously expand our results in rising powers of ic. X c.o8lA:(«i-r4-c) + 7ol (16). x^.8in|<:(ol-,- + c) + T,l (17), x(,.'-i)co.lt(o(-i-+e)+T,l (18). It appears that while [^^,] and ' ■ it 276 SPHERICAL OBSTACLE. [334. still more elevated powers of kc. For a first approximation, thai, we may confine ourselves to the elements of order and 1. Although ['^o] contains a cosine, and ['^j] a sine, they nev^ theless diflfer in phase by a small quantity only. Comparing two of the values of dyftn/dr in (21) § 330 we see that a sin (Arc + ^ nir) — jS cos {kc + ^ nir) = - <- ^>" 1 . 3 . "o^^cTn + 1) •*• ^^''^ P°^^" *>^ ^ identically. Dividing by a cos (Arc 4- \nir\ we get ultimately tan {kc + iwTr) — - = ^^ ^ "^ ^ a acos(A:c + in7r)* 1.3.5 ...(2n + 1)' When n is even, this equation becomes on substitution for a of its leading term from (16) § 323, , I, _fi_ n {kc)^+' ^"""^ a" (n + l)(2n + l){1.3.5...(2n-l)}«-^^^^* For example, if n = 2, tan*c-(^)=-^^ Va/s 3'.o When n is at all high, the expressions tan kc and 13 /a become very nearly identical for moderate values of kc. When n is odd, we get in a nearly similar manner, 3 n (kcy^^ cot kc + - =p — TTTTo — . IX (T Q g To vu^+ (20). a (n+ 1) (2n-f 1) {1.3.5 ... (2n— l)j* ^ ' [From (19) we see that when n is even tan 7, or — /3/a, is approximately equal to — tan kc, and from (20) when n is odd that cot 7 == tan kc. In the first case, by (16) § 323, a has the sign of t"** or of (— 1)*** ; and in the second case a has the sign of t-*»+i or of (— 1 )*<**-*). In both cases the approximate solution may be expressed 7 = -fe;4.^n7r (20^] The velocity-potential of the disturbance due to a small rigid and fixed sphere is therefore approximately, [t.J + M ^(1 +1/*) cos fc(a<-r) • ^(1 + fM) COB* (o«-r) (21), 334.] INTENSITY OF SECONDARY WAVES. 277 if T denote the volume of the obstacle, the corresponding direct wave being [</»] = cos A; (a« + a:) (22). For a given obstacle and a given distance the ratio of the amplitudes of the scattered and the direct waves is in general pro- portional to the inverse square of the wave-length, and the ratio of intensities is proportional to the inverse fourth power (§ 296). In order to compare the intensities of the primary and scattered sounds, we may suppose the former to originate in a simple source, provided it be suflBciently distant (jB) from jT. Thus, if ^^j ^ cos^fc j^-E) ^ ^23), so that at equal distances from their sources the secondary and the primaiy waves are in the ratio -^,(1+|m) (25). The intensities are therefore in the ratio ;^.(l +§/*)» (26), which, in the case of /it = + 1, gives approximately 6r72r» i?x* ^'^^^• It must be well understood that in order that this result may apply, \ must be great compared with the linear dimension of jT, and R must be great compared with X. To find the leading term in the expression for -^n* when kc is small, we have in the first place, /9 i\ P / ^ ^ ^ sinArc _ ni^(kc)r^ i ^ (n-h2)A;'C ) "1.3.6...(2»-l)r 2. n.(2n-h3) ■*■•••] ^^ X 278 FURTHER APPROXIMATION. [334. Again, a« + /8» = Fn (ike) x Fn (- ike) = {1.3.5...(2«-l)(„ + l)(*c)-}.{l + ^-J?-^^^ + ...) (29); 80 that I ^'^' 1.3...(2n-l)(n+l)| 2.(» + l)(2»-l)^ " (30). Hence, from (10), ^„ = C.(fe )"m-"Pn(M) ^i[»,«-r+«+,.] ^» r{1.3.5...(2n-l)}'(n + l)'' {^ " **"' ((2n + 2)(2n-l) + 2n"2n + 3)) "^ ' ' '} ' * '^^^^ When n is even, [since 7 = — Arc + inir approximately,] r, , c (A;c)»»nt"P„(u) r, / . x . , "" r"*"'^((2n + 2)(2n-l)"^2n(^2n + 3))"^ ) ^^^^' while if n be odd, we have merely to replace i^ by i**+* [and cos by sin], the result being then still real. By means of (31) we may verify the first two terms in the expressions for [-^J, [-^a], in (17), (18). To the case of n = 0, (31) does not apply. Again, by (31), , [■^»\ = ^{l-i^ *'c»l y - M sin {k (at^r + c) + 7,} . . .(33), M = gY50^{/^*-f/^' + A}cos{A:(a^-r + c) + 7,} (34). Combining (17), (18), (33), (34), we have the value of [^] complete as far as the terms which are of the order I(^(f compared with the two leading terms given in (21). In compounding tihe partial expressions, it is as necessary to be exact with reqpec the phases of the components as with respect to their ampl ^ut for purposes requiring only on^ V^axmomc element • 334.] PRESSURES ON OBSTACLE. 279 the phase is often of Bubordioabe importance. In such cases we may take 7 = — Ac + i flTT. From (31) or (32) it appears that the leading term in ^}r„ rises two orders Id kc with each step in the order of the harmonic ; and that y}r„ is itself expressed by a series containing only even, or only odd, powers of kc. But besides being of higher order In kc. the leading term becomes rapidly smaller as n increases, on account of the other factors which it contains. This is evident, because for all values of n and fi, P„(fi)< 1; the same is true of «/(« + !); while t" only affects the phase. In particular cases any one of the harmonic elements of [^] may vanish. From (11), (12), since (o' + jS*)"* cannot vanish, we have in such a case ■ ■ \d. ike/ d.kc kc tlie same equation as that which gives the periods of the vibrations of order n in a closed sphere of radius c. A little consideration will shew that this result might have been expected. The table of § 331 is applicable to this question and shews, among other things, that when kc is small, no harmonic element in [if-] can ^-anish. In consequence of the aerial pressures the sphere is acted on by a force parallel to the axis of /l, whose tendency is to set the sphere into^ vibration. The magnitude of this force, if a be the density of the fluid, is given by ^tIk 27rc'ff I (^ + ^) fidfi. which, by the conjugate property of Legendre'a functions, only le term of the first order affects the result of the integration. Now, when r = c, d . ike kc ^. = 3itce^ f,{ikc) d_ d Fi{ikc)dAkc'd.kc 280 SOURCE AT FINITE DISTANCE. [334. In order that the force may vanish, it would be necessary that d_ sin fee , /, {ike) d^ sin kc _ ^ d.kc' kc Fi (ike) {d . kcY kc "" ' which cannot be satisfied by any real value of kc. We conclude that, if the sphere be free to move, it will always be set into vibration. If instead of being absolutely plane, the primary waves have their origin in a unit source at a great, though finite, distance R from the centre of the sphere, we have P f ^ \ ^ sin kc .^^. ''^''[d:tkc)d.kc'~lc^ ^^ On the sphere itself r = c, so that the value of the total poten tial at any point at the surfeu^e is ^ + ^ = -^^^;^2(2n+l)P„(M) Tp/ d \ sin Ac, /"„ (ike) „ / d \ d sin kc ~\ ^ L \dlkc) ~W "^ >„(tifcc)"U. tifccj dJtc HbT J " This expression may be simplified. We have '^ ^'^ = m^-(-^)'''^Mikc) + e^'^/n{-ikc)}. •^" [d . ike) A- -f- G4i) " - isb l<-i)-'r-i'.(*)-.*''f.<- *c: and thus the quantity within square brackets may be written e^ F,,(ikc)fn{- ike) -Fn{- ike) Mike) like Fniikc) which by (6) § 327 is identical with e"" [Fn (ike)]-'. Thus * + ^ = -^ ^<2„ + l)^^ (37). which is the same as if the source had been on the sphere, an the point at which the potential is required at a great distaiu ^^ 828), and is an example of tbi^ g^nec&L Principle, of Bedprooit ^HM-] symmetrical expression. 281 ^3y assiiming the principle, an'! making use of the result (3) of § 328, we see that if the source of the primary waves be at a finite «.li-stance H, the value of the total potential at any point on the s^^jhere ia If A and B be any two points external to the sphere, a unit source at A will give the same total potential at £, ae a unit source at B would give at A. In either case the total potential is made np of two parts, of which the first is the same aa if there "vere no obstacle to the free propagation of the waves, and the second represeuts the disturbance due to the obstacle. Of these "two parts the first ia obviously the same, whichever of the two points be regarded as source, and therefore the other parts must also be equal, that is the value of %f' at fl when ^ is a source is equal to the value oi ^ aX A when B is an equal source. Now when the source A is at a great distance R. the value of ifr at a point B whose angular distance from A is cos~' p, and linear distance fi^ni the centre is r, is (36) p t d \ d sinA'c " Vd.i'tc/ d.kc' kc ' ind accordingly this is also the value of i|<- at a great distance R, 'heji the source is at B. But since i/r is a disturbance radiating mtwards from the sphere, its value at any finite distance R may e inferred from that at an infinite distance by introducing into F each harmonic term the factor _/„ (ilcR). We thus obtain the following symmetrical expression x/.,.i.,./.(*,P.(,-y,4j^» m. which gives this part of the potential at either point, when the other is a unit source. It should be observed that the general part of the argument does not depend upou tJie obstacle being either s 282 INVESTIGATION FOR THE CASE [334. From the expansion of e^>^ in spherical harmonics, we maj deduce that of the potential of waves issuing from a unit simple source A finitely distant (r) from the origin of co-ordinates. The potential at a point B at an infinite distance R from the origin, and in a direction making an angle cos~^ fi with r, will be the time factor being omitted. Hence by the expansion of 6**^'* from which we pass to the case of a finite R by the simple intro- duction of the factor yn(*^-B)- Thus the potential at a finitely distant point J3 of a unit source at A is 336. Having considered at some length the case of a rigid spherical obstacle, we will now sketch briefly the course of the investigation when the obstacle is gaseous. Although in all natural gases the compressibility is nearly the same, we will suppose for the sake of generality that the matter occupying the sphere diflFers in compressibility, as well as in density, from the medium in which the plane waves advance. Exterior to the sphere, ^ is the same exactly, and '^ is of the same form as before. For the motion inside the sphere, if F = 27r/X' be the internal wave-length, (2) § 330, ^n = ^~ [er^'fn (ik'r) - (- 1)- e+*^Vn (- ik'r)l c^^ ^ 2a/Pn ^^+j j^ g.^ ^^^^ satisfying the condition of continuity through the centre. If (T, a' be the natural densities, m, m' the compressifailiti68» lf»|I(*«a'(a,m(W .335.] OF A GASEOUS OBSTACLE. 283 and the conditions, to be satisfied by each harmonic element separately, are d(f>ldr + dyjr/dr (outside) = d^jr/dr (inside) (2), (r[<f> + '^ (outside)} ^a^jr (inside) (3), expressing respectively the equalities of the normal motions and of the pressures on the two sides of the bounding surface. From these equations the complete solution may be worked out; but we will here confine ourselves to finding the value of the leading terms, when Arc, Icfc are very small. In this case, when r=c, yJTo (inside) = - 2ik^ao' \ d^o/dr (inside)=Jifc'»cao'J ^*^' d<f>oldr = - JA«c I (5), -^0 (outside) = tto/c 1 .g. dyjroldr (outside) = — Oq/c* J Using these in (2), (3), and eliminating o^', retaining only the principal term, we find 3 m ^ In like manner for the term of first order, ^1 (inside) = - f a/A^'c/^ ) dir^dr (inside) =' ^a^T^fi j ^ ^' d4H/dr = ikfi J ^^^' ^1 (outside) = Oi/iArc^ . ft ) d^ijdr (outside) = — 201/1^0* . /Lt J "^ which give --^-^^' (")■ At a distance from the sphere the disturbance due to it is expressed by 284 GASEOUS OBSTACLE. [335. If we introduce the relations T = J7rc», k = 27r/X, and throw away the imaginary part, we obtain as the expression for the most important part of the disturb- ance, corresponding to (21) § 334 for a fixed rigid sphere. It appears, as might have been expected, that the term of zero order is due to the variation of compressibility, and that of order one to the vaiiation of density. From (13) we may fall back on the case of a rigid fixed sphere, by making both a and m! infinite. It is not sufficient to make c by itself infinite, apparently because, if m' at the same time remained finite, k'c would not be small, as the investigation has assumed. When m! —m,<T' —a are small, (13) becomes equivalent to corresponding to ^ = cos kat at the centre of the sphere. This agrees with the result (13) of § 296, in which the obst€u;le may be of any form. In actual gases m' = m, and the term of zero order disappears. If the gas occupying the spherical space be incomparably lighter than the other gas, a' = 0, and irT yfr^a — ficoskiat-r) (14), so that in the term of order one, the eflfect is twice that of a rigid body, and has the reverse sign. The greater part of this chapter is taken from two papers by the author " On the vibrations of a gas contained within a rigid spherical envelope," and an " Investigation of the disturbance pro- duced by a spherical obstacle on the waves of soundV and firom the paper by Professor Stokes already referred to. ' Math. Sode^'t Proce€diii9t,UisefSli\4^\ft'»\ Kor. 14, 187! CHAPTER XVIII. SPHERICAL SHEETS OF AIR, MOTION IN TWO DIMENSIONS. 336. In a furmer chapter (§ 135), we saw that a proof of Fourier's theorem might bi; obtained by considering the mechanics of a vibrating string. A similar treatment of the problem of A spherical sheet of air will lead ua to a proof of Laplace's expansion for a function which is arbitrary at every point of A spherical swrface. As in § 333, if -^ is the velocity-potential, the equation of itinuity, referred to the ordinary polar co-ordinates S, ai. takes form, Isin de V 'ty r' Whatever may be the character of the free motion, it can v^ analysed into a series of simple harmonic vibrations, the nature of which is determined by the corresponding functions 1^, considered as dependent on space. Thus, if ^xe"*"', the I equation to determine ^ as a function of 9 and tu is aBd$\ del fon'ddi -|-i't;''V^ = 0,, „(1). Again, whatever function i^r may be, it can be expanded by Bf's theorem' in a series of sines and cosines of the multiples Thus y='^, + '^jCQsw-<ri^i s. L w + ^/^., cos 2(1) -I- -^i' sin 2<d . . . + ^, cos am + 1//,' sin «w + . , ..(2). ) We bete iuttbduce the ccnditioii Uwt it HyolDtioii tound the 286 GENERAL DIFFERENTIAL EQUATION. [336. where the coeflScients '^©i *^i ••• *^/, "^a' ••• are functions of 6 only; and by the conjugate property of the circular functions, each term of the series must satisfy the equation independently. Accordingly. Lmi^'^-s^^-^^*-" <»' is the equation from which the character of '^g or '^Z is to be determined. This equation may be written in various ways. In terms of fi (= cos 0), |;|(' -"'f }+»■*• - r^.*'= » <*'■ or, if i^ = sin d, where A* is written for l^c^. When the original function -^ is symmetrical with respect to the pole, that is, depends upon latitude only, a vanishes, and the equations simplify. This case we may conveniently take first. In terms of /i, (l-^.)'^t._2/^.^,.^. = (6). The solution of this equation involves two arbitrary constants, multiplying two definite functions of n, and may be obtained in the ordinary way by assuming an ascending series and> de- termining the exponents and coefficients by substitution. Thus .{. A' , A«(A»-2.3) , A'(A'-2.3)(/t'-4.5) , 1.2.3.4.5.6 '^ '^^' ^r1 ^'-12 . . (A'-1.2)(fe'-3.4,) . . 1 ... •^^r-T:2:8^+ 1.2.3.4.5 ^-M ^^ in which A and B are arbitrary constants. Let us now further suppose that '^ besides being ' round the pole is also symmetrical with respect t'' {which is accordingly nodal\ or in other won CONDITION TO BE SATISFIED AT POLES. 287 336.] even function of the sine of the latitude (/t). Under these circum- stances it is clear that B must vanish, and the value of ^ be expressed simply by the first series, multiplied by the arbitrary constant A. This value ol' the velocity- potential is the logical consequence of the original differential equation and of the two reatrictiona as to symmetry. The value of A' might appear to be arbitrarj-, but from what we know of the mechanics of the problem, it is certain beforehand that h' is really limited to a series of particular values. The condition, which yet remains to be introduced and by which h is determined, is that the original equation ia satisfied at the pole itself, or in other words that the pole is not a source; and this requires us to consider the value of the series when ^=1. Since the series is an even function of fi, if the pole ;* = + 1 be not a source, neither will be the pole /i = — 1. It is evident at once that if A' be of the form Ti(n + 1), where n is an even integer, the series termi- nates, aud therefore remains finite when /i = 1 ; but what we now want to prove is that, if the series remain finite for /i=l, A' is necessarily of the above-mentioned form. By the ordinary rule it appears at ouce that, whatever be the value of h', the ratio of successive terms tends to the limit /i', and there- fore the series is convergent for all values of fi less than unity. But for the extreme value /i = l, a. higher method of discrimi- Bujtion i I It ia known' that the infinite hypergeometrical series ■at »(a + l)i(i + l) a(i» + lKa + 2)t(i+ l)(t + 2) ..(8) is convergent, if c + d~a-~b be greater than 1, and divergent if c + d — a — b he equal to, or less than 1. In the latter case the value of c + d — ti — b affords a criterion of the degree of divergency. Of two divergent series of the above form, for which the values oi c + d — a — b are different, that one ia relatively infinite for which the value oi c + d — a — b is the smaller. Our present series (7) nu^. t taking h'=n{n + \), wh< to the standard form "♦ttied to be integral. 288 CRITERION OF DIVERGENCY. [336. ^"1.2'* "^ 1.2.3.4 f^ "•" -1 "(» + l) .. , n(n + l)(n-2Kn + 3) "^ TT"'^ ■*■ 1.2.3.4 '*"••• _ 1 . (-_i!0(ln±i) . . C- in)( -in + l)an + i)(in + i + l ) ~ "^ l.i '^■^" 1.2. i. I '^ + .(9), which is of the standard form, if Accordingly, since c + d — a — 6 = 1, the series is divergent for /Li = 1, unless it terminate; and it terminates only when n is an even integer. We are thus led to the conclusion that when the pole is not a source, and '^o ^ &n ^ven function of fi, A' most be of the form n{n-\- 1), where n is an even integer. In like manner, we may prove that when '^o ^ ^^ odd function of /i, and the poles are not sources, ^ = 0, and A' must be of the form w(n4- 1), n being an odd integer. If n be fractional, both series are divergent for /* = ± 1, and although a combination of them may be found which remains finite at one or other pole, there can be no combination which remains finite at both poles. If therefore it be a condition that no point on the surface of the sphere is a source, we have no alternative but to make n integral, and even then we do not secure finiteness at the poles unless we further suppose ^ = 0, when n is odd, and 5 = 0, when n is even. We conclude that for a complete spherical layer, the only admissible values of -^^ which are functions of latitude only, and proportional to harmonic functions of the time, are included under where Pn(f^) is Legendre's function, and n is any odd or even integer. The possibility of expanding an arbitrary function of latitude in a series of Legendre's functions is a necessary con- sequence of what has now been proved. Any possible motion of the layer of gas is represented by the series Vr = ^. + P,(m) [A^ COS iM^ + B, i86.] TRANSITI(»N TO TWO PIMENSiONS. 289 ffhfn(=0. ^|r = A,.+ A,F,{fi) + ... + A,P,(^)+ (U), 1 the value of yjr when ( = is an arbitrary function of latitude. The method that we have here followed has also the advantage f proving the conjugate propertj', £p„(/.)i',„(/*)rf/i=0 (12). where 'I and m are different integers. For the functions P{fi) are the normal functions (§94) for the vibrating system under consideration, and aucordiitglj the expression for the kinetic energy can only involve the squares of the generalized velocities, If (12) do not hold good, the products also of the velocities must enter. The value of -^ appropriate to a plane layer of vibrating gas can of course be deduced as a particular case of the general solu- tion applicable to a spherical layer. Confining ourselves to the ca.HL' where there is no source at the pole (/i= 1), we have to in- vfstigate the limiting form of ^ = CP,(>i), where ii{n+ I) = i-"c', when c* and «' are infinite. At the same time p— 1 and v arc intinitesimal, and cv passes into the plane polar radius {r). so that nv = kr. For this purpose the moat convenient form of P„(ii) is that of Murphy' : pjcofl e) = 1 1' (" -l)«( » + l>(«+2) . . B 1'.2' The limit is evidently 2 ..(13). + = C 1 A'r' ■^2'. 4' 2^4^6" ^...\ = OMkr) (14), ibewing that the Be.isers function of zero order is an extreme case f Legendrc's functions. I When the spherical layer is not complete, the problem re- 8 a different treatment. Thus, if the gaa be bounded by walls ing along two parallels of latitude, the complete integral Iftio arbitrary constants n-ill in general be necessary. Nat. Phil %19,% [;- .'(fl, n I'Jfl.] Todhuntet'e 290 VIBRATIONS OF A SPHERICAL SHEET [336. The ratio of the constants and the admissible values of h* are to be determined by the two boundary conditions expressing that at the parallels in question the motion is wholly in longitude. The valae of fjL being throughout numerically less than unity, the series are always convergent. If the portion of the surface occupied by gas be that indnded between two parallels of latitude at equal distances fix>m the equator, the question becomes simpler, since then one or other of the constants A and J3 in (7) vanishes in the case of each normal function. 337. When the spherical area contemplated includes a pole, we have, as in the case of the complete sphere, to introduce the condition that the pole is not a source. For this purpose the solu- tion in terms of v, i.e. sin 0, will be more convenient. If we restrict ourselves for the present to the case of symmetiy, we have, putting « = in (5) § 336, y{i''^)^'H'^-^^)^'-^h?v^.-^o (1). One solution of this equation is readily obtained in the ordinary way by assuming an ascending series and substituting in the differential equation to determine the exponents and coeflScients. We get^ to = ilj 0.1~/i> ,. (0.1-A')(2.3 ~A«) , ■^ 2« ■*■ 2».4« ^(0,1-A»)(2.3-A»)(4.5-A») 2».4«.6« ' -1" ••• r •••••• \^/* This value of -j^^ is the most general solution of (1), subject to the condition of finiteness when i/ = 0. The complete solution involving two arbitrary constants provides for a source of arbitrary intensity at the pole, in which case the value of '^o ^ infinite when 1/ = 0. Any solution which remains finite when v^O and involves one arbitrary constant, is therefore the most general possible under the restriction that the pole be not a source. Accordingly it is unnecessary for our purpose to complete the solution. The natara of the second function (involving a logarithm of i^) will K trated in the particular case of a plane layer to I presently. 337.J BOUNDED BY A SMALL CIRCLE. 291 By writing n (« + 1) for k' the series within brackets becomes ,_„(^+l)^^(^ -2)„(„tl)(„ + 3) _^_ (3), or, when reduced to the stiindard hypergeometrical form, i+<->4 1.2.1.2 correspooding to a = ~^ti, 6 = ivi + ^, c=l, (1=1. Since c + d — a — b = ^, the series converges for all values of p from to 1 inclusive. To values of tf(=sin~'i') greater than jtt the solution is inapplicable. When n ia an integer, the series becomes identical with Legendre's function Ph(jj.). If the integer be even, the series terminates, but otherwise remains infinite. Thus, when n = l, the series is identical with the expansion of fi, viz. V(l ~ ^)> in powers of u. The expression for ^ in terms of p may be conveniently applied to the investigation of the free symmetrical vibrations of a spheri- cal layer of air, bounded by a small circle, whose radius ia less than the quadrant. The condition to be satisfied is simply d-ifr/dv=0, an equation by which the possible values of /i", or K^i^, are con- nected with the given boundary value of v. Certain particular cases of this problem may be treated by means of Legendre's functions. Suppose, for example, that n = (i, so that A' = 4*0' = 42. The corresponding solution ia ^jf = AP^^J^). The greatest value of /i for which dy^/d/i = is fi= 8302, con-e- Eiing to ^ = 33°53' = '.59137 radians'. F we take cd = r, so that r is the radius of the small circle ured along the sphere, we get Ar = v'(*2) X -59137 = 3-8325, 1 ia the equation connecting the value of k{= 2Tr/X) with the carved radius r, ia the case of a small circle, whose angular radius in 33^ 53'. U the layer were plane (§ 339), the value of kr would bo 8'8317 : so that it makes no perceptible difference in the pitch [•h* gravest tone whether the radius (r) of given length be 292 UNSYMMETRICAL MOTION. [337. straight, or be curved to an arc of 33°. The result of the com- parison would, however, be materially different, if we were to take the length of the circumference as the same in the two cases, that is, replace ctf = r by ci/ = r. In order to deduce the symmetrical solution for a plane layer, it is only necessary to make c infinite, while cv remains finite. On account of the infinite value of h\ the solution assumes the siin{de form ^ = 4il-*^+^-^^-+ 1 (4) or, if we write cv = r, where r is the polar radius in two dimensions, ^=4|l-^ + |^- ^ = AMkr) (5). as in (14) § 336. The differential equation for ^ in terms of v, when c is infinite and cv = r, becomes %h\t*^*-o («> An independent investigation and solution for the plane problem will be given presently. 338. When 8 is different fix)m zero, the differential equation satisfied by the coefficients of sin«a>, cos«a>, is ^(^-^)-£,'+H^'^'^)^'^^f^'i^.-^^.=o (1), and the solution, subject to the condition of finiteness when v = 0*, is easily found to be 8 {8 + \)-h' {8 + 2)(8 + S)-h* ^ ^ 1. • • • r » 2(2«+2) 4(2« + 4) or, if we put h* = n{n+l), p* + ...> (2). (8-n) (g-yi + 2) (g + w + l) (g + n-f 3) 2.4.(2« + 2)(2«4-4) 1 The solution may be completed by the additioii of a second fonelior from (2) by changing the sign of i , which ocean in (1) only as «*, bni a ai' is neoeesaxy, when i is a positive integer. The method of ptoe' ^xempMed praeently in the oaaa of tha ^^haoa Va^ec* :;38.] USSYMMETRICAL MOTION. 293 We have here the complete solution of the problem of the i-ibrations of a spherical layer of gaa bounded by a araall circle whose radius is less than the quadrant. For each value of s, there are a series of possible values of h, determined by the condition d^Jdv = ; with any of these values of n the function on the right-hand side of (2), when multiplied by cossw or isinsai, is a normal function of the system. The aggregate of all the normul functions corresponding to every admissible value of s and ji. with an arbitrary coefficient prefixed to eaeh, gives an expression capable of being identiBed with the initial value of ^, i.e. with a function given arbitrarily over the area of the small circle. When the radius of the sphere c is infinitely great, h- is infinite. If cv = r, Ji'i^ = kfr^, and (2) becomes ■..(2s + 2)^2.4.(28 + 2) (2a + 4) ■"J ^'''' s fanction of r proportional to J, {hr). In terms of /t, the differential equation satisfied by the co- efficient of cos S(u, or sin sa, is ^|(l-.,'^J.A.+.-^^^+, = o (4). Assuming ^, = (1 — /i-)*'^,, we find as the equation for ip. which will be more easily dealt with. To solve it. let a-M') .^e*'. 2(8 + 1);l^^'+|/i=-s(s + 1)|0. = O....(5), '>, = fi' + (t,^^*' + a ■'+... + 0^/4'+' I substitute in (5), The coefficient of the lowest power ■ I «(a — 1); so that a = 0, or a^l. The relation betwet „ and »m found by equating to a aaUftfrJBlrfic ieat of )i'* _ (a + 2/» + « — "' ^^ 294 CONDITIONS TO BE SATISFIED [338. The complete value of <f>, is accordingly given by ^. = 4 |1 + _ ^ + _- _ 1-27374 "'* (s-n)(8-n+2)(»-n + *)(8 + n+l)(8+n+9){8+n+5) 1.2.3.4.5.6 /*+ • ( g-n + l)(g -nH-3)(^4-n + 2)(g H-n4-4) ) + 2.3.4.5 M +...J...(b), where j1 and £ are arbitrary constants ; and ^.^a-M')***. (7). We have now to prove that the condition that neither pole is a source requires that n — « be a positive integer, in which case one or other of the series in the expression for <f>g terminates. For this purpose it will not be enough to shew that the series (unless terminating) are infinite when f& = + 1 ; it will be necessaiy to prove that they remain divergent after multiplication by (1— ft*)*', or as we may put it more conveniently, that they are infinite when ft = ± 1 in comparison with (1 — ft')"^. It will be sufficient to consider in detail the case of the first series. We have (8'-n){8 + n +1) (8 -n) (8 -n+ 2) (8 +71-^ -1) (8+n+S) ^"*" 1.2 ■*■ 1.2.3.4 ■*■ l.i (^g-^7i)(ig-^n+l)(ig-f in+i)(ig-hin + ^4-l) 1.2.J.f "^ which is of the standard form (8) § 336 • ah o(a4- 1)6(64-1) ■^cd'^c(c + l)d(d + l)'^"" if a«i« — iw, 6 = J« + in+i, c = l, d = i. The degree of divergency is determined by the value of a 4 '^ •'bicb is Jbere equal to a— 1. - i 338.J WHEN THE POLES ARE NOT SOURCES. 295 On the other hand, the biDomial theorem gives for the ex- pansion of (1 — ^')~*' i + iV.+*if'f"..* which is of the standard form, if (i=J», c=I, b = d, and makes a + 6 — e — d = i« — 1. Since s— 1 > Ja — 1 , it appears that the series in the expression for if), are infinities of a higher order than (1 —fi')~^, and there- fore remain infinite after multiplication by (1 —/*')*•. Accordingly ■^, cannot be finite at both pules unless one or other of the series tenoinate, which can only happen when Ji — s is zero, or a positive integer. If the integer be even, we have still to suppose B = 0; and if the integer be odd, ^ = 0, in order to secure finiteness at the poles. In either case the value of <j>t for the complete sphere may be t into the form _,, ^;^.(i-^T-'i^' (81. where the constant multiplier is omitted. The complete expres- sion for that part of ^ which contains cos aa or sin aoi as a factor is therefore ^'Zziy-^fi^-^M (3). J A„ is constant with respect to ft and w, but as a function Vtbe time will varj' as ^^(n.n + l)at^^y ..(10), For most purposes, however, it is more convenient to group b terms for which n is the same, rather than those for which s B same. Thus fur any value of n (A, cosaa-^B, sin sto) . ,.(11), i every coefficient A„ B, may be regarded as containing a e fafltor of the form (10). Joitially ^ is cm arbitrary function of ft and a. and therefore l^ei^S represented in the form 296 FORMULA OF DERIVATION. [338. n=ao «-M fl$p /„\ -^^ = 2 Si/* .-f^iA/" COS 8to -f 5,** sin *ai). . .(1 2), which is Laplace s expansion in spherical surface harmonics. From the differential equation (5), or from its general solution (6), it is easy to prove that 4>9 ^^ of the same form as d4>»-ildii, so that we may Write *' = (0"^ <^3). (in which no connection between the arbitrary constants is as- serted), or in terms of yjt by (7), Vr. = (l-/i')*'(|^)>. (14). Equation (13) is a generalization of the property of Laplace's functions used in (8). The corresponding relations for the plane problem may be deduced, as before, by attaching an infinite value to n, which in (13), (14) is arbitrary, and writing nv = At. Since /i« + 1^= 1, '^0 being regarded as a function of v. In the limit fi (even though subject to differentiation) may be identified with unity, and thus we may take -^-^-^yidirrJ'i'' ^''>- When the pole is not a source, -^^ is proportional to Ja{kr). The constant coefficient, left undetermined by (15), may be readily found by a comparison of the leading terms. It thus appears that J,{kr) = {-ikry{j^^'Mkr) (16). a well-known problem of Bessel's functions \ The vibrations of a plane layer of gas are of couiM easily dealt with, than those of a layer of finite our! I have preferred to exhibit the indirect as well method of investigation, both for the sake of the v^ Un = " (3)'- 338.] VIBRATION IN TWO DIMENSIONS. 297 itself with the corresponding Laplace's expansion', and because the connection between BesBel'H and Laplace's functions appears not to be generally understood. We may now, however, proceed to the independent treatment of the plane problem. 339. If in the general equation of simple aerial vibrations V'yff -^ k'l^ = 0, we assume that ^fr is independent of z, and introduce plane polar coordinates, we get (§ 241) ^.i?.'^.*-^- a,; or, if 1^ be expanded in Fourier's series ■^ = ->/'"+'^i+ ■■■+■+■"+ (2). where ^„ is of the form A„ cos n0 + £„ sin nO, dj^ r dr This equation is of the same form as that with which we had to ileal in treating of circular membranes (§ 200); the principal mathematical difference between the two (juestiona lies in the fact that while in the case of membranes the condition to be satkfled at the boundary is ^ = 0, in the present case interest attaches itself rather to the boundary condition dy^jdr = 0, corre- sponding to the confinement of the gas by a ri^d cylindrical envelope'. The pole not being a source, the solution of (3) is f„ = AJ^(h-) (4), and the equation giving the possible periods of vibration within a cylinder of radius r, is J„\Lr) = Q (5). The lower values of kr satiisfying (5) are given in the following table', which was calculated from Hansen's tables of the functions ' I hsTe befii mnch UHiHted hy Heine't Hitnrlhaeh der Kiirirlfitneiionrn, Berlin, t8(il, and by Sir W. Thoniaoo's papon on Lftplsoa'a Theory at the Tides. Phil. r. Vol. u 187S. ■ I here recur Ui tha p* ^nrsUnd Ihat ii cor- ■enda hi tb* « of ' '■ now iuRitiiu. y [Theajw <ndinM lo en I 298 RIGID CIRCULAR BOUNDARY. [339. J by means of the relations allowing J^ to be expressed in terms of «7o And J I. Namber of in- ternal oirea- lar nodes. n = n = l 1-841 n = 2 »-3 3-832 3-054 4-201 1 7-015 5-332 6-705 8-015 2 10174 8-536 9*965 11-344 3 13-324 11-706 4 16-471 14-864 5 19-616 18016 .■ [For the roots of the equation i/n'('') = 0> ^t^^' McMahon^ finds m + 3 4(7m» + 82m-9) *„">=/9'- S/S* 3(8/3')» 82 (83m» + 2075m»- 3039m -f 3527) .. . • - — — ihWf — ' — ^ ^^' where m = 4n*, and ^8^ = 4^ (2n + 4« + 1). It will be found that n = in (6 a) gives the same result as n=l in (4) § 206, in accordance with the identity J^{z) = — Ji (z).] The particular solution may be written i^n ^{AcosnO-{-B sin nO) Jn (kr) cos kat + {Ccoan0 + Dsmn6)Jnilcr)sixik(U (6), where A, B, C, D are arbitrary for every admissible value of 71 and k. As in the corresponding problems for the sphere and circular membrane, the sum of all the particular solutions must be general enough to represent, when t = 0, arbitrary values of '^ and y^. As an example of compound vibrations we may suppose, as in § 332, that the initial condition of the gas is that defined by '^ = 0, y^ = x^rcG8 0. Under these circumstances (6) reduces to y^=i Axcoa 0Ji{kir) COS kiat-^- A^cos OJiik^r) cos k^at-h ^..{7% and, if we suppose the radius of the cylinder to be unity, the admissible values of k are the roots of J^'ik)^0 t8> • jfiwj \ cISbot compodnd vibrations. 299 The condition to determine the coefficients A is that for all values of r from r = to r = 1, r = A,J,{k,r) + A-J,(Lr)+ (9), whence, as in § 332, ^=<-jrrW) <""■ The complete solution in therefore ,2co.«J,(Jt) '''--(i=--T)V,(J)"™''" <">• where the summation extends to all the values of k determined by (8). If we put ( = and r= 1, we get from (0) and (10) I ..(12). "&=-! an equation which mny be verified numencally, or by an analy- tical process similar to that applied in the case of (14) § 332. We may ])rove that log'^i'(') = constant + S '"g [l - Ti). Emce by differentiation m this (12) is derived by putting a = l, and having regard to the fundamental diSerential equation satisfied by J„ which shews that J,"(l) :y,'(l) = -l. [More generally, if Jn'(^) = 0. I 2e^. = i-1 Hitherto we have Supposed the cylinder complete, so that y recurs after each revolution, which requires that n be integral ; hut if instead of the complete cylinder we take the sector included between 0=0 and = 0, fractional values of n wjll in general pre- sent themselves. Since rf-^/dtf vanishes at both limits of ff, -^ must be uf the form ■^ = A cos {kat + e) cos uB J^(hr) (13), Kere n = wl0, v being integral. If ^ be an aliquot part of or w itself), the complete solution involves only integral values 300 ANALOGOUS PROBLEMS FOR WATER WAVES. [339. of w, as might have been foreseen ; but, in general, functions of fractional order must be introduced. An interesting example occurs when /8 = 2^, which corre- sponds to the case of a cylinder, traversed by a rigid waD stretching from the centre to the circumference (compare § 207) The effect of the wall is to render possible a difference of pressure on its two sides; but when no such difference occurs, the wall may be removed, and the vibrations are included under the theory of a complete cylinder. This state of things occors when V is even. But when v is odd, n is of the form (integer + J), and the pressures on the two sides of the wall are different. Id the latter case Jn is expressible in finite terms. The gravest tone is obtained by taking i/= 1, or n = |, when sm lev y^=^ A cos (kdt-^-e), cos ^0.~7Ty-. (14), and the admissible values of k are the roots of tan it = 2it. The first root (after ^' = 0) is A: = 1'1655, corresponding to a tone decidedly graver than any of which the complete cylinder is capable. The preceding analysis has an interesting application to the mathematically analogous problem of the vibrations of water in a cylindrical vessel of uniform deptL The reader may consult a paper on waves by the author in the Philosophical Magazine for April, 1876, and papers by Prof Guthrie to which reference is there made. The observation of the periodic time is very easy, and in this way may be obtained an experimental solution of problems, whose theoretical treatment is far beyond the power of known methods. 340. Returning to the complete cylinder, let us suppose it closed by rigid transverse walls at z^O^ and z=^l, and remove the restriction that the motion is to be the same in all transverse sections. The general differential equation (§ 241) is dr^ ^r dr ^ t^ dO' ^ dz' +A^-U (l). Let y^ be expanded by Fourier's theorem in the series ^ = jffo + -ff 1 cos ^ + H, cos ~ + . . . + Fp cos fp y J + . • . (2), where the ooeffidenta Hp maj Vie taik&tiQiiia of r and 0. This tm ■0.] VIBRATIONS IN A CLOSED CYLINDEll. nOI lUreB the fulfilment of the boundary conditions, when 2 = 0, « = /, I each term must satisfy the differeiitiul equation separately. ,-/ + - J + -, -jTi, + * — P -JT /Id=0 ...... (A). di" r dr r' dff' \, '^ f 1 '' kich is of the same form as when the motion w independent of I* being repliiced by k^ — p'tT'l"-. Thi.- particular solution may ■efore be written = {A„ COS nd + £„ sin n8).cosp~. J„ (Jh'-p'TrH-' . r) cos kat ^(C„cosn8 + D„siaii0)coap'^.J„[Jlc'-p'-7r'l-\r)smkat...(.i). which must be generalized by a triple summation, with respect to all integral values of p and n, and also with respect to all the values of k, detei-mined by the equation. J'.'Uh'-p'w'l-\r) = (5). If r = 1, and K denote the values of h given in the tabk- (§ 339), corresponding to purely transverse vibrations, we have k' = K^ + p'v'll' (6). The purely axial vibrations correspond to a zero value of A". k^pt included in the table. ^^H 341. The complete integral of the equation ■ ^■^i'2"-(-3*-<> <'). ^"when there is no limitation as to the absence of a source at the pole, involves a second function of )■, which may be denoted by ■f^„{kr). Thus, omitting unnecessary constant multipliers, we may take (§ 200) ,fr„-jr Y 2.2 + 2n 2.4.2 + 2n. 4 + 2)1 ■'■[ |lt the second series require* modification, if ti be integral. When bO, the two series become identical, and thus the iinmedi' D^g 71 = in (2) lacks the nee 302 GENERAL SOLUTION. [341. required solution may, however, be obtained by the ordinaiy rule applicable to such cases. Denoting the coefficients of A and B in (2) by/(n), /(— n), we have ^ = il/(n) + 5/(-7i) = (^+5)/(0) + (^-5)/;(0);i + (4+5)/"(0)j^ + .... by Maclaurin's theorem. Hence, taking new arbitrary constants, we may write as the limiting form of (2), ^. = ^/(0) + 5/(0). In this equation /(O) is Jo (kr) ; to find/'(0) we have ( k^r* k*r^ ■^^(^>'^^'"gi^-2T2T^^r. 4.24-2n.4-f2n '"""; ■*■ dw I 2 . 2 + 2w. 2 . 4 . 2 -f 2n . 4 + 2n ""] ' If u denote the general term (involving ?•**) of the series within brackets, taken without regard to sign, ldu_ dlogu _ 2 ^ 2 udn dn 2 + 2n 4 + 2n "* 2m + 2n* so that (-T- j = - w»«o Sm, if fi^m = T + H + Q + — + - (3). 1 2 d m ^ ' Thus /(0)=iogra--2^+2r4«-2r4r6-«+--j 22 2^4« ''"^2«.4«.6« • r and the complete integral for the case n = is ■^^l"F-F7^^'+2r4r:6'^'~-| <*>• For the general integral value of n the corresponding ex- pression may be derived by means of (15) § 338 d V ^'^-^-^y'kmh* ^^ 341.] EXPRESSION BY DESCENDING SERIES. 303 The formula of derivation (5) may be obtained directly from the differential equation (1). Writing z for h* and putting tn = -^~0«..-^ (6), we find in place of (1) ^rf^-^-y- dT"^*'^''^ ^^^' Again (7) may be put into the form "^y + C+^^ + i*--" W' from which it follows at once that ^n = x^^»-i ••••(9); d.^ so that d \« or by (6) ^;=^n^_^y^^ (11), which is equivalent to (5), since the constants in -^o are arbitrary in both equations. The serial expressions for ^^n thus obtained are convergent for all values of the argument, but are practically useless when the argument is great. In such cases we must have recourse to semi- convergent series corresponding to that of (10) § 200. Equation (1) may be put into the form Wn)_ (n-i)(n + i) ^^^^^^^^^^Q (12). whence by § 323 (4), (12), we find as the general solution of (1) (V - 4n') (3' - 4ra») (5' - 4n') 1 1.2. 3. {aikry ■*"'"} (l'-4n')(3'-4n')(5'-4n') . ) ■^ 1 . 2 . 3 . (saT)" ■^•••j ^*'*'- 304 DIVERGENT WAVE. When n is integral, these series are infinite and ull divergent, but (^ 200, 302) this circumstance does not int with their practical utility. The most important application of the complete integral of i is to represent a disturbance diverging from the pole, a probl which has been treated by Stokes in his memoir on the commi cation of vibrations to a gas. The condition that the disturl represented by (13) shall be exclusively divergent is simj i) == 0, as appears immediately on introduction of the time &cl «*"* by supposing r to be very great; the principal difficulty the question consists in discovering what relation between coefficients of the ascending series corresponds to this conditic for which pui-pose Stokes employs the solution of (1) in the form of a definite integral. We shall attain the same object, perhaps more simply, by using the results of § 302. By (22), (24) § 302 I2W I 1.8i> 1.2.(8iz)»^" .i.(Z(.)4.tVo(.)}./;^-g-^ (14). and thus the question reduces itself to the determination of the form of the right-hand member of (14) when z is small. By (5) § 302 and (5) § 200 we have ^ir {K(z)+tJo(z)] =2r + i*V-|- higher terms in z (l.i), so that all that remains is to find the form of the definite integra in (14), when z is small. Putting 's/(/3* + z^)=y —fi, we have Jo V(p'+-2*) Jm y jz y When z is small, 2^l2y is also small throughout the range o integration, and thus we may write The first integral on the right is j^~^dy^jJ^ = -y-log(iz) + ^z + (16V. * Be Moigyi^B Dyferentlol omd Integral CQUnOiM^ i^ 66S. where 7 is Euler's constant (■5772...); and, as we may easily satisfy ourselves by integration by parts, the other integrals do not contribute anj'thing to the leading terms. Thus, when z is verj' small, -{0 1.8i2 i:2.(8ii) 1.2.3.(Si«)""' = 7 + l»g(W+ !•»■+. .(17). Replacing 2 by Icr. and compai-ing with the form assumed by (4) when r is small, we see that in order to make the series identical we must take il = 7 + log i + log /■ + iiV, so that a series of waves diverging fi'om the p>k', whose expression in descending series is t.= [2ikr) *" 1-. 1= P. 3= (18), . i-^^u 2' k'!* 5.-., ..(19). 1.8»:tr"^1.2.{8iAr)» ie represented also by the ascending series In applying the formula of derivation (11) to the descending aeries, the parts containing e""^ and e*'^ as factors will evidently remain distinct, and the complete integral for the general value of n, subject to the condition that the part containing e*^ shall not appear, will be got by differentiation from the complete integral for 7i = subject to the same condition. Thus, since by (5) ■f , = di^,/d»-, ^.=(0' (, -1.3 -1.1 .3.5 l.i.(8ikry "l.2.3.(8ttr)" 306 SOUNDING BOARDS. [341. or, in terms of the ascending series. %kr\{kr k'l^ . k*f* -7 / , tfcr\{kr k'l^ . (7 + log-2-j|2-2r4 + 2».4 2^. v. 6 ....} 2«.4 2'.4».6 (21)1 These expressions are applied by Prot Stokes to shew how feebly the vibrations of a string, (corresponding to the term of order one), are communicated to the surrounding gas. For this purpose he makes a comparison between the actual sound, and what would have been emitted in the same direction, were the lateral motion of the gas in the neighbourhood of the string prevented. For a piano string corresponding to the middle C, the radius of the wire may be about *02 inch, and X is about 25 inches; and it appears that the sound is nearly 40,000 times weaker than it would have been if the motion of the particles of air had taken place in planes passing through the axis of the string. " This shews the vital importance of sounding-boards in stringed instrumenta Although the amplitude of vibration of the particles of the sound- ing-board is extremely small compared with that of the particles of the string, yet as it presents a broad surface to the air it is able to excite loud sonorous vibrations, whereas were the string supported in an absolutely rigid manner, the vibrations which it could excite directly in the air would be so small as to be almost or altogether inaudible." Fig. 64. " The increase of sound produced by the stoppage of lateral motion may be prettily exhibited by a very simple experimeni Take a taning-fork, and boldiug it in the fingers after it has h^ 341.] SYMMETRICAL DIVERGENT WAVES. 307 liiade to vibrate, place a sheet of paper, or the blade of a broad knife, with it« edge pai-aliel to the axis of the fork, and as near to the fork as conveniently may bfj without touching. If the plane of the obiitacle coincide with either of the planes of symmetry of the fork, as represented in section at A or B, no effect is produced ; but if it be placed in an intei'mediate position, such as C, the soand becomea much stronger*." 342. The real expression for the velocity-potential of syin- niotriciil waves diverging in two dimensions is obtained tram (18) § 3-H after introduction of the time factor e'*^ by rejecting the imaginary part ; it is P. 3' 1.2.(8Ar)'"^" P. 3'. 5= ■ 1:2.3. (KiT)""^ ¥ in which, aa usual, two arbitrary constants may be inserted, one as a multiplier of the whole expression and the other as an addition to the time. The problem of a linear source of uniform intensity may also be treated by the general method applicable in three dimensions. Thus by (3) § 277, if /> be the distance of any element dx from 0, the point at which the potential ia to be estimated, and r be the smallest value of p, so that p* = 7'' + af. we may take H:^^<:i:% <^)' which must be of the same form as ( 1 ). Taking y = p~r, we may write in place of (2) > ■Jy.-J{^r-¥y)" ■■(3), from which the various expressions follow as in (14) § 341. When kr is great, an approximate value of the integral may be obtained b^ neglecting the variation of V(2r + j/), since on account of the Lii^id fluctuation of sign caused by the factor er'^ we need attend ■ Phil. TroM. vol. 16B, p. 117. 1B68. 308 LINEAR SOURCE. [34! only to small values of y. Now /"* cos xdx _ r* sin xdx _ /(''''\ Jo V^ "io v^~v u/ ^*^' 80 that <^ = v/(g:)^(l-i) = >v/(|")^''-*^ (5> Introducing the factor tf^\ and rejecting the imaginary part of the expression, we have finally *V(1') cos A: (ct^ — r — JX) (6X as the value of the velocity-potential at a great distance. A similar argument is applicable to shew that (1) is also the expres- sion for the velocity-potential on one side of an infinite plane (§ 278) due to the uniform normal motion of an infinitesimal strip bounded by parallel lines. In like manner we may regard the term of the first order (20) § 341 as the expression of the velocity-potential due to double sources unifonnly distributed along an infinite straight line. From the point of view of the present section we see the significance of the retardation of ^X, which appears in (1) and in the results of the following section (16), (17). In the ordinary integration for surface distributions by Fresnel's zones (§ 283) the whole effect is the half of that of the first zone, and the phase of the effect of the first zone is midway between the phases due to its extreme parts, i.e. \\ behind the phase due to the central point. In the present case the retardation of the resultant relatively to the central element is less, on account of the pre- ponderance of the central parts. [From the formulae of the present section for the velocity- potential of a linear source we may obtain by integration a corresponding expression for a source which is uniformly distributed over a plane. The waves issuing from this latter are necessarily plane waves, of which the velocity-potential can at once be written down, and the comparison of results leads to the evaluation of certain definite integrals relating to Bessel's and allied functions^] ^ On Point-, Line-, and Plane-Sooroes of Sound. Proe. London MaUL Sot. iL xiz. p. 60it 1888. 343.] CYHNDRIOAL OBSTACLE. 309 343. In illiistratioQ of the fonnulte of § 341 we may take the problem of the distiirbance of plane waves of sound hy a cylindrical obstHcie, whose radius i.s small in comparison with the length of the waves, and whose axis is parallel to their plane. (Compare § 335.) Let the plane waves be representeii by ^ = g**l«+« = g,-b,._e''*-™. ^ly The general expansion of ^ in Fourier's series may be readily effected, the coefficients of the various terms being, as might be anticipated, simply the Bessel's functions of corresponding orders. [Thus, as in (12) § 272 a. e*-™'* = J„ikr) + 2iJ, (h-) cos 5 + ... + 2i"J^(kr) cos n$ + ....] But, as we confine oui"8elves here to the case where c the radius of the cylinder is small, we will at once expand in powers of ?■. Thus, when r = c, if e*" be omitted, ^^ ip = l-^k'(.^ + ike. co»0 + (2), m m '' =^ = -4Jfc + a.co8^+ (3). The amount and even the law of the disturbance depends upon he character of the obstacle. We will begin by supposing the material of the cylinder to be a gas of density <r' and compressi- bility m'; the solution of the problem for a rigid obstacle may finally bo derived by suitable suppositions with respect to a', in'. I Xf Ic' be the internal value of k, we have inside the cylinder by the undition that the axis is not a source (§ 339), B that, when r = c, i^(in8ide) = .-4„(l-Ji-'V) + J,c(l-iiV).costf...(4). ^(inside) = -i^„r^c + ^,{I-|i-V)cos5 (5). Outside the cylinder, when r = c, we have by (19), (21) § 341, + --B.(7 + log J-J+-J— (B). ,dr~ c k,f * *• 310 CYLINDRICAL OBSTACLE. [343. The conditions to be satisfied at the surface of separation are thus -il.ArV=.-Jfc»c' + aB, (8X a J il. (1 - i^•'V) = 1 - iA:'C + B, (7 + log *^) ■(9). ^'(l--8 j-^-Jfc^ <^**)' ^^.c(l-^^'^ = ifcc + | : (11). {ix)m which by eliminating -4o, ili we get approximately 5. = i)fc»c(l-^'.5) = i*»c>^,"* .(12). ,(13). Thus at a distance from the cylinder we have by (18) and (20) § 341. Hence, corresponding to the primary wave ^ = cos — {at + x) the scattered wave is approximately (15). 27r.7rc' fm' — m <r — o" ''' r*X»" t 2m' "^(r'n- cosd 27r ■ cos — (a^ — r — ^X) . . ..(16). The fact that y^ varies inversely as X' might have been anticipated by the method of dimensions, as in the corresponding problem for the sphere (§ 296). As in that case, the symmetrical part of the divergent wave depends upon the variation of com- pressibility, and would diaa^^i^QQis m tlv^ ai^^Ucation to an aofen 343.] PASSAGE OF SOUND THROUGH FABRICS. 311 gas ; and the term of the first order depends upon the variation of tifusity. By supposing a' and m' to become infinite, in such a manner that iheir ratio remains finite, we obtain the solution corresponding lo a rigid and immoveable obstacle, »-*\* f cos B) cos — (a( - -m (17). The exceeding smallness of the obstruction offered by fine wires or fibres to the passage of sound is strikingly illustrated in some of Tyndall's experiments. A piece of stiff felt half an inch in thickness allows much more sound to pass than a wetted pocket-handkerchief, which in consequence of the closing of its pores behaves rathei- as a thin lamina. For the same reason fogs, and even itiin and snuw, interfere but libtle with the free propagation of sounds of moderate wave-length. In the case of a hiss, or other very acute sound, the effect would jwrhaps be apparent. [The partial reflections fi\im sheets of muslin may be utilized to illustrate an important principle If a pure tone of high (inaudible) pitch be reflected from a single sheet so as to impinge upon a SGOsitive flame, the intensity will probably be insufficient to produce a visible i-ffect. If, however, a moderate number of such sheets be placed parallel to one another and at such equal distances apart that the partial reflections agree io pheise, then the flame may be powerfully aflected. The parallelism and equidistance of the sheets may be maintained mechanically by a lazy-tocgs arrangement, which nevertheless allows the common distance tn be vaiied. It U then easy to trace the dependence of the action upon the accommodation of the interval to the wave length of the sound. Thus, if the incidence were perpendicular, the flame would be moat powerfully influenced when the interval between adjacent sheets was equal to the A a// wave length; and although the exigencies of experiment make it neces-sarv- to introduce obliquity, allowance for this is readily made'.] > Iridencent Cr)'>tali. Proc. Boii. liut. April ISS'.i. See alao Phil. Hag. vol. k \ CHAPTER XIX. FRICTION AND HEAT CONDUCTION. 341. The equations of Chapter xi. and the consequences that we have deduced from them are based upon the assumption (§ 236), that the mutual action between any two portions of fluid separated by an imaginary surface is normal to that surface. Actual fluids however do not come up to this ideal; in many phenomena the defect of fluidity, usually called viscosity or fluid friction, plays an important and even a preponderating part. It will therefore be proper to inquire whether the laws of aerial vibrations are sensibly influenced by the viscosity of air, and if so in what manner. In order to understand clearly the nature of viscosity, let us conceive a fluid divided into parallel strata in such a manner that while each stratum moves in its own plane with uniform velocity, a change of velocity occurs in passing from one stratum to another. The simplest supposition which we can make is that the velocities of all the strata are in the same direction, but increase uniformly in magnitude as we pass along a line perpendicular to the planes of stratification. Under these circumstances a tangential force between contiguous strata is called into play, in the direction of the relative motion, and of magnitude proportional to the rate at which the velocity changes, and to a coefficient of viscosity, com- monly denoted by the letter /i. Thus, if the strata be parallel to xy and the direction of their motion be parallel to y, the tangential force, reckoned (like a pressure) per unit of area, is 4 w. The dimensions of fi are [ML''^T~'^]. The examination of the origin of the tangential £ ^o molecular science. It IolOB Vjwtl e^x^\fii\ied b^ 344.] FLUID FRICTION. 313 coniance with the kinetic theory of gases ae resultiog from inter- change of molecules between the strata, giving rise to diffusion of mom en turn. Both by theory and experiment the remarkable conclusion has been established that within wide limits the force is independent of the density of the gas. For air at $" Centigrade Maxwell ' found ;*=00l)1878(l + -00366^) (2). the centimetre, gramme, and second being units, 346. The investigation of the equations of fluid motion in which regard is paid to viscous forces can scarcely be considered to belong tj the subject of this work, but it may be of service to eonie readers to point out its close connection with the more generally known theory of solid elasticity. The potential energy of unit of volume of imifonnly strained isotropic matter may be expresseii ' 1 r-JmS= + J»(^+/' + jr'-2/j-2se-2e/+o' + (.' + ■:•) ■ =J«8' + in(2«'+2/' + 2sr'-}8' + a= + 4' + c>) (1), in which S(=e+/+3)is the dilatation, e,/! <7, «, 6, c are the six cnmponeats of strain, connected with the actual displacements a, ff. 7 by the equations "-S- f'dy »-55 <2'' ' „.f + j?, h.-'/*';. , = f-*f (3), as ay ax dz ay ax bd m, n, K are constants of elasticity, connected by the equation «-"—!» (t). f which M meiiaurL-s the ru/idity, or resistance to shearing, and * measures the resistance to change of volume. The components of stress P, Q, R, S, T, U, corresponding respectively to e,f, g, a, h, c, are found irom V by simple <lifferentiation with respect to those |uantitiefi; thus I' = K?>+-2n{e~^h),kc (5), S= «o, &e (6), ^ Friction of AJt ond other Gases. I'k'd. Tnnii. 314 EQUATIONS OF MOTION. [345. If X, Y, Z be the components of the applied force reckoned per unit of volume, the equations of equilibrium are of the form dP dU dT „ ^ . .^, dx dy dz from which the equations of motion are immediately obtainable by means of D'Alembert s principle. In terms of the displace- ments a, P, y, these equations become K^ + ln^ + nV^a + X^O,&c (8), whei-e S=|? + ^ + *y (9). dx dy dz In the ordinarj' theory of fluid friction no forces of restitution are included, but on the other hand we have to consider viscous forces whose relation to the velocities («, v, w) of the fluid elements is of precisely the same character as that of the forces of restitution to the displacements (a, /9, 7) of an isotropic solid. Thus if S' be the velocity of dilatation, so that ''-S+I-1" <'»)■ the force parallel to x due to viscosity is, as in (8), dS' - dS' . rr^ /it\ *d«.+i"d^+"^" <^*^- So far K and n are arbitrary constants ; but it has been argued with great force by Prof Stokes, that there is no reason why a motion of dilatation uniform in all directions should give rise to viscous force, or cause the pressure to differ from the statical pres- sure corresponding to the actual density. In accordance with this argument we are to put /c =3 ; and, as appears from (6), n coincides with the quantity previously denoted by /jl. The factional terms are therefore f_, , d fdu . dv . dw\ &c. and (§ 237) the equations of motion take the form [Du ^\ dp -- . d fdu ,dv dw\ •■••••• 345.] PLANE WAVES. 315 or, if there be no applied forces aod the square of the inutiou be neglected, du dp _„ , d /du dv dw-. „ ,,„, '•a +&-""'"-*" <5(£. + J,+ <;.]-» ("»• We may observe that the dissipative forces here considered correspond to a dissipation fiinctiou, whose form is the same with respect to «, r, w a» that of V with respect to a, 0, y, in the theory of isotropic solids. Thus putting k = 0. we have from ( I) T, , rr f r n /rfuv a f<i-v i* , /dw ,> , idu dv dwy fdv dw\^ /dw dn\* {du dv\'~\ , , , ,,,, + U + ij,) *U+s) +(rf, + £)J'^''!"'' ('*'■ in agreement with Prof Stokes' calculation'. The theory of friction for the case of a compressible fluid was first given by Poisaon*. . 346. We will now apply the differential etjuations to the in- Bestigation of plane waves of sound. Supposing that v and w are Btro and that u, p. &c. are functions of x only, we obtain from E«)S9i5 ■ '^•dl*d>! S eb? ' '■ ^S1>6 equatiou of continuity (3) § 2S8 is in this case f S-£=° '^'' nod the relation between the variable part of the pi-e&tiuio hp and Hbe condensation s is as usual (§ 244) I Sp=ay,^ (3). Hnius, eliminating Bp and « between (I), (2), [}i), we oblaiu I '^'"_„,'''«_*^ rf'«. = (4) ■ dr- dj^ Zp^d^dt ^ '' ■Mluch is the equation given by Stoke.^i'. m- Let us now inquire how a train of hiirinonic vfn?^ '■'' •-■vve- Hteglh \. which are iiiaiutAined at the ori<ri'* ^H ' Cambridne 7'mHinrtiiii'.. vuL tx. { I ^^^ » Journal lie VKeoU Polj/ifi-lmta^^^^^^^^^^^^^^^^^^^^^ 31 G EFFECTS OF FRICTION. [346. as X increases. Assuming that u varies as e*'**, we find as in § 148, y — Ae~*' cm(nt — ^x) (5X where £^ g'- "'"' 9*8 = - *'*?-/?2?- (6\ Where p tf- ^ , iap- ^ (6> a* 4- — — — ci*+ — „r- — In the application to air at ordinary pressures fj^ may be con- sidered to be a very small quantity and its square may be neglected. Thus /3=^ «=l^; (7). It appears that to this order of approximation the velocity of ' sound is unaffected by fluid friction. If we replace n by 2iraX""\ the expression for the coeflScient of decay becomes ""'SiJ^a (^>' shewing that the influence of viscosity is greatest on the waves of short wave-length. The amplitude is diminished in the ratio e : 1, when x = ar^. In c. a. s. measure we may take Po=OOi:3, M= 00019, a = 33200; whence a?=8800X* (9). Thus the amplitude of waves of one centimetre wave-length is diminished in the ratio e : I after travelling a distance of 88 metres. A wave-length of 10 centimetres would correspond nearly to g^^ ; for this case x = 8800 metres. It appears therefore that at atmospheric pressures the influence of friction is not likely to be sensible to ordinary observation, except near the upper limit of the musical scale. The mellowing of sounds by distance, as observed in mountainous countries, is perhaps to be attributed to friction, by the operation of which the higher and harsher components are gradually eliminated. It must often have been noticed that the sound s is scarcely, if at all, returned by echos, and I have found ^ that at a distance of 200 metres a powerful hiss loses its charaotor, even when there is no reflection. Probably this effect alao is dr to viscosity. 1 AooQstioal ObBerraiionB, PkiL Mag, voL m. p. 456, 1B77* 346.] TEANSVEHSE VIBRATIONS, 317 In highly rarefied air the value of « as given in (8) is much ID creased, /i bc-ing constaDt. Sounds even of grave pitch may then be affected within moderate distancea From the observations of Colladon in the lake of Geneva it would appear that in water grave t^ounds are more rapidly damped tlian acute sounds. At a moderate distance from a bell, struck under water, he found the sound short and sharp, without musical character. 347. The effect of viscosity in modifying the motion of air in contact with vibrating solids will be best understood from the solu- tion of the problem for a very simple case given by Stokes. Let us suppose that an infinite plane (j/z) executes harmonic vibrations in a direction (y) parallel to itself. The motion being in parallel t^li-ata, u and w vanish, and the variable quantities are func- tions of a; only. The first of equations (13) § 34-5 shews that the pressure is constant ; the corresponding equation in v takes the ..(1). dt p daf lilar to the equation for the linear conduction of heat, If we f suppose that v is proportional to e'", the resulting equation = .de-'« + iie+"^ (3). V©<'-^' «• ..(2), ■■(5), {the gas be on the jmsitive side of the vibrating plane the inotior I to vanish when 3; = + x. Hence i{ = 0, and the value of i J on rejection of the imaginary part )onding to the motion F=vlcos)(( (6) The veli)city of the fluid in contact with the plane is of the plane itself on tl^^^^l or 318 PROPAGATION OF SOUND [347. apparently sufficient ground that the contrary would imply an infinitely greater smoothness of the fluid with respect to the solid than with respect to itself. On this supposition (5) expresses the motion of the fluid on the positive side due to a motion of the plane given by (6). The tangential force per unit area acting on the plane is if il = 1. The first term represents a dissipative force tending to stop the motion ; the second represents a force equivalent to an increase in the inertia of the vibrating body. The magnitude of both forces depends upon the frequency of the vibration. We will apply this result to calculate approximately the velocity of sound in tubes so narrow that the viscosity of air exercises a sensible influence. As in § 265, let X denote the total traiisfer of fluid across the section of the tube at the point x. The force, due to hydrostatic pressure, acting on the slice between x and ^ 4- So; is, as usual, -S|8. = ava.g (8). The force due to viscosity may be inferred from the investigation for a vibrating plane, provided that the thickness of the layer of air adhering to the walls of the tube be small in comparison with the diameter. Thus, if P be the perimeter of the tube, and V be the velocity of the current at a distance from the walls of the tube, the tangential force on the slice, whose volume is S&x, is by (7) ffJT or on replacing Vhy-^-i-S -PW(i«PM)(f + ^^)-^ (9> The equation of motion /or this period is therefore i ^^^ ui .PSa}fdXl<PX\ . . *X or y^'s MhWiTJ^'-^^--^^"^ 347.] IN NARROW TUBES. 319 I The velocity of sound is approximately «{'-4s\/(a;)l <">• or in the case of a circular tube of radius r, 4)}- <'^>- The result expressed in (12) was first obtained by Helmholtz. «{-yt 348 ^ In the investigation of EarchhoflP, to which we now proceed, account is taken not only of viscosity but of the equally important effects arising from the generation of heat and its communication by conduction to and from the solid walls of a narrow tube. The square of the motion being neglected, the "equation of continuity " (3) § 237 is d9 dM dv ^_/x /,v. dt dx dy dz ^ so that the dynamical equations (13) § 345 may be written in the fonn dt^^ldp^My,^^ M.J?!«_ /2) dt podx po Spodxd>t The thermal questions involved have already been considered in § 247. By equation (4) S"^i^'"« w. where i/ is a constant representing the thermometric conductivity. By (3) § 247 p//^o = 6'(l+« + a5) (4), in which 6 denotes Newton's value of the velocity of sound, viz. V(Po/po)- If we denote Laplace's value for the velocity by a, aV6« = 7 = l + ai8 (5), so that i8 = (a» - 6«)/6«a (6). ^ This and the following §§ appear for the first time in the second edition. The first edition closed with § 848, there devoted to the question of dynamical similarity. ^ Pogg. Aim, toI. cxzxxt., p. 177, 1968. 320 KIRCHHOFF^S INVESTIGATION. [348. It will simplify the equations if we introduce a new symbol 9 in place of 5, connected with it by the relation ff = djfi. Thus (3) becomes f-i'-'-' ('>■ and the typical equation (2) may be written d«+^d^+(«'-*'>d^='*^'"-'*^^ (»)' where fi is equal to fi/po^ !»!' represents a second constant, whose value according to Stokes' theory is \ii!. This relation is in accordance with Maxwell's kinetic theory, which on the intro- duction of more special suppositions further gives " = 1/ (9). In any case yl, yi', v may be regarded as being of the same order of magnitude. We will now, following Kirchhoflf closely, introduce the suppo- sition that the variables u, v, w, 8, ff are functions of the time on account only of the factor e*', where A is a constant to be after- wards taken as imaginary. Differentiations with respect to ^ are then represented by the insertion of the factor A, and the equations become du/dx + dvldy + dw/dz + h8==0 (10), hv-yV^v==-dP/dy \ (11), hw-fjiV^w^^-dP/dz J P = (6«4.A/')« + (a«-6*)^ (12), 8^ff^{vlh)V*ff (13). By (13), if « be eliminated, (12) and (10) become P = (a'-hAAA'0^-^(6' + V)V^^ (14). du . dv dw ,^, -_,^ ^ ., ,. ;7:; + ;77. + ;77 + *^"''^^ = ° (i^)- ax ay az By differentiation of equations (11) with respect to a?, y, ^, with subsequent addition and use of (14), (15), we find as the equation in ff EFFECTS OF HE:AT CONDDCTION. 348.] A solution of (16) may be obtained in the fomi e' = A,Q, + A,Q^ (17). where Q,, Q, are functions satisfying respectively ^'Q,=\Q.. V-Q, = X,Q, (18). Xi, X- being the roots of h^- {a^ + h (^' + fi" + v)\\+ ~\b-- + 1> (^^' + ^^'■)]\'=0...(l9). while A,, Aj denotf arbitrary constants. Id correspondence with this value of 0', particular solutions of equations (11) are obtained by equating w, v, w to the differential coefficients of BiQ. + B,Q„ taken with respect to x, y, s. The relation of the constants 5, , S, to j4,. At appears at once from (15), which gives v^ {B,q, + B,q^) + (h - ^V') {A,q, + ^,g,) = o. »o that by (18) "■'-'.('-x;)- *=^-("-i) (»'• More general solutions may be obtained by addition to k, v, w respectively of w', «', «^, where i/, o', w' satisfy M ;* Thm «.u-+i(,ciQ,/(fa + B,(i8,/ii« »-.■ +B,dQ.ldy + B,dQ,ld!/ u. = lo' + B,dQJdi + B,dQJdt where iJ,, it, have the values above given. By substitution in (15) of the values of it, w, it npjiears that ~,»'...(21). ■ (22), |& do' dy d,'"- J specified in (22) (23), 349. These results are first applied by Kirchhotf to the case of plane waves, supposed to be propagated in infinite space in the direction of +x. Thus v" and v/ vanbh, while u', Q,, Q, are independent of y and t. It follows from (23) § 348 that u' also The equations for Qi and Q, are 322 PLANE WAVES. [349. so that we may take Q,^e'^'f\ Q^^e-'"^ (2), where the signs of the square roots are to be so chosen that the real parts are positive. Accordingly u = ilAi* (^ - 1') «-'^"' + ^V (^- I') «-*^^ (3). ff = ^,tf-*^^. + ^tf-«^*« (4), in which the constants Ai, A^ may be regarded as determined bf the values of u and ff when x^O. The solution, as expressed by (3), (4), is too general for our pi*eseDt purpose, providing as it does for arbitrary communicaticMi of heat at a? = 0. From the quadratic in X, (19) § 348, we see that if /Lt', fi\ V be regarded as small quantities, one of the values of X, say Xi, is approximately equal to A'/a*, while the other X^ is very great. The solution which we require is that corresponding to Xi simply. The second approximation to Xi is by (19) § 348 h^ y^Xx* _ A* f A(/i^-f Ai^^+y) ) vh'h* ^"a« + A(/ + /' + i')"*" Aa« ~a«j «' I «' ' so that ±'^^\-^l,[l^' -^ 1^" -^^O^-^l^')] (5). If we now write in for A, we see that the typical solution is It = e-^'* e»» <'-«/«» (6), where ^' = ^.|^' + /' + .(l-D| (7). In (6) an arbitrary multiplier and an arbitrary addition to t may, as usual, be introduced ; and, if desired, the solution may be realized by omitting the imaginary part. These results are in harmony with those already obtained for particular cases. Thus, if i/ = 0, (7) gives n« in agreement with (7) § 346, where On the other hand if viscosity be left out of account, so tittil ^' s fif' = 0, we fall back upon (18) § 247. It is unnecessaiy to stf anything to the discussions already given. 349.] SYMMETRY ROUND AN AXIS. 323 In tbe case of spherical waves, propagated in the direction of + r, Kirchhoff iiods in like manner as the expression for the radial velocity ^.~-''"-"" w. vfhere m' has the same value (7) as before. 3B0. We will now pass on to the more important problem and titippose that the air is contained in a cylindrical tube of circular section, and that the motion is symmetrical with respect to the axis of X. If y' + J* = )■•, and v = q.ylr, w=.g.zlr, l/ = q'.l//r, w'^q.zlr, then u, u', q, q', Qi, Q^ are to be regarded as functions of tt and r. We suppose further that as functions of x these quantities are proportional to e"**, where w is a complex coustaat to be deter- mined. The equations (18) § 348 for Q,, Qt become dr* r dr \ -:~(\-m')Q, (1), *^if=(^.-™•>ft <^). For u', q' equations (21), (23) give d?+r3F -U--'"7« ''>• H^Vi-i-f^--^^ <«■ •"■-lA"' ; <^)- These three equations are satisBed if u! be determined by means of the first, and q' is chosen so that , m dti »^— *F^^-* *''■ a relation obtained by subtracting from (4) the result of differen- tiating (5) with respect to r. The solution of (3) may be written 'i = AQ, in which .d is a constant, and Q a function of r satisfying 21—2 1 324 CIRCULAR TUBE. [350. 1 Thus, by (20), (22) § 348, ti = ^Q-^,m(^-r)Q,-^m(^-i.)g, (S\ ff^A,Q,^A,Q, ...(10). On the walls of the tube u, q, ff must satisfy certain conditions. It will here be supposed that there is neither motion of the gas nor change of temperature ; so that when r has a valae equal to the radius of the tube, u, q, ff vanish. The condition of which we are in search is thus expressed by the evanescence of the determi- nant of (8), (9), (10), viz. : _m«A /I iNdlogQ fh_ \ d log Q i -(^')'-^-<>- <"> The three functions Q, Qi, Q,, which are required to remain finite when r = 0, are Bessel's functions of order zero (§ 200), so that we may write in the usual notation Q=/o{rV(m«-A/M')!) Q, = ^.{rV(m'-XO} (12). Q, = /o{rV(wi«-X,)} ) In equation (11) the values of \i, Xg are independent of r, being determined by (19) § 348. In the application to air under normal conditions /t', /a", p may be regarded as small, and we have approximately X,^hya\ \^-^ha*lub\ (13). A second approximation to the value of Xi has already been given in (5) § 349. It. is here assumed that the velocity of propagation of viscous effects of the pitch in question, viz. \/(2/t'n), § 347, is small compared with that of sound, so that infi'/a\ or hfjufa\ is a small quantity. In interpreting the solution there are two extreme cases worthy of special notice. The first of these, which is that considered by Kirchhoff, arises when fi\ fi', v are treated as very small, so small that the layer of gas immediately affected by the walls of the tube is but an insignificant fraction of the wbok contents. When /a' &a vanish, we have ;150.] VISCOSITY SMALL. 325 so that r^{m^ — \,) is h<?re to be regarded tu* small. On the other baud rV('«' — A//*')' '''^(^^~\) are large. The value of J,{t), when z is small, ia given by the ascending aeriea (5) § 200 ; from which it follows at once that d\i}gJ,{s)lcU = -^z. When s is very large and snch that its imaginai'y part is positive, (10) § 200 gives d log Ja (z)i'dz = — tan (z - Jjt) = - i. Thus, if we retain only the terms of highest order, d\<igQ/dr=^{h/^') \ dlogQ,/rfr=ir(X,-m') (14). d\ogQ,ldr = ^(ha'lpb') ] Using these in (11) with the approximate values of X,, \ from (13), we find "-::('-^) <-). where y' = ^(1.' + (a/b-b/a)-/v (16), i the sign of VA ia to be 80 chosen that the real part is positive. We now write l> = ni (17). MO that the frequency is nj2w, Thua VA = Van).(l + (18); ind /«= ±(m' + im") (19). |where by (1.5) "'~^2.ar- "* -a^s/2.ar ''^"^■ [f we restore the hitherto suppressed factors dependent upon x and t, we have where B is an arbitrary constant, and R, R', R" are certain functions of r, which vauish when r is equated to the radius of K, and which for points lying at a finite distance from the ^anme the values 326 VELOCITY OF SOUND. [350. The realized solution for w, applicable at points which lie at a finite distance from the walls, may be written u = Cj^'* sin (n< + m''x + S,) + C,e-»'* sin {nt - m'^w + 8^). . .(21X where Ci, C^, ii, if denote four real arbitrary constants. Ac- cordingly mf determines the attenuation which the waves suffer in their progress, and m' determines the velocity of propagation. This velocity is in harmony with (12) § 347. The diminution of the velocity of sound in narrow tubes, as indicated by the wave-length of stationary vibrations, was observed by Kundt (§ 260), and has been specially investigated by Schnee- beli^ and A. Seebeck*. From their experiments it appears that the diminution of velocity varies as r~S in accordance with (22), but that, when n varies, it is proportional rather to nr^ than to n~^. Since /t is independent of the density (p), the effect would be increased in rarefied air. We will now turn to the consideration of another extreme case of equation (11). This arises when the tube is such that the layer immediately affected by the friction, instead of merely forming a thin coating to the walls, extends itself over the whole section, as must inevitably happen if the diameter be sufficiently reduced. Under these circumstances hr'/fi is a small, and not, as in the case treated by Earchhoff, a large quantity, and the argu- ments of ail the three functions in (12) are to be regarded as small. One result of the investigation may be foreseen. When the diameter of the tube is very much reduced, the conduction of heat from the centre to the circumference of the column of air becomes more and more free. In the limit the temperature of the solid walls controls that of the included gas, and the expansions and rarefactions take place isothermally. Under these circumstances there is no dissipation due to conduction, and everything is the same as if no heat were developed at all. Consequently the coefficient of heat-conduction will not appear in the result, whkdi ^ Pogg, Atm, rol. oxxzyi. p. 296, 1S69. * Pogg, Ann, vol. oszzn. p. 104, 1S70. EXCEKDINGLY NARROW TUBES. 350.] will involve, moreover, the Newtonian sound, and not that of Laplace (n). aliie (lij of the vtilooity ol* Wheni JoU)=l- ' 2'.4'' P SO that approximately dlogJ,(^)/d^ = -^z{l + ^i') (23). When the results of the application of (23) to Q, Q„ Qt are introduced into (II), the equation may be divided by Jr, and the left-hand member will then conniat of two parts, of which the first is independent of r and the second is proportional to r". The first part reduces itself without further approximations to v(X, — X,). For the second part the leading terms only need be retained. Thus with use of (13) v(\,-K) a'l^ im' fH't^-]^] -'-u-'}'0, %e ratio of the second term to the first is of the order In-'jv, by mppositioD a small quantity, so that we are to take simply ■ b'l^ ' _ Sin'n .■(•2i). ¥ the solution applicable under the supposed conditions. Before leaving this question it may be worth while to consider briefly the corresponding problem in two dimensions, although it is of less importance than that of the circular tube treated by Kirchhoff, The analysis is a little simpler; but, as it follows practically the same course, we may conteut ourselves with a mere indication of the necessary changes. The motion is supposed to be independent of s and to take place between parallel walls at ±y,- The equations (1) to (11) of the preceding investigation may be regarded as still applicable in the present problem, if we write tp for q and y for r, with omission of the terms where r occurs in the denominator. The general solution of the equations corre- inding to (1), (2), (7) contains two functior" -it sines and cosines of multiples of v 328 TWO DIMENSIONS. [351. absence of the sine functioD, so that in (12) we are simply to replace the function J© by the cosine. In the case where yd &c. are regarded as infinitely small we have as in (14), when y^jfu dXogqidy^^ihlii,') ) d log Q,ldy ^ ^Hha^lvb') ] ^''•'^' but in place of the second of equations (14) d log QJdy^y,{\^m') (261 When these values are substituted in (11), the resulting equation is unchanged, except that r is replaced by 2yi. The same substi- tution is to be made in (15), (20), (22). The latter gives for the velocity of sound '{'-iml-- <">• It is worth notice that (27) is what (11) § 347 becomes for this case when we replace Vm' by 7' ; and we may perhaps infer that the same change is sufficient to render that equation ap- plicable to a section of any form when thermal effects are to be taken into account. In the second extreme case where the distance between the walls (2y,) is so small that hyi^/v is to be neglected, we have in place of (23) . d logcos-?/d-? = -^(l + i^) (28). The equations following are thus adapted to our present purpose if we replace Jr" by Jy,'. The analogue of (24) is ac- cordingly -^=S^=^^r ...(29). 351. The results of § 350 have an important bearing upon the explanation of the behaviour of porous bodies in relation to sound. Tyndall has shewn that in many cases sound penetrates such bodies more freely than would have been expected, although it is reflected from thin layers of continuous solid matter. On the other hand a hay-stack seems to form a very perfect obstaicda It is probable that porous walls give a diminished refleotioa^ m that within a building so bounded resonance is less prediOl 'fan if the walU were formed ot coxi\>m\usvsA TiA.^t^T« . . POnoUS WALL. When we inquire into the mechanical tjuestioa, it is evident that; aoiind is not destroyed by obstacles as such. In the absence of dissipative forccB. what is not transmitted must be reflected. Destruction depends upon viscosity and upon conduction of heat; but the influence of these agencies is enormously augmented by the contact of solid matter exposing a large surface. At such a surface the tangential as well as the normal motion Ls hindered, and a passage of heat to and fro takes place, as the neighbouring air ia heated and cooled during its condensations and rarefactions. With such rapidity of alternation as we are concerned with in the case of audible sounds, these influences extend to only a very thin ■ layer of the air and of the solid, and are thus greatly favoured by ^b fine state of division. ■ Let us conceive an otherwise continuous wall, presenting a fiat iace, to be perforated by a great number of similar narrow channels, uniformly distribnted, and bounded by surfiices every- where perpendicular to the face of the wall. If the channels be sufficiently numerous, the transition, when sound impinges, from simple plaue waves on the outside to the state on the inside of aerial vibration corresponding to the interior of a channel of unlimited length, occupies a apace which is small relatively to the wave-length of the vibration, and then the connection between the condition of things inside and outside admits of .simple ex- pression. Considering first the interior of one of the channels, and taking the axis of :c parallel to the axis of the channel, we suppose that as functions of at the velocity components ii, v, w and the condensation « are proportional to e'*', while as functions of t everything is proportional to e'"^, n being real The relationship between k and « depends upon the nature of the gas and upon the size and form of the channel, and has been determined for certain important cases in § .350, ik being there denoted by m. Supposing it to be known, we will go on to shew how the problem of reflection is to be dealt with. For this purpose consider the equation of contiiiinty ii- integrated over the cross-section er of the channel Siurr ih>' walls of the channel are imi>euetrable. 330 REFLECTION OF SOUND [351. This equation is applicable at points distant from the open end more than several diameters of the channel. Taking now the origin of x at the face of the wall, we have to form corresponding expressions for the waves outside ; and we may there neglect the effects of viscosity of conduction of heat. If a be the velocity of sound in the open, and k^ « n/a, we may write for waves incident and reflected perpendicularly s^^e^ + Be-^^)^ (2), u^a('-e^ + Be-^^)e^ (3); so that the incident wave is g^gimt+K^ ^4) or, on throwing away the imaginary part, 8 ^^ cos (nt + kgx) (5). These expressions are applicable when x exceeds a moderate multiple of the distance between the channela Close up to the face the motion will be more complicated ; but we have no need to investigate it in detail. The ratio of u and « at a place near the wall is given with sufficient accuracy by putting a? = in (2) and (3), ^ = 5^+1 <^>' We now assume that a space, defined by parallel planes one on either side of a? = 0, may be taken so thin relatively to the wave-length that the mean pressures are sensibly the same at the two boundaries, and that the flow into the space at one boundary is sensibly equal to the flow out of the space at the other boundary, and yet broad enough relatively to the transverse dimensions of the channels to allow the application of (6) at one bounding plane and of (1) at the other bounding plane. The equality of flow does not imply an equality of mean velocities, since the areas concerned are different. The mean velocities will be inversely proportional to the corresponding areas — that is in the ratio a-ia-^a, if <r' denote the area of the unperforated part of the wall corresponding to each channel. By (1) and (6) the connection between the iiudde and outside motion is expressed by n a(jB-l), ^ . 351. J AT THE FACE OF A I'OROU« WALL. 331 We will denote the ratio of the imperforated to the perforated parts of the wall by g, so that g = a'ja. Thus I- ■«=-*•- (7) If g = fi. k = k^, that is, if the wall be abolished, or if it be reduced to iufinitely thin partitioua between the channels while at the same time the dissipative effects are neglected, there is no reflection, If there are no perfomtions ($ = X'), then .8 = 1, Cing total Inflection. G(?nenilly in place of (7) we may write is the solution of the problem proposed. It is understood .„».v .vaves which have once entered the wall do not return. When dissipative forces act, this condition may always be satistied by supposing the channels to be long enough. The necessary length of channel, or thickness of wall, will depend upon the properties of the gas and upon the size and shape of the channels. Even in the absence of dissipative forces there must be reflection, t except in the extreme case ,"7 = 0. Putting k = k\ in (8), we Jexce ^biav€ "'ih <">■ If 5=1, that is if half the wall be cut away, B = \, B' = ^, so that the reflection is but small. If the channels be circular and arranged in square order as close as possible to one another, g = (4; — tr)l'!r, whence B = -121, B'=015, nearly all the motion , being transmitted. If the channels be circular in section and so small that nr'lv wy be neglected, we have, (24) § 3.50, -^-""'^ ('»>^ Ithat (21) the wave propagated into a channel is proportional to e"''6in(«t + ni"d;+S,) (U). ,«'=-■ ' M/*V" 02), 332 NUMERICAL EXAMPLES. [351. To take a numerical example, suppose that the pitch is 256, so that n = 27r X 256. The value of it! for air is "16 C.G.S., and that of V is '256. If we take r = j^ cm., we find nr^/Sp equal to about Y^' If r were ten times as great, the approximation in (10) would perhaps still be sufficient. From(12), ifn = 27rx256, m' = m" = -00115/r (13); so that, if r = Y^^ cm., m' = l*15. In this case the amplitude is reduced in the ratio « : 1 in passing over the distance l/mf, that is about one centimetre. The distance penetrated is proportional to the radius of the channel. The amplitude of the reflected wave is by (8) ^~m'(l+^)(l-i) + io' or, as we may write it, ^ = Jf + l-iJf ^^*>' where if =(1 +g)m'/ko (15). If / be the intensity of the reflected sound, that of the incident sound being unity, 2M«-23f+l 2if» + 2if+l ^*'*'- The intensity of the intromitted sound is given by i — T— ^^ nT\ 2if' + 2JJf+l ^ ^• By (12). (15) i./ = 2iL±4vV7) (,8). r »Jn ^ If we suppose r = j^ cm., and ^r = 1, we shall have a wall of pretty close texture. In this case by (18), ilf=47-4 and 1— / = *0412. A loss of 4 per cent, may not appear to be im- portant; but we must remember that in prolonged resonance we are concerned with the accumulated effect of a large number of reflections, so that a comparatively small loss in a single re- flection may well be material. The thickness of the porous layer necessary to produce this effect is less than one centimetre. Again, suppose r=y^cm.,^=l. We find ilf= 4-74, 1— /«- d the necessary thickness would be less than 10 ceati' I 352.] RESONANCE OF BUIJ.DINGS. 333 If r be much greater thau j^ cm., the exchange of heat between the air and the sides of the channel is no longer sndi- ciently free to allow of the use of {24i) § 350. When the diameter is so great that the thermal and viscous ejects extend through only a small fraction of it, we have the case discussed by KirchholT .) § 350. Hero ^^'lU+rS}^] ,19). ich value is to be substituted in (8), If for simplicity we put 0, we find ^7 (1 -t) / = 7'7't'^" (21). The supposition that j = is, however, inconsistent with the circular section ; and it is therefore preferable to use the solution corresponding to (27) § 350, applicable when the channels assunie the form of narrow crevasses'. We have merely to replace r in (19). (20), (21) by %i. 2j, being the width of a crevasse. The incident sound is absorbed more and more completely as the width of the channels increases ; but at the same time a greater length of channel, or thickness of wall, becomes necessary in order to prevent a return from the further side. If g = 0, there is uo theoretical limit to the absorption; and, as wo have seen, a moderate value of g does not of itself entail more than a com- paratively small reflection, A loosely compacted hay-stack would seem to be as effective an absorbent of sound as anything likely to be met with. In large spaces bounded by Don-porous walls, roof, and floor, and with few windows, a prolonged resonance seems inevitable. The mitigating influence of thick carpets in such cases is well known. The applicAtiou of similar material to the walls and to the roof appears to ofJer the best chance of further improve- ment. 3C2. One of the most curious consequences of viscosity is the generation in certain cases of regular vortices. Of this an example, Bcovered by DvoMk, has already been mentioned in § 260. Id ■ It mo? be reouu'ked (hat eveu io the Iwo-diineaaionBl problMt a f=(t ioTolvM %a mOnits upaculgf tiff, ft 334 TWO-DIMENSIONAL EQUATIONS [352. a theoretically inviscid fluid no such effect could occur, § 240 ; and, even when viscosity enters, the phenomenon is one of the second order, dependent, that is, upon the square of the motion. Three problems of this kind have been treated by the author' on a former occasion, but here we must limit ourselves to DvoHk's phenomenon, further simplifying the question by taking the case of two dimensions and by neglecting the terms dependent npoo the development and conduction of heat. If we suppose that /> = a'/>, and write 8 for \og(p/f^X the fundamental equations (12) § 345 are ^ds du du da , ,-, , ,, d (du , dv\ ,- . with a corresponding equation for v, and the equation of continuity §238 du dv ds ds ds ^ ,^ dx dy dt dx dy ^' Whatever may be the actual values of u and v, we may write — ^4.^ _d0_d^ dx dy' dy dx ^ '' in which — . du dv *- . du dv ^**=di+^' ^'^=^-di (*)• From (1), (2) / . „d\d8 du ^ ,_, du du „ d f d8 ^ d8\ / , „d\d8 dv ^ ,— . dv ^ dv ^ „ d ( ds ds\ dx dy '^ dy\ dx dy) ^ ^' Again, from (5), (6), / . / d . // d \ -, d?8 d [ ds ^ ds\ / / if\ri*f <^ . <^\ d ( du ^ du\ d ( dv ^ dv\ (7). I On the Circalation of Air oboerved in Kandt's Tabes, and on aoiiM idBM 35'1.] TO A SECOND APPROXIMATrON. 335 For the first approximatioo the terms of the second order in 11, V, 8 are to be omitted. If we aseurao that as functions of ( all the periiMiic quantities are proportional to e"", and write q for a' + tH^' + tn;t", (7) becomes 9V*s + «'«=0 (8). Now by (2), (4.) V"^ = ~ins=i (g/n) V's. s.. that (f> = iqs/n... (9)', and M=-^ T- + 7- . v = ^ j- - -/- (10). n ax ay n ay da ^ SubBtituting in (5), (6), with omission of the terms of the second order, we get in view of (8), ► 'hence (/V'-tH)-f = (11). If we eliminate s directly from etiuationa (1), we get "^ T-,\ . d / dii du\ d f dv dv\ ! .-e,. fi t;\ . d f dii dii\ d I dv dv\ I If we now assume that as functions of x the quantities a, ^, &c. fe proportional to «***, equations (8), (11) may be written (dVrfy'-r')« = (13). here t-"' = 1^— n'/q, (d?id>f-k'^)^ = Q (U). here i'' = k? + ijt/p,'. If the origin for y be in the middle between the two parallel bounding planes, j must be an even function of y, and -^ must be . an odd function. Thus we may write 8 = A cosh k"y . e^"* . e***, \^ = B sinh k'y . e'"' . e***. ..(15), M = { — kqfn . A cosh f'y + k'B cosh k'y) e'"* .e'*' ) .. „. V = {iqk"ln . A sinh k^'y - ikB sinh k'y) «*"' . e*** | > It la niiiieoe»HaT7 to iidd a gompleincntai? raoction ^' Hatiafj'iiig t*#'=0, for ■ notion correBpoDiIing thereto may be regarded bh coveted ^^ •!/. 336 MOTION BETWEEN PARALLEL WALLS. [352. If the fixed walls are situated at y = ± yi, « and v must vanish for these values of y. Eliminating from (16) the i-atio of ^ to A, we get as the equation for determining k, iftanh k'y, = ifc'A"tanh Fy, (17), where k\ k" are- the functions of k above defined. Elquation (17) may be regarded as a modified and simplified form of (11) § 350, modified on account of the change from symmetry about an axis to two dimensions, and simplified by the omission of the thermal terms represented by v. The comparison is readily made. Since X, =s 00 , the third term in (11), involving Q„ disappears altogether, and then Xr^ divided out. In (11), (12) r is to be replaced by y, and Jo by cosine, as has already been explained. Further, We now introduce further approximations dependent upon the assumption that the direct influence of viscosity extends through a layer whose thickness is a small fraction only of y^. In this case ]^ r= n'/a* nearly, so that Wy^ is a small quantity and k'y^ is a laige quantity, and we may take tanh fc'yi = ± 1 , tanh k"y^ = ± }c"yx . Equation (17) then becomes k^^k'W'hf,.,.,..: (18), or, if we introduce the values of fc', k" from (13), (14), Thus approximately (19). -^.1 1+ '-' 2y,V(2n/M')] • * in agreement with the result already indicated in § 350. In taking approximate forms for (16) we must specify which half of the symmetrical motion we contemplate. If we choose that for which y is negative, we replace coshA:'y and sinhJfy by ^e"*'*'. For cosh k^'y we may write unity, and for sinh k"y simply k''y. If we change the arbitrary multiplier so tliat the maximum value of li is tio and for the present take Uq equal to unity, we have w=:(-i+e-*'<y+yi))e^c<*< . ) v = tifc/ik'.(y/yi + c-^<«'+y'0^«*^J ^ ^ icil, of course, « and « ^oBOftVi ^Vi«ii ^ ^ - ^v - - .. 352.] FIRST AITBOXIMATION, 337 If in (20) we change k into ~k and then take the nieim, we ..(21). u = (- 1 + e-**+*.>) COB kx e'*" v = - kjlf . {yly, + «-*■ "^''■' ) sin kx e-'"' Although k is nut absolutely a real quantity, we may consider il to be BO with sufKcient approximation for our purpose. We may also take in (14) fc' = V(i-n//) = y9(l + ») (22). li = -Jinjifi.'). Using this approximation in (21). we get in tt-rms of real quantities, '( = cos kx [- cos nt + fl-'u^i'.i COB \nt - |S (y + y,)]] \ tsin kx Ty cos (»( — i it) ^V2 |_y, — v'"-*'v L...(2:)). + e-tiw*<i.> COB In* - Jtt - ^ (y + y,)) 1 It will shorten the expressions with which we have to deal if wi- measure j from the wall (on the negative side) instead of, as hitherto, from the plane of symmetry, for which purpose we must write y for y + ^i. Thus «, = co«fca:[-cosTii + e-'>'cos(ji(-|ey)] \ tBinfcery,-y ^ 4V2 L y. PS {nt -\-ir)- e-'" cos (iit - l^r - ^y) -.(24), the subscripts indicating the order of the terms. These are the values of the velocities when the square of the motion is neglected. In proceeding to a second approximation we require to form expressions for the right-hand members of (7) and (12), which for the purposes of the first approximation were neglected altogether. The additional terms dependent upon the square of the motion are partly independent of the time and partly of double frequency involving 2nt The latter are not of tiiiich interest, so that we shall confine ourselves to the non- periodic part. Further simplifications are admissible in virtue of the small thickness of the retarded layer in proportion to the idth of the channel (2;/,) and still more in proportion to the nve-length (X). Thus i/^ is a small quantity and may usually glected. _ 338 MOTION BETWEEN PARALLEL WALLS [351 From (24) V«t^i = )9V2.co8A»?6-^ysm(ne-i7r — ySy) (25), dui/dx + dvi/dy = k sin kx cos nt (26), + terms in 2nt (27), (£ "^ t) ^'"^^ = - i*'^ ^^ *^ ^^"^ (^^ )9y + cos ySy) + terms in 2nt (28). Thus for the non-periodic part of yft of the second order, we have from (12) V*i^, = - *^,sin ikxer^y {sin/3y + 3cos)9y- 2e-^y}...(29). In this we identify V* with (djdyY, so that ^, = ^-le^^' {sin /9y + 3 COS /9y + Je-'i"'} . . . .(30). to which may be added a complementary function, satisfying V*-^, = 0, of the form ^' = Ty? ^^ '"^^ ^* (yi - y) + 5 (y, - y) cosh 2A: (y, - y)} . . .(31), or, as we may take it approximately, if yi be small compared with X, Equations (30), (32) give the non-periodic part of '^ of the second order. The value of « to a second approximation would have to be investigated by means of (7). It will be composed of two parts, the first independent of ty the second a harmonic function of 9i|l^^^ In calculating the part of dj>ldx independent of t bom V»^ = — dsjdJt — uds/dx — v da/dy, we shall obtain nothing fit>m da/dt. In the remaini^ the right-hand side it will be sufficient to employ t of the first approximation. From da/dt ^-du/dos'-di^i 352.] TO A SECOND APPEOSIMATION. 339 iu conjunction with (26), we get s = — uja.ain kx sin nt, whence ^ipjd (^y)' = ht^jla^ . cos^ kx e"*" sin Qy. b'rom thiij it is easily seen that the part of u, resulting from d^jda: in (3) ia of ordor h'lQ' in comparison with the pait (33) resulting from ^j, and may be omitted. Accordingly by (30), with introduction of the value of ^ and (in order to restore homogeneity) of m„', 2ku,H isiny3y+3co3/3,v + ie-*''f...(34); iind from (32) 8,8« 2hi^ cos 2kw 'sj, -l^'(y>-y)+'<'(y>-yn ..(36). The complete value of the terms of the second order in u, v are given by addition of (33), (35) and of (34), (36). The constants A', B are to be detennined by the condition that these values vanish when i/ =0. We thus obtain as the complete' expression of the terms of the second order .n 2kx I , Jg-B!l (4 -sin j3y + 2 cos fly + e-S") + ^ - Jf (y.-y)"! k Outside the thin tilni < tioD we may put c"*' -- + ^3 (>/,-!/) - ^(y.-^n influenced by the 340 VORTICES. [35l' From (39) we see that v^ changes sign as we pass from tk boundary y = to the plane of symmetry y = yi, the critical val« of y being y, (1 - VJ), or •423y,. The value of tt, from (24) corresponding to (39) is Wi= — tioCosA:a:cosn/ (41), so that the loops correspond to kx=^0, 7r, 27r, . . . , and the nodes correspond to fcc= Jtt, fir, .... The steady motion represented by (39), (40) is of a very simple character. It consists of a series of vortices periodic with respect to x in the distance ^X. From (40) it appears that v is positive at the nodes and negative at the loops, vanishing of course in eacii case both at the wall y = and at the plane of symmetry y=y^ Fig. 66. C Z> i t * t ♦ A * * * * * JS J»r n ^n Sir In the figure AB represents the wall, CD the plane of symmetry, and the directions of motion in the vortices are indicated by arrows. It is especially to be remarked that the velocity of the vortical motion is independent of fi\ so that this effect is not to be obviated by taking the viscosity infinitely small In that way the tendency to generate the vortices may indeed be diminished, but in the same proportion the maintenance of the vortices is facilitated, so that when the motion has reached a final state the vortices are as important with a small as with a large viscosity. The fact that when viscosity is neglected from the first no such vortices make their appearance in the solution shews what extreme care is required in dealing with problems relating to the be- haviour of slightly viscous fluid in contact with solid bodies. In estimating the mean motion to the second order there ii another point to be considered which has not yet been upon. The values of u^ and Vi in (24) are, it is periodic, but the same property does not attach thereby defined of the particles of the A' ' ' not the velocity of any individual pf» particle; whichever it may be, that p2.] CIRCULAB TUBE. 341 i occupies the position w, y, (§ 237). If ai + f, y + ij define i actual position &X time t of the particle whose mean position pring several vibrations is {x, y), then the actual velocities of the irticte at time ( are, not u,, v^. but dn. „ du, dv, ^ dv, \ thus the mean velocity parallel to a: is not necessarily zero, t is equal to the mean value of ^di(,ldx + r}du,ldy (42), I which again ^=fu,dt, r}=I*)idt (43). a the present case the mean value of (42) is -u„'l4a.»m2k-xe-'^(e-"'-coady) (44), which ia to be regarded as an addition to (37). However, at a abort distance from the wall (44) may be neglected, so that (39) remains adequate. We have seen that the width of the direct current along the wall ^ = is '423^1, and that of the return current, measured up to the plane of symmetry, is '577^,. The ratio of these widths is not altered by the inclusion of the second half of the channel lying beyond the plane of symmetry ; so that the direct current is distinctly narrower than the return current. This disproportion will be increased in the case of a tube of circular section. The point under consideration depends in fact only upon a comple- mentary function analogous to (32), and is so simple that it may be worth while briefly to indicate the steps of the calculation. »The equation for i|f, is' (£-'i-*^)V-=° 1*^)^ but, if we suppose that the radius of the tube is small in compari- "^ X, ^*' may be omitted. The general solution is ^,= \A+ Br'' + B'r'\ogr+Cr*] sin ±/cx (46), ^^■Jn/r=|2B + fi'(2 1ogr + l) + 40r'|8m2ia:...(47), k, Fhil. Bot.y vol. IS. 1B£6: BuMt'i Hydrodynamht, 342 DIRECT AND RETURN CURRENTS. [352. whence fi' = 0, by the condition at r = 0. Again, t;, = -.d'^^rda: = -2Jfcl-4r->+5r-hCr'} cos 2Jbc . . .(48), whence ^ = 0. We may therefore take ii,= {25+4(7r«}8in2Jb? v,= - 2Jt {Br-hCf*]coB2kx If, as in (40), r, = 0, when r = jB, B-hCiP = 0, and ii,= 2C(2r«-i?)8in2Jb? (50). Thus Uj vanishes, when r = iJ/V2 = -707/2, iZ - r = -29312. The direct current is thus limited to an annulus of thickne^ *293i2, the return current occupying the whole interior and having therefore a diameter of 2 x -707 iJ, or 1-4 14 iJ. } (49). 363. The subject of the present chapter is the behaviour of inviscid incompressible fluid vibrating under the action of gravity and capillary force, luore especially the latter. In vij-lue of the first condition we may assume the existence of a velocity-potential (i^), which by the second condition must satisfy (§ 241} the equation ^*<f> = (1), thi^oughout the interior of the fluid, In terms of tp the equation for the pressure is {§ 244) hp!p = R-d4>jdt (2). if we assume that the motion is ao small that its square may be neglected. The only impi-e.ssed force, acting upon the interior of the fluid, which we have occasion to consider is that due to gravity ; so that, if r be measured vertically downwards, R=gz, and (2) becomes hplp=gz-d4,!dt (3). ^k Let us now consider the propagation of waves upon the hori- ^nontal mir&ce (z = 0) of water, or other liquid, of uniform depth I, limiting our attention to the case of two dimensions, where the motion is confined to the pltUM tK> The general solution of (1) under this condition, and.4lM^M|IMn)Uoii ia proportional to «^. is ' ^with regan^ t- ^'CnHl] velocity mnrt iali at. ■ 344 WAVES ON WATER. [3! If the motion be proportional also to e****, and we throw away imaginary part in (4), we get as the expression for waves pro| gated in the negative direction if) ^ C cosh k{z'- 1) cos(nt-\'kx) (5), in which it remains to find the connection between n and k. I{ h denote the elevation of the water surface at the point <,! and T the constant tension, the pressure at the surface due to' capillarity is -Td^kldx^ and (3) becomes or, if we differentiate with respect to t and remember that dh/dt = - d<l>ldz, T d^<l> _ d^_^ .. ■ pdx'dz'^ dz dP ^ ^ Applying this equation to (5) where ^ = 0, we get for the velocity of propagation F« = nVA» = (^/& + rA;//>)tanhW (!)\ where, as usual, it = 27r/X (8). In many cases the depth of liquid is sufficient to allow us to take tanh kl^^l] and then "-2^ + -^ W gives the relation between V and X. When \ is great, the waves move mainly under gravity and with velocity approximately equal to *J(g\/27r). On the other hand, when \ is small, the influence of capillarity becomes predominant and the expression for the velocity assumes the form r=y/{2wT/p\) (10). Since X = Vr, the relation between wave-length and periodic time corresponding to (10) is XVT«=27rr/p (11). Except as regards the numerical factor, the relations (10), (11) can be deduced by considerations of dimensions from the £sw^t that the dimensions of T are those of a force divided by a line. ^ A more general formula for the Telocity of propagation (ii/ft) $A ^ between two liquids is given in (7) § 36o. ' Kelvin, Phil, Mag, vol. xui. p, 876, 1871. 5.] MINIMUM VELOCITY. 345 \ If we inquire what values of \ correspond to a giveu value of Kve obtalD from the quadratic (9) X-TrFVslT/j.VC'-tJ'sW (12). kh ahews- that for no wave-length can T"" be less than V^. r.-{iT<,ip)i (13). The values of X and of r corresponding to the minimum ^ocity are given by K = 2w(T/ffp)K T„ = 2ir(r/4^p)l (14). If we take in co.s. measure (? = 981, and lor water p=i, -76, we have F'„ = 23-l, X, = l-71, 1/t=13-6. The accompanying table gives a few corresponding values of e-Iength, velocity, and frequency in the neighbourhood of the J point : — Wftve-iength Velocity Frequency ■5 10 1-7 2-5 3-0 O-O 31-5 24-7 23.1 23-9 2*.9 29-6 63-0 M-7 n-s 9-6 8.3 5.9 A comparison of Kelvin's formula (9) with observation has been eBfected by Matthiessen', the ripples being generated by touching the surface of the vaiious liquids with dippers attached to vibrating forks of known pitch. Among the liquids tried were water, mercury, alcohol, ether, bisulphide of carbon; and the agreement was found to be satisfactory. The observations include frequencies as high as 1832, and wave-IengthH an small as ■04 cm. .Somewhat similar experiments have been carried out by the author' with the view of determining T by a method independent of any assumption respecting angles of contact between fluid and solid, and admitting of application to surfaces purified to the utmost from grease. In order to see the waves well, the light was made iutennittent in & perioil equal to that •>{ the waves t§ 42), and Foucault's optical method was employed for i-endering TJaible small departures from truth in plane Qg Mdiwioal reflecting 1846 T DETERMINED BY RIPPLES. tail B«arfeceN. From the measured values of t and X, T may be dcA Btnined by (11), corrected, if necewi&iy, for any sniall effect ■^a\ity. The values thus found were for clean water 74"0 C.as. ■ for a surface greasy to the puiut where camphor itiotions ntarii ■ cease 330, for a surlioci' saturated with olive-oil 41'U, and for Itatiirated with oleate of «udii 250. It should be remembered thi: vthe teusioD of contaminated surfaces '\» liable to variations depes- ■ dent upon the extension which has taken place, or is takii^ ■place; but it is not necessary for the purposes of this work (• ■enter further upon the (question of "superficial ■viscosity." I 364, Another way of generating capillary wav««8, or t tions as they were termed by Faraday, depends upon the principle discussed in § 68 fi. If a glass plate, held honzontally and mufe to vibrate as for the piixluctiou of Chladiu's figures, be covered vith a thin layer of water or other mobile liquid, the phenomena I in question may be readily observed'. Over those parts of die plate which vibrate sensibly the surface is ruffli'd by minute wavei the degree of fineness iucreasing with the frequency of vibration. The same crispations are observed upon the sur&ce of liquid i large wine-glass or finger-glass which is caused to vibrate in usual manner by carrying the moistened finger round the circum- ference (§ 234). All that Is ossential to the production of' crispatioufl is that a body uf liquid with a free surface be constrwned to execute a vertical vibration, It is indifferent whether the origin of the motion be at the botU)m, as in the first case, or, as in the second, be due to the alternate advance 1 and retreat of a lateral boundary, to accommodate itself to which I the neighbouring surface must rise and fall. More than sixty years ago the nature of these vibrations I examined by Faraday' with great ingenuity and success, Tht I conditions are simplest when the motion of the vibrating h< I plate on which the liquid is spread is a simple up 1 motion without rotation. To secure this Faraday at iplate to the centre of a strip of glass or lath of dea], I at the nodes, and caused to vibrate by fricti^m. Still C I venient is a large iron bar, maintained in vibmi. I which the plate may be attached by cement, ■ On the CiigpatioiiB d( Floid retting' upon u Tibnit< . »vi, p. 50. 1688. 354.] FABAUAV'S CRISPATIONS. 347 The vibrating liquid standing upon ihe plate presents appear- ances which at first are rather difficult to interpret, and which vary a good deal with the nature of the liquid in respect of transparency and opacity, and with the incidence of the light. The vibrations are too quick to be followed by the eye ; and thus the etfeet observed is an average, due to the superposition of an indefinite number of eleoientarj- impressions corresponding to the various phases. If the plate be rectangular, the motion of the liquid consists of two sets of statiooarj' vibrations superposed, the ridges and furrows of the two sets being perpendicular to oae another and usually parallel to the edges of the plate. Confining our attention for the moment to one set of stationary waves, let us consider what appearance it might be expected to present. At one moment the ridges form a set of parallel and equidistant lines, the interval being X. Midway between these are the lines which represent at that moment the position of the furrows. After the lapse of a J peiiod the surface is flat ; after another J period the ridges and furrows are again at their maximum developement, but the positions are exchanged. Now, since only an average effect can be perceived, it is clear that no distinction is i-ecognizable between the ridges and the furrows, and that the observed effect must be periodic within a distance equal to JX, If the liquid on the plate be rendered moderately opaque by addition of aniline blue, and be seen by diffused transmitted light, the lines of ridge and furrow will appear bright in comparison with the intermediate nodal lines where the normal depth is presen'ed throughout the vi- bration. The gain of light when the thickness is small will, in accordance with the law of absorption, outweigh the loss of light irbich occurs half a period later when the furrow is replaced by a 'he actual phenomenon is more complicated in consequence of k ooexistCQCe of the two sets of ridges and furrows in perpendi- r directionii (a, y). In the adjoining tigiire (Fig. 66) the thick R represent th« lidKOs, and the thin lines the furrows, of the of maximum excursion. One quarter \nd one half period later the ridges 'he placvs of maximum elevation .of the thick lines with one 348 RESOLUTION INTO PROUBESSIVE WAVES. [35ll guishable by ordiaary vision. They appear like holes in the «heei of colour. The nodal lines where the normal depth of colour ■ preserved throughout the vibration are shewn dotted ; they an inclined at 45*, and pass through the intersectioiia of the tlm lines with the thick lines. The pattern is recarrent in ikt Kt 66. ■., ..• .,• ■., ,• ,.• '• '•-. ,-' ^\ ■ '•■ ■■. ■■■ directions both of x and y, and in each case with an interval equal to the real wave-length (\). The distance between the bright spots measured parallel to a: or y is thus \ ; but the shortest distance between these spots is in directions inclined at 45°, and is equal to X/V2. As in all similar cases, these stationary waves may be tesolved into their progressive components by a suitable motion of the eye. Consider, for example, the simple set of waves represented by 2 cos kx cos n( = cos (iit + kx) 4 cos (u/ — kx). This is with reference to an origin fixed in space. But Ist Of^ij refer the pheuomenon to an origin moving forward with the TslotiMil (njk) of the waves, so as to obtain the impression that- produced npon the eye, or in a phob^^phio c forward in this manner. Writing ka^+»t for LlSSAJOtJS PHENOMENON. 349 bw the average effect of the first term is independent of af. so pt what is seen is simply that set of progressive waves which jves with the eye. I In order to see the progressive waves it is not tieceaaary to »ve the head aa a whole, but only to tuni the eye as when we How the motion of a real object. To do this without assistance ■ not very easy at first, especially if the ai^ea of the plate be mewhat small. By moving a pointer at various speeds until the ■ht one is found, the eye may be guided to do what is required f it ; and after a few successes repetition becomes easy. Faraday's assertion that the waves have a period double that of Hhe support has been disputed, but it may be verified in various ways. Observation by stroboscopic methods is perhaps the most satisfactory. The violence of the vibrations and the small depth of the liquid interfere with an accurate calculation of fi-equency on the basis of the observed wave-length. The theory of vibrations iti the Hub-octave has already been considered (§ 68 b). 386. Typical stationary waves are formed by the superposi- tion of e'lual positive and negative progressive waves of like frequency. If the one set be deiived from the other by reflection, the equality of frequencies is secured automatically; but if the two sets of waves originate in different sources, the unison is a matter of adjustment, and a question arises as to the effect of a slight We may take as the expression for the two sets of progressive waves of equal amplitude and of approximately equal frequency co&{kx — nt) -H co&(k'x+ n't), , which is the same, 2 cos li (it + A-') 'K + !("'-") i! X f^os [i (*"-*) '^ + i ("■ + ") 'i (D- If n' = H, k' = 1; the waves ai-e absolutely stationary; but we have now to interpret (1 ) when (»(' — h), (k The position at ?"•■ ''t^" t "^ 'h" ' nearly stationary ^< ^of ai Rffrroi "prog fr :M inU;^.;r .^ k) are merely small. and hollows of the ,,'iven by "T (2), di.,.l:,.-.,n..n, IJ is 350 STANDING WAVES [35l| or approximately U^in^nyik HI from which it appears that in every case the shifting takes plm in the direction of waves of higher pitch, or towards the souroeoC graver pitch. If F be the velocity (n/k) of propagation of tk progressive waves, (3) may be written £r/F=(w-n')/2w (4)l The slow travel under these circumstances of the places nhm the maximum displacements occur is a general phenomenon, M dependent upon the peculiarities of any particular kind of waves; but the most striking example is that afforded by capillaiy waves and described by Lissajous'. In his experiment two nearly unisonant forks touch the surfeice of water so as to form approxi- mately stationary waves in the region between the points of contact. Since the crests and troughs cannot be distinguiahedi the pattern seen has an apparent wave-length half that of the real waves, and it travels slowly towaitls the graver fork. A firequency of about 50 will be found suitable for convenient observation. If the waves be aerial, there is no difference of velocity ; but (4) still holds good, and gives the rate at which the ear must travel in order to remain continually in a loop or in a node. 366. One of the best opportunities for the examination of capil- lary waves occurs when they are reduced to rest by a contrary movement of the water. Waves of this kind are sometimes described as standing waves, and they may usually be observed when the uniform motion of a stream is disturbed by obstacles. Thus when the sur&ce is touched by a small rod, or by a fishing-line, or is displaced by the impact of a gentle stream of air &om a small nozzle, a beautiful pattern is often displayed, stationary with respect to the obstacle. This was described and figured by Scott Russell', who remarked that the purity of the water had much to do with the extent and range of the phenomenon. On the up-stream side of the obstacle the wave-length is short, and, as was first clearly shewn by Kelvin, the force governing the vibtm* 1 Phil. Mag. voL xvi. p. 67, 18S3. 3 Compt. Rend. toI. Lxvn. p. 1187, 1868. s Brit. Am. Rep. 1844, p. 875, Plate 57. See also Ponoelet, Ann. tf, CMm. vol. XLYi. p. 5, 1831. B is principally cohesion. On the dovni-stream side the waves a longer and are governed principally by gravity. Both sets of Jlraves move with the same velocity relatively to the water (§ 353) ; mely. that required in order that they may maintaiii a fixed sition relatively to the obstacle. The same condition governs the velocity and therefore the wave-lengths of those parts of the pattern where the fronts are oblique to the direction of motion. [ the angle between this direction and the normal to the wave- ront be called ff, the velocity of propagation must be equal tu j^cos^, where Un represents the velocity of the water. If i'„ be less than 23 cm, per sec, no wave-pattern is possible, • no waves can then move over the suiface so slowly as to Attain a stationary position ivith respect to the obstacle. When lexceeds 2H cm. per sec, a pattern is formed ; but the angle ff has limit defined by v^ cos = 23, and the curved wave-front has a responding asymptote. It would lead us too far to go further into the matter here, but. it may be mentioned that the problem in two dimensions admits of analytical ti-eatment', and that the solution explains satis- factorily one of the peculiar features of the case, namely, the limitation of the smaller capillary waves to the up-stream side, aud of the larger (gravity) waves to the down-stream side of the obstacle. 367. A large class of phenomena, interesting not only in themselves but also as throwing light upon others yet moi-e olMcure. depend for their explanation upon the transformations undergone by a cylindrical body of liquid when slightly displaced from its equilibrium configuration and then left to itself. Such a cylinder is formed when liquid is-iues under pressure through a circular orifice, at least when gravity may be neglected : and the behaviour of the jet, as studied experimentally by Savart. Magnus, Plateau and others, is substaTitially independent of the forward motion common to all its parts. It will save repetition and be more in accordance with the general character of this work if we t commence our investigation with the theory oi'aii inliniii' rylinder tliqnid, considered as a system in equilibriui] i. r :l, "■uon B the tonn of Standing W&rei on thu Surfp fl. Math. Soe. nl xv. fi. 69, 1683. 352 LIQUID CYLINDER, [I of the capillary force. With a solution of this mechanical most of the experimental results will easily be connected. Taking cylindrical coordinates s, r, ^, the equation of tkl slightly disturbed surface may be written r = a,+/(^,^) (l\ in which /(^, z) is always a small quantity. By Fourier's theoml the arbitrary function /may be expanded in a aeries of temail! the type a^ cos n^ cos kz ; and, as we shall see in the course of tk investigation, each of these terms may be considered independent^ of the others. Either cosine may be replaced by a sine ; and dx summation extends to all positive values of k and to all podtivc integral values of n, zero included. During the motion the quantity Oo does not remain absoluteh constant ; its value must be determined by the condition that the enclosed volume is invariable. Now for the surfctce rssOo^OnCOsn^coBkz (2), we find Volume = yjr^d<t>dz = z {ira^ + ^'ira^) ; so that, if a denote the radius of the section of the undisturbed cylinder, whence approximately rto = a(l-ianV«') (3). This holds good when n=l, 2, 3.... If ?i = 0, (2) gives in place of (3) a, = a(l-i«,VaO (4). The potential energy of the system in any configuiation, due to the capillary force, is proportional simply to the surface. Now in (2) so that by (3), if a denote the surface corresponding upon the avemge to unit of length. I POTEKTIAI, ENERGY. 17.] The potential energy due to ;h aiid from the conjuration i ^pill&rity, estimated per unit if equilibrium, is accordingly P = i7r7'(i-o' + «»-l)a„V« (6). T denoLiij)^. its usual, the superficial tension. Id (G) it is supposed that k and n are not zero. If k be zero, (i|) requires to be doubled in order to give the potential energy i;i:ir responding to ,-.o. + ..co.„+ (7); and again, if ii be zero, wc are to take tP = i^ra'a* - 1) «,'/" ^sponding to r = a, + a„ coat^.... •(8), ..(9). From (6) it appears that when » is unity or any greater ;ger, the value of i* is positive, shewing that for alt displace- its of these kinds the original equilibrium is stable. For the of displacements symmetrical about the axis (n = 0), we see (8) that the equilibrium is stable or unstable according as ku greater or less than unity, i.e. according as the wave-length (2vjk) of the symmetrical deformation is less or greater than the circumference of the cylinder, a proposition first established by Plateau. If the expression for r in (2) involve a number of terms with various values of n and k. and with arbitrary substitution of sines for cosines, the corresponding expression for P is found by simple addition of the expressions relating to the component terms, and it contains the squares only (and not the products) of the quantities a. We have now to consider the kinetic energy of the motion. Since the fluid is supposed to be invisoid, there is a velocity- potential ^, and this in virtue of the incompressibility satisfies Laplace's equation. Thus, (4) § 241, d'^fr \d^ ld^^|r d'ylr_ rfr" rdr'^1^ d^" dz' ' or, if in order to correspond with (2)we assuni part is proportional to cos n<f) cos kz. ' that the variable 354 KINETIC ENERGY. [] The solution of (10) under the condition that there ii introduction or abstraction of fluid along the axis of is § 200 y^^finJu (ikr) cos n^ cos ks (11). The constant /9» is to be found from the condition that Ail radial velocity when r^a coincides with that implied in ^| Thus ikfinJn(ika)^dan/dt (12^ If p be the density, the kinetic energy of the motion is If Green's theorem (2) § 242 ip Ilbkd^ldrl^a ad<t>dz = ^Trpz.tka.Jn (ika) J^'(ika).fin' \ so that by (12), if K denote the kinetic energy per unit of leogd, '-i''»'^^ir)(^)' w When n = 0, we must take in place of (13) '-i^-v^^ir,©' w The most general value of if is to be found by simple summt- tion from the particular values expressed in (13), (14). Since the expressions for P and K involve the squares only, and not the products, of the quantities a, daldt, and the corresponding quanti- ties in which cosines are replaced by sines, it follows that the motions represented by (2) take place in perfect independence of one another, so long as the whole displacement is small. For the free motion we get by Lagrange's method from (6), (13) which applies without change to the case n == 0. Thus, if o^ varies as cos {pt — e), T ika. JJ (ika), . ... ,. ^,^v- giving the frequency of vibration in the cases of stability. If nsO, and ka<l, the solution changes its form. suppose that Oo varies as el^, ^ T ika.Jo'iika) ^ pa* Jo(ika) ^ Proe. Boy. 8oe. ^<JL (1-**' E 17.] FREQUENCY EQUATION. 355 ^ When II ia greater than unity, the circumstances are uaually such that the motion is approximatfly in two dimensions only. We may then iidvantageously introduce into (16) the supposition that i-a is small. In this way we get, (5) § 2O0, ;,.-.(»'-l +i.„.)^. [l + .-i£^] (18), \f if ka be neglected altogether, p--(.--,)^. (19), I ^phe two-dimensional formula. When m = ], there is no force of resti- tution for a displacement purely in two dimeaeions. If \ denote the wave-length measured round the circumference, X = 2Tra/n. ' Thus in (19), if n and a are infinite. :(?)■• ..(20), I agreement with the theory of capillary waves upon a plane Compare (7) § 353. A similar conclusion may be reached f the consideration of waves whose length is measured axially. Jkus, if X = 2Tr//:, and as x, n = 0. (Iti) reduces to (20) in virtue |the relation, ^ 302, 350, Limit,^„ iV/(w)//„(i>) = 1. 368. Many years ago Bidone investigated by experiment the behaviour of jets of water i,=wuing horizontally under considerable pressure from orifices in thin platea If the orifice be circular, the section of the jet, though diminished in area, retains the circular form. But if the orifice be not circular, curious transformations ensue. The peculiarities of the orifice are exaggerated in the jet, but in an inverted manner. Thus in the case of an elliptical aperture, with major axis horizontal, the sections of the jet taken at increasing distances gradually lose their ellipticity until at a certain dist«nco the section is circular. Further out the section again assume.it ellipticity, but now with major axis vertical, and '' the circumstances of Bidoue's experiments) the ellipticity _ the jet is reduced to a ilat sheet in the vertical ^k I thin. This sheet preserves its continuity to ^^^ % six ieet) from the orifice, where finally ^^^^^H ' orifice ^e in %\ve lovnv i^l ^"o. €»^- 356 OBSERVATIONS BY BIDONE AND MAGNUS. [Jii] lateral triangle, the jet resolves itself into three sheets dispoBJ symmetrically rouud the axis, the planes of the sheets bafl perpendicular to the sides of the orifice; and in like maimffi the aperture be a regular polygon of any number of sides, thsj are developed a corresponding number of sheets perpendicalani| the sides of the polygon. Bidone explains the formation of these sheets by reference t»' simpler cases of meeting streams. Thus equal jets, moTing in tk same straight line with equal and opposite velocities, flatten thoB- selves into a disc situated in the perpendicular plane. If the ai« of the jets intersect obliquely, a sheet is formed symmetrically o the plane perpendicular to that of the impinging* jets. ThoK portions of a jet which proceed from the outlying parts of a singk unsymmetrical orifice are regarded as behaving in some degxt like independent meeting streams. In many cases, especially when the orifices are small and the pressures low, the extension of the sheets reaches a limit. Sections taken at still greater distances from the orifice shew a gradual gathering together of the sheets, until a compact form is regained similar to that at the first contraction. Beyond this point, if the jet retains its coherence, sheets are gradually thrown out again, but in directions bisecting the angles between the directions of the former sheets. These sheets may in their turn reach a limit of developement, again contract, and so on. The forms assumed in the ciise of orifices of various shapes including the rectangle, the equilateral triangle, and the square, have been carefully investigated and figured by Magnus. Phenomena of this kind are of every day occurrence, and may generally be observed whenever liquid falls from the lip of a moderately elevated vessel. As was first suggested by Magnus^ and Buff '', the cause of the contraction of the sheets after their first developement is to be found in the capillary force, in virtue of which the fluid behaves as if enclosed in an envelope of constant tension; and the re- current form of the jet is due to vibrati07is of the fluid colmnn about the circular figure of equilibrium, superposed upon Ik general progressive motion. Since the phase of the vibniti initiated during passage through the aperture, depends npf ^ HydraulUohe Untcanuohungi&ii, Pogg* Awu toL kov, p. 1| V • Pogg. Ann. \o\. o, p, IWAWl. \ |8,] FURTHER EXPERIMENTS. 357 B elapsed, it is always the same at the same point in space, thus the motion is steady in the hydrodynamical sense, and the ndary of the jet is a fixed surface. Relatively to the water the •* waves here concerned are progressive, such as maybe compounded p{ two stationary systems, and they move up stream with a velocity Hual to that of the water 8o as to maintain a fixed position rela- r»ly to external objects, § 356. j If the dejKirture from the circular form be small, the vibrations t those considered in § 357, of which the frequency is determined I equations (16), (18), (19). The distance between consecutive responding points of the recurrent figure, or, as it may be called, ( wave-length of the figure, is the space travelled over by the I during one vibration. Thence results a relation between bre-length and perifxl. If the circumference of the jet be small 1 comparison with the wave-length, so that (19) §357 is appli- ible, the periodic time is independent of the wave-length ; and ten the wave-length is directly proportional to the velocity of B jet, or to the square root of the pi-essure. The elongation of ^ftve-length with iucreasing pressure f/as remarked by Bidone and DJ Magnus, but no definite law was arrived at, In the experiments of the author' upon elliptical, triangular, and square apertures, the jets were caused to issue horizontally in order to avoid the complications due to gravity ; and, if the pressure were not too high, the law above stated was found to be verified. At higher pressures the observed wave-lengths had a marked tendency to increase more rapidly than the velocity of the jet. This result points to a departure from the law of isochronous vibration. Strict isochronism is only to be expected when vibra- tions are infinitely small, that is when the section of the jet never deviates more than infinitesimally from the circular form. Under the high pressures in question the departures from circularity were very considerable, and there is no reason for expecting that such vibrations will be executed in precisely the same time as vibrations ■ if infinitely smalt amplitude. The increase of amplitude under high preeenre is easily ex- plained, inasmuch as the lateral velocities to which the vibrations » mainlv due vary in direct proportion to the longitudinal Consequently the amplitude varies approxi- 358 CX)MPARI80N WITH THEORY. [j mately as the square root of the pressure, or as the wave-l In general, the periodic time of a vibration is an eTen.ftinctiBif amplitude (§67); and thus, if A represent the head of liquid,! wave-length may be expected to be a function of A of the tal (M+Nh) VA, where M and N are constants for a given apertat It appears from experiment, and might perhaps have beena*] pected, that N is here positive. For a comparison with theory it is necessary to keep within tk range of the law of isochronism ; and it is convenient to employ ii the calculations the area of the section of the jet in place of tk mean radius. Thus, if il =ira\ (19) § 357 may be written p = 7r«r*p-*^-*>/(n'-«) (1), in which il is to be determined by experiments upon the rate of total discharge. For the case of water (§ 353) we may take in c.as. measure 7=74, p^l; so that for the frequency of the gravest vibration (n = 2) we get from (1) p/27r = 7-91^-* (2). For a sectional area of one square centimetre there are thus about 8 vibrations per second. A pitch of 256 would correspoDd to a diameter of about one millimetre. For the general value of n, we have p/27r = 3-23^-* V(w» - 71) (3). If A be the head of water to which the velocity of the jet is due and X the wave-length, yf(2gh),A^ 3-23V(w'-n) ^'^ In one experiment with an elliptical aperture (n=2) the observed value of \ was 3*95 while the value calculated fiiom (4) is 3*93. In the case of a triangular aperture (nsS) the observed value of \ was 2*3 and the calculated was 2*1. the observed value for a square aperture (n = 4) was 1*85 calculated 1'78. The excess of the observed over the values in the last two cases may perhaps have been sive departure frx)m the circular figure. The general theory, imrestricted to doubtless involve great compUcations ; ©8.] LARGE AMPLITUDES. 359 MJting it may be obtained with facility by the method of nensions. If the shape of the orifice be given, \ may be re- [ded as a function of T, p, A, and H the pressure under which p jet escapes. Of these 7 ts a force divided by a line, so that its teensions are 1 in mass. in length, and ~ 2 in time ; p is of s 1 in mass, — 3 in length, in time : .^ la of dimensions 9 in mass, 2 in length, iu time ; and finally if is of dimensions I in mass, — 1 in length, and — 2 in time. If we assume XxVpyA'H". x + y-t-u = 0, -3(/4-22 ■ 1, -2x-2u = Q, -.T. .v = 0, 2 = i(l-:r); XxA^iTA'^H-'f. The exponent x is here undetermined ; and, since any number K>f terms with different values of x may occur simultaneously, all that we can infer is that \ is of the form or, if we prefer it. X=Ai./(TA~iH-'). \= T-iRiA' . F{HAiT-').. ..(5), where _/" and F are arbitrary functional symbols. Thus for a given liquid and shape of orifice there is complete dynamical similarity if the pressure be taken inversely proportional to the linear dimension. The simple case previously considered where the departures from circularity are small, and the vibrations take place approximately in two dimensions, corresponds to ^ = coD9tant. The method of determining T by observations upon \ is scarcely delicate enough to compete with others that may be employed for the same purpose when the tension is constant. But the possibility of thus experimenting upon surfaces which have been formed but a fraction of a second earlier is of oonsi- » derable interest. In this way it may be proved with great ease ^■Ibat the tension of a soapy solution immediately after the forma- ^^ua of a fi-ee surface differs comparatively Utile from that of pure ^^V-'Rv whopoaq -when a few seconds hn\-e elapsed the difference 1 Liquid Siirbces, Proc. Roy. See. vol. 360 SYMMETRICAL DISTURBANCES. [SU Hitherto it has been supposed for the sake of simplicitjtk| the jet after its issue from the nozzle is withdrawn from the i of gravity. If the direction of projection be vertically downing] as is often convenient, the velocity of flow (9) continually increMe^j while at the same time the area of the section diminishes^ tk' relation being t*il =:£ constant. But, so far as regards X, tibefrl turbance which thus ensues is less than might have been expectei, for the changes in 1; and A compensate one another to a eoi- siderable extent. By (1) X « v/p X t^ X A*, if h denote the whol^ difference of level between the sorfiifie d liquid in the reservoir and the place where \ is measured. 369. In § 358 the motion of the liquid is regarded as steady, every portion as in turn it passes the orifice being simihrif affected. Under these circumstances no term corresponding to 71 = can appear in the mathematical expressions; but it mast not be forgotten that for certain disturbances of this type the cylindrical form is unstable and that therefore the jet cannot loDg preserve its integrity. The minute disturbances required to bring the instability into play are such as act differently at different moments of time, and have their origin in eddying motions of the fluid due to friction, and especially in vibration communicated to the nozzle and of such a character as to render the rate of discharge subject to a slight periodic variation. If v be the velocity of the jet and t the period of the vibration, the cylindrical column issuing from a circular orifice is launched subject to a disturbance of wave-length (X) equal to vt. If this wave-length exceed the circumference of the jet (27ra), the disturbance grows exponentially, until finally the column of liquid is divided into detached masses separated by the common interval X, and passing a fixed point with velocity v and frequency 1/t. Even though no regular vibration has access to the nozzle, the instability cannot fidl to assert itself, and casual disturbances of a complex character bring about disintegration. It will be convenient to ^' the first place somewhat in detail the theory of U>* in (16), (17) § 857, and then to consider its beautiful phenomena desmbed by Savart i explained by Plateau. ^ 'iPW' If ka = z, and we introduce the notation of § 221 a, (17) § T ;!/,(.) a-'*)- ..(1). " pa- IM ' In this equation Ii{z) and /((?) are both positive, so that as z decreases (or as X increases) q first becomea real when z=\. At this point instability commences, and at first the degree of in- stability is infinitely small. Also when t is very small, or X is , T 2" very great, 5' = ^ - pa 'I ultimately, so that q is again small. For some vahie of z between and 1, 5 is a maximum, and the investigation of this value is a matter of importance, because, as has already been shewn § 87, the unstable equilibrium will give way by preference in the mode so characterized. The function to be made a maximum is '(i-^)/,W//.W (2). or, expanded in powers of z, ence, to find the maximum, we obtain on differentiation o»' + VJ t 1-7 100 «•+ = 0. ■f the last terms be noglected, the quadratic gives r" = -4914. If this value be substituted in the small terma. the equation becomes ■98928 -ff'-t-^j' = 0, whence i» = -*86. z = -679'. The values of expression (2), or of its square root, to which i[ is proportional, may be calculated from tables of /» and /,, § 221 o. We have . IWj' . l(«l' 0-0 ■(©OO 0^6 ■3321 0-1 ■0703 f»'7 ■3433 0-S ■1362 0^8 ■3269 0-3 ■2012 0^9 ■2647 0-4 ■2667 m ■0000 05 ■3015 1 On the Instability of Jeta. ffO(^. Lond. Math, Soc. Tot.«,i6. 7,1878. 362 MAXIMUM instabh-ity. [Sftj From these values we find for the maximum by Lagmpil interpolation formula ^sb'696, corresponding* to X = 27ro/-f = 4-51 X 2a (8). Hence the maximum instability occurs when the wave-kogll of disturbance is about half as great again as that at wU instability first commences. Taking for water in c.G.S. units T^78, p » !» we get ftr it case of maximum instability ?- = 73J^ = -^^^(2a). (4> This is the time in which the disturbance is multiplied in tbe ratio e : 1. Thus in the case of a diameter of one centimetre the disturbance is multiplied 2*7 times in about ^ second. If tbe disturbance be multiplied 1000 fold in time t, qt^3log^l0^6'i, so that t = *828 (2a)i For example, if the diameter be one milli- metre, the disturbance is multiplied 1000 fold in about ^ second. In view of these estimates the rapid disintegration of a jet of water will not cause surprise. The above theory of the instability of a cylindrical sur&oe separating liquid from gas may be extended to meet the case where the liquid is outside and the gas, whose inertia is neglected, is inside the surface. This represents a jet of gas discharged under liquid ; and it appears that the degree of maximum in- stability is even higher than before, and that it occurs when \ = 6*48 X 2a\ But it is scarcely necessary for our purpose to pursue this part of the subject further. 360. The application of our mathematical results to actual jets presents no great difficulty. The disturbances, by which equilibrium is upset, are impressed upon the fluid as it leaves the aperture, and the continuous portion of the jet represents the distance travelled over during the time necessary to produce disintegration. Thus the length of the continuous portion neces- sarily depends upon the character of the disturbances in respect of amplitude and wave-length. It may be increased considerably, as Savart shewed', by a suitable isolation of the reservoir fronoi 1 On the Instabilify of Cylindrioal Fluid Sorfues, PkiL Mag. toL zmv, p. ITT* 1898. < Ann. de Ckimie, un, p. Wl, \%»&. 8AVABTS LAWS. m tremorB, whether due to external sources or to the impact of the 1 jet itself in the vessel placed to receive it. Nevertheless it does not appear possible to carry the prolongation very far. Whether the residual disturbances are of external origin or are due to r friction, or to some peculiarity of the fluid motion within the I reser\oir, haa not been satisfactorily determined. On this point Plateau's explanations are not very clear, and he sometimes expresses himself as if the time of disintegration depended only upon the capillary tension without reference to initial dis- turbances at all. Two laws were formulated by Savart with respect to the length of the continuous portion of a jet, and have been to a certain extent explained by Plateau'. For a given fluid and a given oriflce the length is approximately proportional to the square root of the head. This follows at once from theory, if it can be assumed that the disturbances remain always of the same character, so that the time of disintegration is constant. When the head is given, Savart fuiind the length to be proportional to the diameter of the orifice. From (4) § 359 it appears that the time in which a small disturbance is multiplied in a given ratio varies not as u, but as u.). Again, when the fluid is changed, the time varies as pT"'. But it may well be doubted whether the length of the continuous [wrtion obeys any very simple laws, even when external disturb- ances are avoided as far as possible. When a jet falls vertically downwards, the circumstances upon which its stability or instability depend are continually changing, more especially if the initial velocity be very small. The kind of disturbance to which the jet is most sensitive as it leaves the nozzle is one which impresses upon it undulations of length equal to about 4^ times the initial diameter. But as the jet falls, its velocity increases, with consequent lengthening of the undulations, and its diameter diminishes, so that the degree of instability soon becomes much reduced. On the other hand, the kind of disturb- ance which will be effective in a later stage is altogether ineffective in the earlier stages. The change of conditions during fall has thus a protective influence, and the continuous port tends to become longer than would be the case were the velocity constant, the initial disturbances being unaltered. iiiienl«l« et Ib^rique dea Liqnidee loDiiiii ftox senlM forosa 364 plateau's theory. [3*1 When the circumstances are such that the reservoir i influenced by the shocks due to the impact of the j^ tk disintegration often assumes a complete regularity and is attenU by a musical note (Savart). The impact of the regular series d drops, which at any moment strike the receivin^r vessel, det^nuH the rupture into similar drops of the portion of the jet at the mm moment passing the oriflce. The pitch of the note, thonj^ wi definite, cannot ditfer greatly from that which corresponds to tk division of the column into wave-lengths of maximum instabili^; and in fact Savart found that the frequency was directly as tk square root of the head, inversely as the diameter of the orifice, and independent of the nature of the fluid — laws which follow immediately from Plateau s theory. From the observed pitch of the note due to a jet of given diameter, and issuing under a given head, the wave-length of the nascent divisions can be at once deduced. Reasoning from some observations of Savart, Plateau found in this way 4'38 as the ratio of the length of a division to the diameter of the jet. Now thst the length of a division can be estimated a priori^ it is preferable to reverse Plateau s calculation and to exhibit the frequency of vibration in terms of the other data of the problem. Thus frequency = ^-:^JL_ (1), and in many cases, where the jet is not too fine, v may be replaced by 's/{2gh) with sufficient accuracy. But the most certain method of attaining complete regularity of resolution is to bring the reservoir under the influence of an external vibrator, whose pitch is approximately the same as that proper to the jet. Magnus* employed a Neef 's hammer, attached to the frame which supported the reservoir. Perhaps an electrically maintained tuning-fork is still better. Magnus shewed that the most important part of the effect is due to the forced vibration of that side of the vessel which contains the orifice, and that but little of it is propagated through the air. With respect to the limits of pitch, Savart found that the note might be a fifth above, and than an octave below, that proper to the jet. Accor *' there is no well defined lower limit ; while, OJ external vibration cannot be efficient if it te» ^ Pogg. Aw fio.] VIBRATIONS OF LOW FREQUENCY. 365 rfr vhose length is less thau the cii'cu inference of the jet. This gives K fiw the interval defining the upper limit tt : +'51, or about a fifth. I the case of Plateau's numbers (tt : i^SS) the discrepancy is a j^tle greater. The question of the influence of vibrations of low [uency is difficult to treat experimentally in consequence of I complications which arise fi-om the almost universal preseiice taooic overtones. It is evident that the octave, for example, ihe principal tone, though present in a very subordinate degree. nevertheless be the more important agent of the two in termining the behaviour of the jet, if its pitch happen to lie [the neighbourhood of that of maximum instability. In my own Kriments' tuning-forks were employed as sources of vibration, I in every case the behaviour of the jet on its horizontal course I examined not only by direct inspection, but also by the Ithod of intermittent illumination (§ 42) so arranged that Sbere was one view for each complete period of the phenomenon to be observed. Except when it was important to eliminate the octave as far as possible, the vibration was communicated to the reservoir through the table on which it stood. The forks were either screwed to the table and vibrated by a bow, or maintained electrically, the former method being adequate when only one fork was required at a time. The circumstances of the jet were such that the pitch of maximum sonaitiveness, as determined by calcu- lation, was 259, and that fonning the transition between stability and instability 372. With pitches varjing downwards from 370 to about 180, the observed phenomena agreed perfectly with the unambiguous pre- dictions of theory. B'rom the point — decidedly below 370 — at which a regular effect was first observed, there was always one drop for each complete vibration of the fork, and a single stream, each drop breaking away under precisely the same conditions as its predecessor. After passing 180 it becomes a question whether rii./ octave of the fork's note may not produce an effect as well as ■ )••: prime. If this effect be sufficient, the number of drops is 'loubled, and when the prime is very subordinate indeed, there is 'noble stream, alternate drops breaking away under different r^poa and (under the action of gravity) taking sensibly 366 FILES OF DROPS [Stll different courses. In these experiments the influence of tk' prime was usually sufficient to determine the number of dzofil even in the neighbourhood of pitch 128. Sometimes* hoven^ the octave became predominant and doubled the number of dnfi When the octave is not strong enough actually to doaUe Ai drops, it often produces an effect which is very apparent to ■ observer examining the transformation throug^h the revoliiiig holes. On one occasion a vigorous bowing of the fork, widek favours the octave, gave at first a double stream, but this ate a few seconds passed into a single one. Near the point d resolution those consecutive drops which ultimately coalesce m the fork dies down are connected by a ligament. If the octm is strong enough, this ligament subsequently breaks, and (he drops are separated ; otherwise the ligament draws the half-fbiined drops together, and the stream becomes single. The transition from the one state of things to the other could be watched with facility. In order to get rid entirely of the influence of the octave a different arrangement was necessary. It was found that the desired result could be arrived at by holding a 128 fork in the hand over a resonator of the same pitch resting upon the table. The transformation was now quite similar in appearance to that effected by a fork of frequency 256, the only differences being that the drops were bigger and twice as widely spaced, and that the spherule, which results from the gathering together of the liga- ment, was much larger. We may conclude that the cause of the doubling of a jet by the sub-octave of the note natural to it is to be found in the presence of the second component from which hanlly any musical notes are free. When two forks of pitches 128 and 256 were sounded together, the single or double stream could be obtained at pleasure by varying the relative intensities. Any imperfection in the timing is rendered very evident by the behaviour of the jet, which per- forms evolutions synchronous with the audible beats. This observation, which does not require the aid of the strobosoopic disc, suggests that the effect depends in some degree upon the relative phases of the two tones, as might be expected a In some cases the influence of the sub-octav^ ' making the alternate drops unequal in xr *-cting them into very diSero^* DOUBLED B\' OVERTONES. 3G7 L Betumiug now to ihe case of a single fork screwed to the table, 9 found that as the pitch was lowered below 128, the double 1 was regularly established. The action of the twelfth (85J) Jdw the principal uote demands special attention. At this pitch I might expect the first three components of a compound note to Buence the result. If the thii-d component were pretty strong, ifould determine the number of drops, and the result would be ;-fold stream. In the case of a fork screwed to the table the Mimponent of the note must be extremely weak if not alto- gether missing ; but the second (octave) component is fairly strong, and in fact determined the number of drops {190J). At the same time the influence of the prime (85^) is sufficient to cause the iiltemate drops to pursue dififerent paths, so that a double stream is obser\'ed. By the addition of a 25G fork there was no difficulty in obtaiuing a triple stream ; but it was of more interest to examine whether it were possible to reduce the double stream to a single one with only 85J drops per second. Id order to secure as stroDg and as pure a fundamental tone as possible and to cause it to act upon the jet in the most favourable manner, the air space in the reservoir (an aspirator bottle) above the water was tuned to the note of the fork by sliding a plate of glass over the neck so as partially to cover it (§ SOS). When the fork was held over the resonator thus formed, the pressure which eicpels the jet was rendered variable with a frequency of 8.5J, and overtones were excluded as far as possible. To the unaided eye, however, the jet still appeared double, though on more attentive examination one set of drops was seen to be decidedly smaller than the other. With the revolving disc, giving about 85 views per second, the rtial slate of the case was made clear. The smaller drops were the spheruies, and the stream was single in the same sense as the streams given by pure tones of frequencies 128 and 256. The increased size of the spherule is of course to be attributed to the grc&ter length of the ligament, the principal drops being now thrt-e a widely spaced as when the jet is under the influence of p S56 fork. ffjth still graver forks screwed to the table the number of i continued to correspond to the second component of the B double octave of the principal uote (64) gave 1 28 drops ) influence of the pvirae -Hoa wj feeVit "Oaa^ "OMfc 3(38 bell's experiments. [3{l| duplicity of the stream was only just recog^sable. Below 64tk observations were not carried, and even at this pitch attempUte] attain a single stream of drops were unsuocessful. 362. Savart 8 experiments upon this subject have been fxn&a developed by Mr C. A. Bell, who shewed that a jet may be made to play the part of a telephonic receiver^ The external vihntiMi may be conveyed to the nozzle through a stringy telephone (§156i^ An india rubber membrane, stretched over the upper end of i metal tube, receives the jet and communicates the vibration die to the varying impact to the cavity behind, with which the €tf may be connected. The diameter and velocity of the jet requR to be accommodated to the general character as to pitch of tbe sounds to be dealt with. " When the membrane is held dose under the jet orifice, no sound will be audible in the ear-piece ; bat as the receiving tube is gradually withdrawn alon^ the jet path, t sound will be heard corresponding in pitch and quality to the dis- turbing sound — provided, of course, that the jet is at such pressure as to be capable of responding to all the higher tones to which the disturbing sound may owe its timbre. The intensity of this sound grows as the distance between jet orifice and membrane is in- creased. Finally, while the jet is still continuous above the membrane, a point of maximum intensity and purity of tone will be reached ; and if the membrane be carried beyond this point the sound heard will at first increase in loudness, becoming harsh in character at the same time, and at a still lower point will de- generate into an unmusical roar. In the latter case the jet will be seen to break above the membrane." From the fact that small jets travelling at high speeds respcmd equally to sounds whose pitch varies over a wide range Mr Bell argues that Plateau's theory is inadequate, and he looks rather to vortex motion, dependent upon unequal velocity at the centre and at the exterior of the column, as the real cause of the phenomena presented by these jets. As an example of a jet self-excited, the interrupter of J^ may be referred to. In this case the machinery by effect is carried back to the nozzle is electria mechanical devices answer the purpose equally duction of a resonator, such as the fork of § 2F COLLISIONS BETWEEN DROPS. 369 e which may be made to take its place, if the telephone be inght in coutact with the nozzle, gives greater regularity to the 8, and usually allows also of a greater latitude in respect of It should not be forgotten that in all these cases of self- fctation a certain condition as to phase needs to be satisfied, Ji>p instance in the interrupter of § 23.i r, supposed to be working ■1, the platinum points be displaced through half the interval weeu consecutive drops, it is evident that the action will cease Stil some fresh accommodation is brought about. 363. When a small jet is projected upwards in a nearly viTtical direction, there are complications dependent upon the iillisions of the drops with one another. Such collisions are iij.jvitable in consequence of the different velocities acquired by the drops as they break away irregularly (rom the continuous portion of the column. Even when the resolution is regularized by the action of external vibrations of suitable frequency, the drops must etill come luto contact before they reach the aummit of their parabolic path. In the case of a continuous jet the "equation of continuity" shews that as the jet loses velocity in aiscending, it must increase in section. When the stream consists of drops following the same path in single tile, no such increase of section is possible ; and then the constancy of the total stream demands a gradual approximation of the drops, which in the case of a nearly vertical direction of motion cannot stop short of actual contact. Regular vibration has, however, the effect of postponing the collisions and consequent scattering of the drops, and in the case of a direction of motion less nearly vertical may prevent them altogether. The behaviour of a nearly vertical fountain is influenced in an extraordinary manner by the neighbourhood of an electrified body. The experiment may be tried with a jet from a nozzle of 1 mm. diameter rising about 50 centime. In its normal state the jet resolves i|.itself into drojis, which even before pasi^ing the summit, and still e after pa».iiug it, aii? scattered through a considerable width. I feeblv electrilied body is presented to it, the jet undergoes ifonnation, and appears to become coherent; il electrical action the scattering becomes 1 effect is readily attributed 370 INFLUENCE OF ELECTRICITY. [SUl feeble electricity in producing apparent coherence depends upoii different principle. ' It has been shewn by Beetz^ that the coherence is appmtj only, and that the place where the jet breaks into drops is u' perceptibly shifted by the electricity. By screenings varioas piife with metallic plates connected to earth, Beetz further proved th^ contrar}' to the opinion of earlier observers^ the seat of senshife* ness is not at the root of the jet where it leaves the orifice, but il the place of resolution into drops. As in Lord Kelvin's water- dropping apparatus for atmospheric electricity, the drops cam away with them an electric charge, which may be collected h receiving them in an insulated vessel. It may be proved by instantaneous illumination that tk normal scattering is due to the rebound of the drops when they come into collision. Under moderate electrical influence there is no material change in the resolution into drops nor in the subse- quent motion of the drops up to the moment of collision. The difference begins here. Instead of rebounding after collision, as the unelectrified drops of clean water generally do, the electrified drops coalesce, and thus the jet is no longer scattered about*. An elaborate discussion of this subject would be out of place here. It must suffice to say that the effect depends upon a difference of potential between the drops at the moment of collision, and that when this difference is too small to cause coalescence there is complete electrical insulation between the contiguous masses. When the jet is projected upwards at a moderate obliquity, the scattering is confined to the vertical plane. Under these circumstances there are few or no collisions, as the drops have room to clear one another, and moderate electrical influence is without effect. At a higher obliquity the drops begin to be scattered out of the vertical plane, which is a sign that collisions are taking place. Moderate electrical influence will reduce the scattering to the vertical plane by causing coalescence of drops which come into contact. If, as in Savart's beautiful experiments, the resolution into drops is regularized by external vibrations of suitable freqi ' Pogg, Ann, vol. cxlit. p. 448, 1872. ^ The influence of Eleotrioity on Colliding Water IV zxTiix. p. 406, 1879. principal drops follow the same course, and unless the >jection is nearly vertical thei'e are no collisions between them, lut it aometimea happens that the spherules are thrown out hterally in a distinct stream, making a considerable angle with !&e main stream. This is the result of collisions between the ipherules and the principal drops. It may even happen that the brmer are reflected backwards and forwards several times until at aat they escape laterally. In alt cases the behaviour under beble electrical influence is a criterion of the occurrence of ^Uisions. In an experiment, due to Magnus', the Bphemles are diverted St>m the main stream without collisions by electrical attraction. Advantage may be taken of this to obtain a regular procession ?f drops finer than would otherwise be possible. ■ 364. The detached masses of liquid into which a jet is ireeolved do not at once assume and retain a spherical figure, but execute a series of vibrations, being alternately compressed and elongated in the direction of the axis of symmetry. When the itesolution is effected in a perfectly periodic manner, each drop is RL the same phase of its vibration as it passes through a given Jjoint of space ; and thence arises the remarkable appeai-ance of khernate swellings and contractions described by Savart. The interval from one swelling to the next is the space described by the drop during one complete vibration about its figure of equi- bbriiim, and is therefore, as Plateau shewed, proportional cceteris paribus to the square root of the head. The time of vibration is of course itself a function of the nature of the fluid (T, p) and of the size of the drop, to the calculation of which we now proceed. It may be remarked that the argument from dimensions is sufficient to shew that the time (t) of an infinitely small vibration of any type is proportional bo f/ipV/Tf), where V is the volume of the drop. In the mathematical investigation of the small vibrations of a tiquid mass about its spherical figure of equilibrium, we will Bonflne ourselves to modes of vibration symmetrical about an axis, frhich suffice for the problem in hand. These modes require for expression only Legendre's functions P„ ; the more genei 372 MATHEMATICAL THBOBY OF [S6i] problem, involving Laplace's functions, may be treated in Ai same way and leads to the same resulta The radius r of the surfieice bounding the liquid may be expanded at any time t in the series (§ 336) r-a, + aiPiOi) + ... + a»P»(/*)+ (1), where ai, a^... are small quantities relatively to a«, and /a repre- sents, as usual, the cosine of the colatitude (0)l For the volume (F) included within the Borbce (1) we have F = f^r J"" VdA* = JTro.' [1 + 32 (2n + l)-» OnVo,*] ....(2X the summation commencing at n =: 1. Thus, if a be the radios of the sphere of equilibrium, a = ao[l + 2(2n + l)-»anVa**] (3). The potential energy of capillarity is the product of the tension T and of the surface S, To calculate S we have For the first part I r^d^ = 2ao* + 22 (2n + 1)-» On^ For the second part The value of the quantity on the right may be found with the aid of the formula r+i dP dP r+i in which m is an integer equal to or different firom n. Thus = \ln (n + 1) a„« [ P^^dfi = ln(n^ 1){9h Accordingly iSf = 47rao* + 2w 2 (2n + 1)-* '-* «47ra« + 2ir2(n-l)(r by (S). 374 INSTABILITY DUE TO ELECTRICITY. [i cent, per second was broken up under the action of a fork 128 vibrations per second* Neglecting the mass of the spherules, we may take for the volume of each principal 19*7/128, or '154 cub. cent. Thence by (11), putting p=l 7=74, we have r « 0494 second. This is the calculated nhl By observation of the vibrating jet the distance between the ibs| and second swellings, corresponding to the maximum oblatenesi the drops, was 16*5 centims. The level of the contraction midisl between the two swellings was 36*8 centims. below the snrfiiMeaE the liquid in the reservoir, corresponding to a velocity of 2(1 1 centims. per second. These data give for the time of vibratioD T « 16-5/269 = 0612 second. The discrepancy between the two values of t is probably attrita- table to excessive amplitude, entailing a departure fix>m the law of isochronism. Observations upon the vibrations of dr(^ delivered singly from pipettes have been made by Lenard^ The tendency of the capillary force is always towards the restoration of the spherical figure of equilibrium. By electrifying the drop we may introduce a force operative in the opposite direc- tion. It may be proved' that if Q be the charge of electricity in electrostatic measure, the formula corresponding to (9) is ^.n<nrl){(„+2)r-£.} (^ It T> Q*/167ra', the spherical form is stable for all displace- ments. When Q is great, the spherical form becomes unstable for all values of n below a certain limit, the maximum instability corresponding to a great, but still finite, value of n. Under these circumstances the liquid is thrown out in fine jets, whose fineness, however, has a limit. Observations upon the swellings and contractions of a regularly resolved jet may be made stroboscopically, one view corresponding to each complete period of the vibrator ; or photographs may be taken by the instantaneous illumination furnished by a powerful electric spark'. 1 Wied. Ann, yol. xxx. p. 209, 1887. < Phil. Mag, toI. ziv. p. 184, 188S. * Some Applioaiions of Pfaotographj, Proe, Ray, 8oe. ImtL w&L zm " ^^1 ; Naturtt voL zliy. p. 849, 1891. ^•- In the mathemaCical investigations of this chapter no account * has been taken of viscosity. Plateau held the opinion that the - difference between the wave-length of spontaneous division of a ' jet {4-0 X 2a) and the critical wave-length (ir x 2a) was an effect ■ if viscosity; but we have seen that it ia sufficiently accounted for l>y inertia. The inclusion of viscosity considerably complicates the mathematical problem', and it will not here be attempted. The result is to shew that, when viscosity is paramount, long I hre.ads do not tend to divide themselves into dropa at mutual <iisitance8 comparable with the diameter of the thread, but rather to give way by attenuation at few and distant places. This appears to be in agreement with the observed behaviour of highly viscous threads of glass, or treacle, when supported only at the terminals. A separation into numerous drops, or a varicosity ]iointing to such a resolution, maj' thus be taken as evidence that the fluidity has been sufficient to bring inertia into play. A still more general investigation, in which tbe influence of electrification is considered, has been given by Basset'. ' Fhil. Mag. toI. hut. p. 1*5, 1892, > AvUT. Joum. qf Math. voL xvi. No. 1. CHAPTER XXI. VORTEX MOTION AND SENSITIVE JBTS. 365. A LARGE and important group of acoustical phenomeoft have their origin in the instability of certain fluid motions of the kind classified in hydrodynamics as steady. A motion, the same at all times, satisfies the dynamical conditions, and is thus in t sense possible; but the smallest departure from the ideal so defined tends spontcmeously to increase, and usually with great rapidity according to the law of compound interest. Examples of such instability are afforded by sensitive jets and flames, seolian tones, and by the flute pipes of the oi^n. These phenomena are still very imperfectly understood ; but their importance is such as to demand all the consideration that we can give them. So long as we regard the fluid as absolutely inviscid there is nothing to forbid a finite slip at the surface where two masses come into contact At such a surface the vorticity (§ 239) is infinite, and the surface may be called a vortex sheet. The existence of a vortex sheet is compatible with the djmamical conditions for steady motion ; but, as was remarked at an early date by v. Helmholtz^ the steady motion is unstabia The simplest case occurs when a plane vortex sheet separates two masses of fluid which move with different velocities, but without internal relative motion — a problem considered by Lord Kelvin in his investigation of the influence of wind upon waves'. In the following discussion the method of Lord Kelvin is applied to determine the law of falling away from steady motion in some of the simpler cases of a plane surface of separation. 1 Phil. Mag. toI. xxxn. p. 887, 1868. * PhU. Mag. toI. xuu p. 868, 1871. See also Proe. Math. 8oc. Tol. jl » 1878; Basaet'i Hydrodynamio, % ^^\, \%^\ XmebX? % H^earoa q |iittiifa>^\tt< : SLIP. : as Buppose that below the plane 2 = the 6uid is of at density p and moves parallel to x with velocity V, and bove that plane the density is p' and the velocity V. As >3, let s be measured downwards, and let there be rigid walls bounding the lower Huid sX z = l and the upper fluid at z = —l'. The disturbance is supposed to involve te and ( only through the factors e**", e'"'. The velocity potential (Fd.' + .J) in Wtie lower fluid satisfies Laplace's equation, and thus by the Ktndition at 2 = I takes the form ^=CcoHhfc(2-0c""'+»"' (1); hnd a similar expression, ^' = C'coshi(2 + i')-e'''"'**" (2), jdies to the lower fluid, if the whole velocity-potential be there f'ai+^'). The connection between and the elevation (h) at fee common surface is sothat,if /i = Ue^ini+iz, (3j_ kCsmhk-l = i(n + kV)H (4). In like manner, -kC siniikl' = i(n+kV')H (5). We have now to express the condition relating to pressures at 2 = 0. The general equation (2), § 244, gives for the lower fluid = -gk-wt}>-ikV<f>, squares of small quantities buing neglected. In like manner for the upper fluid at 2 = ^' = -gh-i,><f,'-iicr<t,'. I be no capillary tension, Sp ao( r tension be T, the difference is D that gip-p') h + k-'Tk = ip-in^-kV) 4 AND CAPILLARITY. When the values of ^, ^' at ^ = are introduced from (1). (2 (4), (5). the condition becomes p') + Ic'T = tp ( F 4 nik)' coth H + hp' ( V + n/Ar)* coth H' -in This is the equation which determines the values of h/X:. the roots of the quadratic are real, waves are propagated with ths corresponding real velocities; if on the other hand the roots a imaginary', exponential functions of the time enter into th^ solution, indicating that the (steady motion is unstable. Tbt criterion of stability is accordingly (p coth *■( + p' coth kf) {g {p - p') + Tle\ -kppco%]\kUoihkl'(V-Ty>Q (8). If g and T both vanish, the motion is unstable for all disturb- ances, that is, whatever may be the value of k: If T vanish, tha operation of gravity may be to secure stability for certain valuel of k, but it cannot render the steady motion stable on the wholQ For when k is infinitely great, that is, when the corrugations a infinitely fine, coth W = coth Ar = 1, and the term in g disappean fi-om the criterion. In spite of the impressed forces tending t stability the motion is necessarily unstable for waves of infini tesimal length ; and this conclusion may be extended to vort sheets of any form and to impressed forces of any kind. If T be finite, then on the contmry there is of necesatt stability for waves of infinitesimal length, although there may 1 instability for waves of finite length. For further examination we may take the simpler conditio! which arise wben I and V are infinite. The criterion of stabili^ then becomes (p + p-)[g(p-p-)+Tk^]~kpp'iV-Vy>Q (9). and the critical case is determined by equating the left- member to zero. This gives a quadratic in k. If the roota of tl quadratic are imaginary, the criterion (9) is satisfied for all inte mediate values of k, as well as for the infinitely small and i finitely large values by which it is satisfied in all cases, provid that p > p\ The condition of complete stability is thus 'ist.p-p' ... (I0> CHARACTER OF INSTABILITY. city (§ 353) of « Let W denote the mmimuin veU ^0, ^ = 0. Then by (7) ip+prW' = 4r^{p~p-)T (11), id (10) may be written 1 W'>^- ..(12). (15). If (V—V) do not exceed the value thus determined, the eodj motion is stable for all diiiturbancee ; otherwise there will i some finite wave-lengths for which disturbances increase ex- DDentially. If we now omit the terms in (7) dependent upon gravity and ion capillarity, the equation becomes p (.1 + kV)' coth kl + p'{n + kV)' coth kf = (13). When I = I', or when both these quantities are intiaite, we (ve simply p{n + kV)' + p'{n + kry = (14). » ^ pv+p'r + is/(pp').{v-r) k p+p' We see from (15) that, as was to be expected, a motion tnmon to both parts of the liquid has no dynamical significance. equal aildition to V and V is eqiiivalent to a deduction of [e amount from njk. Ifp=p', (15) becomes nlk.-i(.v+r)±inr-r) (m The essential features of the case are brought out by the nple case where V = —V. so that the steady motions of the two ksses of fluid are equal and opposite. We have then „llr=±iV (17); d for the elevation, A = ^e-*"cos(iyr+ e) (18). corresponding to h = Hcoe{iiii: + e) (19). tiatiy. If whfu ( = 0, d/'ldt = 0, h = HcoshkVtcos(nx-¥ e) (20), waves upon the surface of separation are tmplitude with the tiiae atya!."t»lss.^\» I I INSTABILITY OF JBT8 the law of the hyperbolic cosine. The rate of increase of the tero with the positive expooent is extremely rapid. Since k = 27r/Xi the amplitude is multiplied by e*, or about 23, in the occupied by either stream in passing over a distance X. If V = V, the roots (16) are er^ual, but the general solutioi may be obtained by the usual method. Thus, if we put F=r(l+a), where a is ultimately to vauish, and k = tf**-*^' f^el«f''- + 5«-i-**"-), where A, B are arbitrary constanta. Passing now to the limit where a = 0. and taking new arbitrary constants, we get or in real (juantitiea, k = {C + Dt] COB k(x-Vt + t). If initially h = coskx, dhjdt=Q, h = aosk{Vt-!i!)-^kYt&ak(yt-m) (21> " The peculiarity of this case is that previous to the displacemenl there is no real surface of separation at all. The general solution involving I and V may be adapted toj represent certain cases of disturbance of a two-dimensional jet o(| width 2i playing into stationary fluid. For if the disturbance h aymmetricai, ao that the median plane is a plane of symmetry, thol conditions are the same as if a fixed wall were there introduced^ If the surrounding fluid be unlimited. I' = x , coth if=l; the equation determining n becomes, if 1^ = 0, p' = p, (n + kV)'cothkl + n^ = (22), of which the solution is -1 ±i^{tRnhkl) '^'+ tanh kl -(23). 1 + tanh A/)' -(24). ^L wben v^(tanh kl) ..(25> S-] WITH FINITE SLIP. 81 i represents the progression of sjTnmetrical disturbances in a I of width 2/ placing into a stationary environment of the |ae density. If kl be very small, so that the wave-length is large in com- irison with the thickness of the jet, k = Iie''"^'>-^^ cos i:\x-Vt] (26). I The investigation of the asymmeti-ical disturbance of a jet iquires the solution of the problem of a single vortex sheet when the couflitioD to be satisfied at z = ^ is i^ = 0, instead of as hitherto d<f>idi = 0. The value of tf> is ^ ^ ' kcosU kl ..(27); from which, if as before d^'!dt='Q when z = — l', p(n + JfcV)'tanh/,-Z+p'(n ^kVycolhkV = ... (28). ui=x,p-=p, r=o, (n + kV)*t&nhkl + n^ = (2S). This is applicable to a jet of width 2/, moving with velocity V iu still fluid and displaced in such a manner that the sinuosities of its two surfaces are parallel. When kl is small, we have appro^ciraately h = He'^-""'-*'"cosk{x~kl. Vt) . . (30). By a combination of the solutions represented by (26), (30), we may deterraiue the consequences of any displacements in two dimensions of the two surfaces of a thin jet moving with velocity V in still fluid of its own density. 366, The investigations of § 365 may be considered to aflbrd an adequate general explanation of the sensitiveness of jets. In the ideal case of abrupt transitions of velocity, constituting vortex sheets, in frictionless fluid, the motion is always unstable, and the degree of instability increases as the wave-length of the disturb- iiuce diminishes. ' MlEa result to actual jets would lead veness increases indefinitely j^ certain flames, the M liar from the Thfl direct 382 TENDENCY OF VISCOSITY. [366. upper limit of human hearing; but there are other kinds of sensitive jets on which these high sounds are without effect, tad which require for their excitation a moderate or even a grm pitch. A probable explanation of the discrepancy readily suggests itselC The calculations are founded upon the supposition th&t the changes of velocity are discontinuous — a supposition that cannot possibly agree with reality. In consequence of fluid friction a surface of discontinuity, even if it could ever be formed, would instantaneously disappear, the transition from the one velocity to the other becoming more and more gradual, until the layer of transition attained a sensible width. When this width is comparable with the wave-length of a sinuous disturbance, the solution for an abrupt transition ceases to be applicable, €uid we have no reason for supposing that the instability would increase for much shorter wave-lengths, A general idea of the influence of viscosity in broadening a jet may be obtained from Fourier's solution of the problem where the initial width is supposed to be infinitesimal Thus, if in the general equations v and w vanish, while u is a function of y only, the equation satisfied by u is (as in § 347) di'pd^^ ^^^• The solution of this equation for the case where u is initially sensible only at y=0 is "=^'2V(^) <2)' where v = fi/p, and IT, denotes the initial value of Judy, When y« = 4i^, the value of u is less than that to be found at the same time at y = in the ratio e : 1. For air i/=16 cga, and thus after a time t the thickness (2y) of the jet is comparable in magnitude with 1'6^/t; for example, after one second it may be considered to be about 1^ cm. There is therefore ample foundation for the suspicion that ibf phenomena of sensitive jets may be greatly influenced hj H friction, and deviate materially from the results of oalw based upon the supposition of discontinuous changes €^ nder these oircaTDStaufieB \t W^xck»i^ msgrntsAt to ^■66.] GENERAL EQUATION. 383 sl the character of the equilibrium of stmtilied motion in cases more i nearly approachiug what is met wntb in practice. A complete JDveBtigatioti which should take account of all the effects of osity would encounter many formidable difficulties. For the sent purpose we shall treat the fluid as fHctionless and be ptent to obtain solutions for laws of stratification which are free 1 discontinuity. For the undisturbed motion the component jcities V, w are zero, and u is a function of y only, which we I denote by U. A curve in which U is ordinate and y is \ represents the law of stratification, and may be called for vity the velocity curve. The vorticity Z (§ 239) of the steady lotion is equal to jfdUfdy. If in the disturbed motion, assumed to be in two dimensions, uie velocities be denoted by U-t-u, v, and the vorticity by Z+^ the general equation (4), g 239, takes the form at ' dx dy in which dZldt = 0, dZjdx = 0. Thus, if the square of the disturbances be neglected, the equation may be written f.pf + .f.O (3); at d.v dy ' and the equation of continuity for an incompressible fluid gives 'Mr" '*^ If the values of Z and ^ in terms of the velocities be sub- Btituted in {li), (d ., d\idii dv\ . d^U _ We now introduce the supposition that as functions of j; a I and V are proportional to e*"" . c***. From {4) 1 if this value of u bo o if*") 384 CASE OF STABILITY. [SCl| In (7) A; may be regarded as real, and in any particiikl problem that may be proposed the principai object is to detenul the corresponding value of n, and especially whether it is rod t| imaginary. One general proposition of importance relates to tk case where d^U/d^ is of one sign, so that the velocity curfei wholly convex, or wholly concave, throughout the entire spM between two fixed walls at which the condition v >■ is sati^ Let n/k = p + %q, t;s=a + »/8, wherep,}, a,i9 are reaL SubstitatiBf in (7) we get or, on equating separately to zero the real and imagpnaiy parts, (ii^-*^*+ dy (p + irr+r ^^^ Multiplying (8) by /9, (9) by a, and subtracting, we get ad}a _d'^ d f^da _dfi\_iPU q(cf+/3*) .... ^df "df-dyV^dy 'dy)~di/' {p+Uy + ^"'^^'^'- At the limits v, and therefore both a and /8, are by hypothesis zero. Hence integrating (10) between the limits, we see that q must be zero, if d^U/dy^ is of one sign throughout the range of integration. Accordingly n is real, and the motion, if not abso- lutely stable, is at any rate not exponentially unstable. Another general conclusion worthy of notice can be deduced from (7). Writing it in the form we see that, if n be real, v cannot pass from one zero value to another zero value, unless d^Ujdy* and (n + fclT) be somewhere of contrary signs. Thus if we suppose that U is positive and d^U/dj/* negative throughout, and that Fis the greatest value <rf U, we find that n + kV must be positive. 367. A class of problems admitting of fairly simple solutioa is obtained by supposing the vorticity Z to be constant thnwy^ out layers of finite thickness and to change its Tslne m -it^ = 0.. ..(1). 17.] LAYERS OF 1 tssing a limited uiiniber ol' planes, fur each of which y is constant. > such cases the velocity curve is composed of portions of straight aes which meet one another at finite angles. This state of things supposed to be disturbed by bending the surfaces of transition, Throughout any layer of constant vorticity {?t/ydy' = 0, and ma by (7), §366, wherever n+kU ia not equal to zero, . . ^~ which the aohition is V = A^ + Be-''" (2). If there are several layers in each of which Z is constant, the uioua solutions of the form (2) arc- to be fitted together, the bitrary constants being so chosen as to satisfy certain boundary nditions. The first of these conditions is evidently the conti- Bi^ of V, or aa it may be expressed, A«=0 (3). The other necessary condition may be obtained by integrating '), § 366, across the surface of transition. Thus (r.- ")-©--©■"- <«■ These are the conditions that the velocity shall be continuous I the places where dUjdy changes its value. In the problems which we shall consider the fluid is either lunded by a fixed plane at which y is constant, or else extends I infinity. For the former the condition is simply v = 0. If lere be a layer extending to infinity in the positive direction, A iiist vanish in the expression (2) applicable to this layer; if a fer extend to infinity in the negative direction, the correspond- ig B must vanish. Under the first head we will consider a problem of some generality, where the stratified steady motion takes place between fixed walls at y = and at y = hi + b' + bi. The vorticity is constant throughout each of the three layers bounded by i/ = 0, y = i, ; y = bi, y = b, + b'; y = b, + b\ y = 6j+t' + 6, (Fig. 67). There are thus two internal surfaces where the vorticity changes. The values of U at these surfaces may be dt-noted by U,, U^. Fig, 07. B.U. 2fi 386 LAYERS OF UNIFORM VORTICITY. [367. In conformity with (3) and with the condition that v = when y = 0, we may take in the first layer i; = Vj = sinh ^'y (5); in the second layer i; = Vj = Vi H- 3fi sinh k{y — b^ (6) ; in the third layer t;=st;, = t;a4"itfs8inhA;(y — 6i — 6') (7). The condition that t; = 0, when y = 6i 4- 6' + ij, now gives = if, sinh kht + M^ sinh k (6, + 6') 4- sinh k (6, + 6' 4- 6i). . .(8). We have still to express the other two conditions (4) at the surfaces of transition. At the first surface V = sinh hbi , A {dvjdy) = kMi ; at the second surface v = ifi sinh kV + sinh k{hi + b% A {dvjdy) = kM^ . If we denote the values of A(dU/dy) at the two surfaces respectively by Aj, A,, our conditions become (n + kU,)M,-A,smhkbi=-0 (9), (n + ifc[7,)if,-A,{3fisinhAA'4-sinhA;(6i + 6')}=0...(10). By (8), (9), (10) the values of ifi, M^, n are determined. The equation for n is found by equating to zero the determi- nant of the three equations. It may be written ^n«4-J5n+(7 = (11), where il«sinhifc(6, + 6'4-6i) (12), B^k(V,+ U,)smhk(b^'^V'hb,) + At sinh kbt sinh k (bi + 6') 4- Ai sinh kbi sinh k (62 4- &')• • • (13), (7 = ifc«I7il7,sinhJfc(6,4-6'4-ii) 4- kUi^ sinh kb^ sinh k (bi 4- b') 4- kU^^i sinh kbi sinh k (6, 4- b') + AiAssinh&&i sinhA:6a sinh^*6' (14). To find the character of the roots we have to form the expression for B" — 4-4(7. On reduction we get B«-4ilC={ifc([7i-[7,)sinhJfc(6,4-6'4-6i) 4-Aisinhit&i sinhit(&a + 6')* ^^^^^t sinh A; (&i + &0}* ^41A sinh* ifefcx sinh* fcbt (16). |67.] FIXED WALLS. Hence if Aj, Aa have the same sign, that is, if the velocity |cur\'e (§ 366) be of one curvature throughout, S" — iAC is positive, I iffid the two values of n are real. Under these circumstances the ^disturbed motion is stable. We will now suppose that the surfaces at which the vorticity I changes are symmetrically situated, so that b,= b^=b. In this case we find tA = sinhk(2b + b') (16), ES =k{U,+ U,)Biiihki2b+b')+(X+iit)mahkb sinhfr(6+6')...(17), C=h'U,U,^ahk(ib + b')-\-k(UA,+ U^^,)smhkbsinhk{b + b') + AA8iiihU-(» ainhi-&' (18), -WC = 4AAsinb*A-& +{A-(rr,-y,)sinhfr(26+6')+(a,-A,)8iiiht6sinhi(6+6')l'"-(19)- Under this head there are two sub-cases which may be lecialty noted. The first is that in which the B-TBlues of U are the same on both sides of the ftmediau plane, so that the middle layer is a ■-legion of constant velocity without vorticity, ■and the velocity curve is that shewn in Fig. 68, We may suppose that tJ=V in the middle layer, and that (7= at the walls, without loss of generality, since any constant velocity ((/,) superposed upon this system merely alters n by Fig. 68. >tfae corresponding quantity —kU.,, as is evident from (7), § E Thus U,^U,= y, d, = A, = A=-7'/6; »Dd B'~^AC = 4 A' sinh' kb. HeUue ii-rn,' — ,- -■- uVToi ~ i'\ b siuh A-(2b + ti) As was to be expected, since the curvature of the velocity curve la of one sign, the values of n in (20) are real. It is ea^ pom the symmetry to see that the two normal disturbances are |Dch that the values of v at the surfaces of separation are either )qual or opposite for a given value of x. In the first case the jnrfacBs are bent towards the same side, and (as may be found 1 the equations or inferred from the particular case presently h1^ the corresponding value of n in (20) has the 388 LAYERS OF UNIFORM VOBTICITY. [367. upper sign. In the second case the motion is symmetrical with respect to the median plane which behaves as a fixed wait. If the middle layer be absent (6' = 0), one value of n, that corresponding to the Sjrmmetrical motion, vanishes. The remain- ing value is given by , ■- 2einh'A:& T''tanhA:& ,„., "+*''-rinrs b— (">• The other case which we shall consider is that in which the velocities U on the two sides of the median plane are opposite to one another ; so that U, = -0,= V, A,= -Ai--mT^ (22). Here B = 0. and C= - i»F* Binh k (26 + 6') - afc/iF" sinh jt* sinh k(b + b') — ^'F^sinh'ifct sinhW. For the sake of brevity we will write hb = ^, kb' = ff ; so that the equation for n becomes n' j!i'sinh(2j3+ff')+2A^Hinhgainh(jg+ff)+/t'6inh'jgsinh'ff' *fF*" jfsinh(2^ + ^) _ {p, sinhff sinh ^ + k sinh (g + ^)\* - k* sinh*g . f sinh ^ sinh <2j8 + ^) ^^^'■ Here the two values of n are equal and opposite ; and, since Ai, A) are of opposite signs, the question is open as to whether n is real or imaginary. It is at once evident that n is real if /i be positive, that is, if A, and V are of the same sign as in Fig. 69. Even when fi is negative, n» is necessarily positive for great values of k, that is, for small wave-lengths. For we have ultimately from (23) n~±kV. Fig.fiS- We may now inquire for what values of fi n* may be negt^MjU when k is very small, that is, when the wave-length is very n|^H Equating the numerator of (23) to zero, w ' hyperbolic sines, we get as a quadratic in M'6*6' + 2^(6 + y)+S" r&enoe /»»l/t, ot |b« 367.] FIXED WALLS. 389 When li lies between these limits (and then only), n* is nega- tive, and the disturbance (of great wave-length) increases expo- nentially with the time. We may express these reaults by means of the velocity V^ at the wail where y = 0. We have „i+i6' + £i.,h=V ' h + ^U ^). The limiting values of V., are therefore bVj^b' and 0. The velocity curve corresponding to the first limit is shewn in Fig, 70 by the line QPOP'Q', the point Q being found by drawing a line AQ parallel to OP to meet the wall in Q. If 6' = 26, QP is parallel to OA, or the velocity is constant in each of the extreme layers. At the second limit I in Fig. 71. 1^0 = 0. and the velocity curve is that Sie. 70. Fig. 71. It is important to notice that motions represented by velocity Mrves intermediate between these limits are unstable in a manner not possible to motions in which the velocity curve, as in Fig, 68, is of one cuirature throughout. According to the first approximation, the motion of Fig. 71 13 on the border-line between stability and instability for disturb- anceti of great wave-length; but, if we pursue the calculation, ^we find thftt ■■ ■- •v-l'- .■■r^fable Taking in (23) . _ 1 /, _ 2/fc', ..(25). r1. M^-l^)"^-'-? (2«)^ it is that represented in Fig. 70. If PQ be bent more downwards than is there shewn, as for example in Fig. 71, the steady motion is certainly unstable. Reverting to the general equations (11), (12), (13), (14), (15), let us suppose that A, = 0, amounting to the abolition of the corresponding surface of discontinuity. We get 5 = Jfc(I7i+t^,)sinhifc(6,4-6'4-6i) + AisinhJfc6i sinh ^• (6, + 6')» B»-4-4C«{ifc(t7|-l7,)8inhA;(6, + 6' + 60 + A, sinh kbi sinh k(b^ + 6')j« ; so that n^-kU^ (29), _ _irj Aj sinh kbi sinh k (6j + 6^ ^ ^ sinh k (6i 4- Vlt^b^) The latter is the general solution for t** vortidty of breadths bi and V+b%. ^"^ 'obtained by supposing in (II) i 390 LAYERS OF UNIFORM VORTICITY. [367.136] From the second form of (23) we see that, whatever may be the value of Jb, it is possible so to determine fi that the distuilK ance shall be unstable. The condition is simply that fju must ))c| fro between the limits ^ , sinh i (6 4 b') ± si nh kb sinhA;6 sinh ^6' ' |t| or -A;{cothfc64-cothiA*'}, -Jb {cothfc6 + tanhiA:6'}...(26), of which the first corresponds to the superior limit to the numeri- col valve of fi. When k is very lfiu:^e, the limits are very great and very dose. When A; is small, they become -1/6-2/6' and -1/6, as has already been proved. As k increases from to oo , the numerical value of the upper limit increases continuously from 1/6 + 2/6' to 00 , and in like manner that of the inferior limit from 1/6 to CO . The motion therefore cannot be stable for all values of k, if /uk (being negative) exceed numerically 1/6. The final condi- tion of complete stability is therefore that algebraically /A>-l/6 (27). In the transition case The occurrence of (29) suggests that any value of —kW is nissible as a value of n, and the meaning of this iw apparent 2 &oin the fundamental equation (7), § 366. For, at the place where n+ kU=0, (1) need not be satisfied, that is, the arbitrary con- stants in (2) may change their values. It is evident that, with the prescribed values of n aTid k, a solution may be found satisfy- - ing the required conditions at the walls and at the surfaces where tiU/dtf changes value, as well as equation (3) at the plane where n + kU=0. Id this motion an sidditional vorticity is supposed to be communicated to the fluid at the plane in question, and it moves with the fluid at velocity U. We may inquire what occurs at a second place in the fluid where the velocity happens to be the same as at the first place of added vorticity. The second place may be either within a layer of originally uniform vorticity, or upon a surface of transition. In the first case nothing very special presents itself. If there be no new vorticity at the second place, the value of v is definite aa usual, save as to une arbitrary muitiplyer. But, consistently with the given value of », there may be new vorticity at the second as well as al the first place, and then the complete value of ti for the given n may be regarded as composed of two parts, each propor- tional to one of the new vorticities and each affected by an arbitral^ muitiplyer. If the second place lie upon a surface of transition, it follows from (4) that v=0, since A{rff/"/rfy) is finite. From this fact we might be tempted to infer that the surface in question behaves like a fixed wall, but a closer examination shews that the inference would be unwarranted. In order to understand this, it may be well to investigate the relation between v and the displacement of libe surface, supposed also to be proportional to e'"' . e'**. Thus, if e equation of the surface be J' = j-Ae'''"*'^ = (31), e condition to be satisfied is' ilF ,.dF dF „ ,„,, -i, + U-j- + v-j:=0 (32), at ax ay -ih{n-i-km + v=0 (33) ' Lamb's ILj-lfHlyiw 392 LAYERS OF UNIFORM VORTICITY. [367. is the required relation. A finite h is thus consistent with an evanescent v. 368. In the problems of § 367 the fluid is bounded by fixed walls ; in those to which we now proceed, it will be considered to be unlimited. As a first example, let us suppose that on the upper side of a layer of thickness b the undisturbed velocity U is equal to 4- F, and on the lower side to — F, while inside the layer Fig. 72. Fig. 73. Fig. 74. it changes uniformly, Fig. 72. The vorticity within the layer is V/b, and outside the layer it is zero. The most straightforward method of attacking this problem is perhaps on the lines of § 367. From y = — oo to y = 0, we should assume an expression of the form Vi = ^, satisfying the necessary condition when y = — oo . Then fix)m y = to y = 6, v, = Vi + Ml sinh ky ; and irom. yas6toys= + Qo, v, = Vj + Mi sinh k(y— b). But by the conditions at +00,^, must be of the form e""*^, so that l+Mi + M^e-^^O. The two other conditions may then be formed as in § 367, and the two constants Mi, M^ eliminated, giving finally an equation for 71. But it will be more appropriate and instructive to follow a different course, suggested by vortex theory. If we write the fundamental equation ( in the form d^/dy'^kh)^Y (2), we see that, if F= from y = — 00 to y = + oo, then v = 0. Any value that v may have may thus be regarded as dependent ttf ^ and forther, in virtue of the linearity, as compounded by ouD <doii of the yaluoB con^i^Yidit^ to the partial tsIq 368.] INFINITELY EXTENDED FLUID. 393 In the applications which we have in view Y vanishes, except at certain definite places — the surfaces of discontinuity — where alone d'Uldy' differs from zero. The complete value of v may thus be found by summation of partial values, each corresponding to a single surface of discontinuity. To find the partial value corresponding to a surfece of dis- continuity situate at H = yi, we have to suppose in (2) that Y vanishes at all other places, while v vanishes at ± « - Thus, when y>yi, v must be proportional to e"**""'', and when y<yu V must be proportional to e+*"'-i'ii. Moreover, since v itself must be continuous at y = yi, the coefficients of the exponentials must be equal, so that the value may be written r=Ce'**-i'.' (3), when C is some constant. In the particular problem above proposed there are two surfaces of discontinuity, at y = and at y = b; and accordingly Foomplete value of v may be written in the form v = Ae"^-\-Be^'y-'" (4). We have now to satisfy at each surface the equation of condi- (4), § 367. When .y = 0. we have from (4) v, = A+ Be-"'. A {dvldy), = - 2kA. • while t7 = -7, A{dUldy) = + 2Vib; twheny = ii, Vi = Ae~>^+B. Aidvldy)t,~-2kB. 3 U = + V. A{dUjdy) = -2Vlb. 'he conditions to be satisfied by B : ^ and n are thus A{n-kr+rib]+B{Ve-"'/b]^Q (5). A\Ve-^lb\-B{>i + i:V-Vlb]=0 (6): which by elimination of B : A, n-=^^[(kb-iy-e-^] (7). Vhen kb is small, that is, when the waulJaBlirsil-'ewat in comparison with 6, the case approximateOf^^^^^^^^PiCfldon transition from the velocity — V t* from (7) 394 LAYERS OF UNIFORM VORTICITY. [368. in agreement with the value already found (17), § 365. In thk case the steady motion is unstable. On the other hand, when ti is great, we find from (7) n« = Jfc»F« (9); and, since the two values of n are real, the motion is stable. It appears, therefore, that so far from the instability increasiDg indefinitely with vanishing wave-length, as happens when the transition from — F to 4- F is sudden, a diminution of wave-length below a certain value is accompanied by an instability which gradually decreases, and is finally exchanged for actual stability. The following table exhibits more in detail the progress of bW/V^ as a frmction of kb : — kb bht^lV^ kb 6V/K=» •2 -4 -6 -8 - -03032 - -08933 - -14120 -•16190 1-0 1-2 1-3 2-0 1 - -13534 - -05072 + -01573 + -98168 We see that the instability is greatest when A;6 = "8 nearly, that is, when X = 8&; and that the passage frx)m instability to stability takes place when kb = 1'3 nearly, or X = 56. Corresponding with the two values of n, there are two ratios of B : A determined by (5) or (6), each of which gives a normal mode of disturbance, and by means of these normal modes arbi- trary initial circumstances may be represented. It will be seen that for the stable disturbances the ratio B : A is real, indicating that the sinuosities of the two surfaces are at every moment in the same phase. We may next take an example from a jet of thickness 26 moving in still fluid, supposing that the velocity in the middle of the jet is V, and that it falls uniformly to zero on either side, (Fig. 73). Taking the origin of y in the middle line, we may write [r=F(l:py/6) (10), in which the - sign applies to the upper, and the + sign to the lower half of the jet (Fig. 73). There are now three sur&c^ y =r — 6, y ss 0, y s + 6, at which the form of v suffSers diaoon^ 48 in (4) we may take 368.] INFINITELY EXTENDED FLUID. tio that, when M-A + Bri' + Ca--"'. Aid«ld!l)---2iA; y»0, U-r. l^i,dUldy)~-2Vlb. i).^e-'» + B + Oe-", A((;»/ij) = -2M; i)-jle-»' + a-" + C, Al(J»/cit/)»-2iC. be introduction of these values into the equations of condition p, § 367 gives mA+iB-i--fC = !> (12). ■)A + ^%-l,m-kb)B■^^C-0 (13), •i'A-t-tB+mC = (14), which are the equations determining A ; B\0 and n. By the symmetries of the case, or by inspection of (12), (13), (14), we aee that one of the normal disturbances is defined by B-0, A-i-C = 'S (16), and that the corresponding value of in is 7". Thus for the symmetrical disturbance "-"B*'-"""' <"**■ dicating atability, a« far as this mode is concerned. The general determinant of the system of three equations may ^ put into the form (m-7")[»«'+(y + 2A-6-3)m+7'(l + 2A6))=0...(17). I which the first fiictor corresponds to the symmetrical disturb- ! already considered. The two remaining values of n are , if (y + 2A6-3)'-47'(l + 2*A)>0 (18), t not otherwise. When it is infinite. t = 0, and (18) is satis- 80 that the motion is stable when the wave-length of rurbance ia small in comparison with the thickness (2 6) of the On the other hand, as may be proved without difficulty by J f, or e"*", in (18), the motion is unstable, when the [th is great in comparison with the thickness of tbe \it. 396 LAYERS OF UNIFORM VORTICITY. [368. The values of the left-hand member of (18) can be more eaaflj computed when it is thrown into the form (5 + 2Jfc6 - e-^)» - 16 (1 4- 2Jfc6) (19). Some corresponding values of (19) and 2kb are tabulated below:— 2kb (19) nb (19) •5 -•054 2-5 -•975 10 -•279 3^0 -•794 1^5 -•599 3-5 -•263 2^0 -•876 4 1 + •671 The imaginary part of n, when such exists, is proportional to the square root of (19). The wave-length of maximum instability is thus determined approximately by 2kb = 2*5, or X = 2'o x 26. The critical wave-length is given by 2kb = 3*5 nearly, or X = 1"8 x 2i, smaller wave-lengths than this leading to stability, and greater wave-lengths to instability. In these respects there is a fistirly close analogy with cylindrical columns of liquid under capillary force (§ 357), although the nature of the equilibrium itself and the manner in which it is departed from are so entirely different. One more step in the direction of generality may be taken by supposing the maximum velocity V to extend through a layer of finite thickness V in the middle of the jet (Fig. 74). In this layer accordingly there is no vorticity, while in the adjacent layers of thickness b the vorticity and velocity remain as before. Taking, as in (11), four constants A, B, 0, D to represent the discontinuities at the four surfaces considered in order, and writing 7 = 6"**, 7=e~*^', we have at the first surface [7=0, A(di7/dy) = +F/6, v = ^+7J5-f77'C + 7«7'A A (dv/dy) = - 2*4 ; at the second surface U^V, A(dUldy)^-V/b, t; = 7il + J5-h7'C4-ry'A A (dv/dy) ^ - 2kB ; at the third surface v«77'it + «^ B.] INFINITELY EXTENDED FLUID. 397 ^he fourth surface P = 0, A(dUldi/) = -i-V/b, v^'fy'A+yy'B + yC + D, A(rit;/rfy) = -2iA fang these values in (4) | 367, we get A{l + 2^IV]+yB + yy'C + 'fy-D = (20), yA + B{l-2b{k + nlV)]+y'0 + r/D = (21), yy-A + y'B+ C {1 -2b{k + nlV)]+'rD = (22), 7V^+77'fi + 7C+i>{l + 2frn/F|=0 (23). The elimination of the ratios A : B : C : D would give a bi- quadratic io n, which, however, may be split into two quadratics, one relating to symmetrical disturbances for which A + D = 0, B + = 0; and the other to disturbances for which A~D = 0, B—O=0, The resulting equation in n may be written ± y - 1 + 2H + y (1 :j: y :j: 2kby') = (24), In (24) the upper signs of the ambiguities correspond to the symmetrical disturbances. The roots are real, and the correspond- ing disturbances are stable, if {±y':fy'j' + -2nby-i[±y'-l+2kb + y^l:fy-:f2k-by-)]...i2b), be positive. ^B^ In what follows we will limit our attention to the symmetrical ^Hsturbances, that is, to the upper signs in (2o), and to terms of orders not higher than the first in &'. The expression (25) may then be reduced to (\~y'-2i:by + 2kb'{l+'^){l-y'~2kb) (26). Villi be very small, this becomes ik'b'-iikb'.H' (27). rft' IB zero (27) is positive, and the disturbance is stable, as we UtA before ; but, if b and b' be of the same order of magnitude I both small compared with X. it follows from (27) that the kurbance is unstable, although it be symmetrical. ftif in (24) we suppose that b' = 0, we fall back upon the euppo- 398 INFINITIES OCCURRING [368, sitions of the previous problem. For the symmetrical disturbance putting 7' = 1 in (24), we get shewing that the values of ibn/V are 7* — 1 and — 2A6. The former agrees with (16), and the latter gives nH-itT^=0. We have abready seen that any value of —A? fT* is a possible solution for n. If on the other hand we suppose that 6 s 0, we fistll back upon the case of a jet of uniform velocity V and thickness 6' moving in still fluid. The equation for n becomes, after division by 6*, n= + ( 1 ± 7') Jfc F . n + i ( 1 ± 7 ) i-" ^' = 0, or (n + A;F)«J-^ + w« = (281 In (28) ^^, = coth ^kb\ ^^ = tanh JA*' ; so that the result is in harmony with (22), (29), § 365, where I corresponds with ^6'. Another particular case of (24), comparable with previous results, is obtained by supposing b' to be infinite. 369. When cPU/dy^ is finite, we must fall back upon the general equation § 366 from which the cun^e representing t; as a function of y can theoretically be constructed when n (being real) is known. In fact we may regard (1) as determining the curvature with which we are to proceed in tracing the curve through any point. At a place when n + kU vanishes, that is, where the stream-velocity is equal to the wave-velocity, the curvature becomes infinite, unless V vanishes. The character of the infinity at such a place (suppose y = 0) would be most satisfactorily investigated by means of the complete solution of some particular case. It is, however, sufficient to examine the form of solution in the neighbourhood of y » 0. for this purpose the differential equation may be simplifi' 'lien y is small, n+ fcU may \i% \»t^\«A 369.] WHEN n + kU=0. ri-Ujdy^as appro simatelj' constant. In comparison with the large ti-'no, jftf* may be neglected, and it suffices to consider d'v/df + i/-'v=0 (2), a known constant multiplying y being omitted for the s;vke of brevity. This falls under the head of Ricati'a equation d?v!dy'' + y-v = Q (3), of which the solution is in general (hi fractional)' v = >Jy.\AJ„,{^) + BJ_^{^)\ (4), where m=l/(/i + 2), ^ = -2niif-'^ (5). When, aa in the present case, m is integral, J_„ (f) is to be replaced (I 34-1 ) by the function of the second kind Y„, (f). The general solution of (2) is accordingly v = ^y.[AJ,{i^/y) + BY,{_Uy)\ (C). In passing through zero y changes sign and with it the character of the functions, If we regard (6) as applicable on the positive side, then on the negative side we may ^v^^te r=^/y.|C/,(2^/y)^-i)^,(2^/^/)i (7), the ai'giiment of the function.^ in (7) being pure imaginaries. From the known fonns of the fiinctions (^Sil) wu may deduce, as applicable when y is .small, ^ +B!4(I-j + lji')-lo«(2Vy).(j-ij-) + j-}y-l (8); ^■R) that ultimately ■ "-i"' $-^-4»'»s». J'-'i-ifir' (9). ^^B remaining finite in any case. ^^H We will now shew that any value of —kU is an admissible ^^Rlae of n in (1). The place where n-\-kU = Q\B taken as origin ^^oT y; and in the first instance we will suppose that n-¥kU vanishes nowhere else. In the immediate neighbourhood of y = the solutions applicable upon the two sidea are (6), (7), and they are nbject to the condition that v shall be continuous. Heucc by ifl, Sluditn Bber die Betiel'tcheu Functiontn j 81. Leipsig, 186B; Qny ^J^rcfO"'. P- 383, 1895. 400 SENsmVE TtkiSES. (9), B = D, leaving three constants arbitrary. The which the functions start from y = being thtis ascertained, tlurl flirther progress is subject to the original et)UHtion (I), whui I completely definiis them when the three arbitraries are known, b I the present case two relations are given by the cooditions to le I satisfied at the fixed walla or other bomidaries of the fluid, MU I thus is determined the entire form of v, save as to a CMiKtan ' multiplyer. If B and D are finite, there is tnfuiite vorticity at lii' origin. Any other place-s at which n + i:U=0 may be treated in i similar manner, and the most general solution will cont^ain m many arbitrary constants as there are places of infinite vortidij. But the vorticity need not be infinite merely because n + k[l=9: and in fact a particular solution may be obtained with only oae infinite vorticity. At any other of the critical places, such fat exaniple as we may now suppose the origin to be, B and D t vanish, so that v = 0, (i'v/di/' =■ A, or C, From this discussion It would seem that the infinitiea whiej present themselves when n + lcU=0 do not seriously interfere v the application of the general theory, so long as the square of t disturbance from steady motion is neglected. A large part of the preceding paragraphs is taken from certtun papers by the author'. The reader should also consult Lo^J Kelvin's writings* in which the etTects of viscosity are dealt with,^| 370. It remains to describe the phenomena of seosJtii^| flames and to indicate, so far as can be done, the application ^H theoretical principles. In a sense the combination of flame ai^| resonator described in § 322 h may be called sensitive, but in th^| case it in rather the resonator to which the name attaches, tll^| office of the fiame being to maintain by a periodic supply of he^H the vibration of the resonator when once started. Followiq^f Tyndall, we may conveniently Umitjh^erni lo naked Hamea »^| jets, where the origin of the ^^^^BRv ^ nnduiiigflM|^^^| found in the instability which Wi vtirtei^B^^^^^^I The earliest obsorvatmn u ^M 1 Prm:. Math, Soe., vol. xr. p. aT, 18W. ^M k to commaoicate b aupplemect. ^H L I fhu. JToti. vol vat, iffi. va&,a:i^.\, ^1 401 ■ 370.] EARLY OBSERVATIONS. Leconle ', who noticed the jumping of the 8ame from an ordinary tishtail burner in response to certain notes of a violoncello. The sensitive condition denmnded that in the absence of sound the flame should be im the point of flaring. When the pressure of gas was reduced, the sensitivenesR was lost. An independent observation of the same nature drew the attention of Prof. Barrett to sensitive Hames; and he investigated the kind of burner best suited to work with the oi-dinary pressure of the gas mains ^ "It is formed of glass tubing about | of an inch (1 cm.) in diameter, contracted to an oritice -^ of an inch (■16 cm.) in diameter. It is very essential that this orifice should be slightly V-shaped.. ..Nothing is easier than to form such a burner; it is only uecessary to dmw out a piece of glass tubing in a gas flame, and with a pair of sciasoi's snip the contraction into the shape indicated." But the must striking by far is the high-pressure flame employed by Tyndall. The gaa is supplied from a special holder under a pressure of ^ay 25 cm. of water to a pinhole steatite burner, and the flame rises to a height of about 40 cm. Under the influence of a sound of suitable (verj- high) pitch the flame roars, and drops down to perhaps half its original height'. Tyndall shewed that the seat of sensitiveness is at the root of the flame. Sound coming along a tube is ineffective when presented to the flame a little higher up. and also when caused to impinge upon the burner below the place of issue. It is to Tj-ndall that we owe also the demonstration that it is not to the flame as such that these extraordinary elfects are to be ascribed. Phenomemi substantially the same are obtained when a jet of unigiiited gas, of carbonic acid, hydrogen, or even air itself, issues from an orifice under proper pressure. They may be rendered visible in two ways, By association with smoke the whole course of the jet may be made apparent; and it is found tbat suitable smoke jets can surpass even flames in delicacy. """ I notes here effective are of much lower pitch than those ! most efficient in the case of flames." Another way of jc the seasitivene.«s of an air-jet visible to the eye is to cause oflnence of Uiuiaftl Sounds on (be Pluno ot ft Jet ol Coat-gSB. Phil. 3SS, 185S. I. p. 216. 1^67. an. pp. 93, 375. iaC7 ; Sound, 8rd ed. ob. n. 402 VABICOSB OR SINUOUS ? T^^ it to impinge upon a flame, such at* a candle flame, which pUal merely the pari of an indicator. I In the sensitive flame of Prof, Uovi ■ and of Mr Barry ' thu p.- \ is unigntted at the burner, but catches fire on the further sid<- wire-gauze hold at a suitable distance. On the same prijicipii an arrangement employed by the author". A jet of coal gas frr: a pinhole burner rises vertically in the interior of a cavity frji; which air ia excluded. It then passes into a brass tube a (f inches long, and on reaching the top burns in the open. t\<' front wall of the cavity is formed of a flexible membrane of tl^u' - paper, through which external soundfi can reach the burner. In these cases the sensitive agent is the unignited part of the jci ' Used in this way a given burner requires a much less pn.«sun? il gas than is necessaiy when the flame is allowed tu reach it, ait>] the sounds which have the most influence are graver. Struck by the analogy between these phenomena and thi»i' of water-jets investigated by 8avait and Plateau, the earlier ob- servers seem to have leaped to the conclusion that the manner n^ disintegration was also similar — symmetrical, that is, about th' axis; and Prof. Leconte went so far as to deduce the existence '•{ A cohesive force in gases. A surface tension, however, r^uires <t very abrupt transition between the properties of the matter on the two sides, such as could have only a momentary existence when there is a tendency to mix, so that it appears extn-inuly imlikely that capillarity plays here any sensible jwrt. The question of the manner of disintegration, whether it be by gradually increasing varicosity or by gradually increasing sinuonty, is of the greatest importance, and the answer is still, perhaps, i^J some cases opeii to doubt. But that the latter is predominant ^H general follows from a variety of argimient& The necessity, ^H remarked by Barrett, for an unsymmetrical oriflcc points strui){^^| in this direction. The same conclusion is drawn by Ridotit* Gk^^| the results of some ingenioiis experiments. The latter obBen^^f found further that tishi ' '' 1 '' '■'-• Tiniuu^ a&tn^^l angle of jetB from tw< > ") ^H ■ Toritio, ^H k Mature, \q\~ ^^^^^^^^^^^^^^^^^^^^^H 370.J NODES AND LOOPS. 403 sensitiveneas depeudent upon the direction of the sound. If this direction lie in the plane of symmetry containing the flame (that perpendicular to the plane of the nozzles), there is nu response. Even in the case of the tall high-presaure flame from a pin- hole burner, where to all appearance both the nozzle and the flame (when undisturbed) are perfectly symmetrical, there is reason to believe that the manner of disintegration is sinuous, or iins^-m metrical. Perhaps the easiest road to this conclusion ia by L-xamining the behaviour of the flame when exposed to stationary -sonorous waves, such as may be derived by supei-posing upon direct waves from a source giving a pure tone the waves reflected perpen- dicularly from a flat obstacle, e.g. a sheet of glas.s. According to the analogy with capillary jets, an analogy pushed further than it will bear by most writers upon this subject, the flame should be excited when the nozzle is situated at a node, where the pressure varies most, and remain unaffected at a loop where the pressure does not vary at all. There was no diflSculty in proving experi- mentally' that the fact* arc precisely the opposite. The source of sound was a bird-call (§ 371), and the observations were made by moving the burner to and fro in front of the reflector until the positions were found in which the flame was least disturbed. These positions were very well defined, and the measurements shewed distances from the reflector proportional to the series of numbers 1, 2, 3, &c., and therefore correapouding to nodes. If the positions had coincided with loops, the distances would have formed a series proportional to the odd numbers 1, 3, 5, &c. The wave-length of the sound, determined by the doubled interval between consecutive minima, was 31 2 mm., corresponding to pitch /Jt". A few observations were made at the same time on the litions of the silences as estimated by the ear listening through a tube. As was to be expected, they coincided with the loops, bisecting the intervals given by the flame. When the flame was in a position of minimum eS'ect, aud the free end of the tube was >ae to the burner at an equal distance from the reflector, und heard was a maximum, and diminished when the the tube was displaced a little in either direction. It was ishod that the flame is affected where the ear would !ted. and vice versd. J. vol. yll. p. 1153, 1S79. — to I Posi 404 OBSTRUCTION PREJUDICIAL. [370. Flames from pinhole burners, which perform well in other respects, seem always to shew a marked difference according to the direction in which the sound arrivea If, while a bird-call ii in operation, the burner be turned steadily round its axis, two positions differing by 180° are found, in which there is little or no response. This peculiarity may sometimes be turned to acooant in experiments Thus after such an adjustment has been made that the direct sound has no effect, vigorous flaring^ may yet result from the impact of sound from the same source after reflection frx)m a small pane of glass, the pane being held so tiaX the direction of arrival is at dO"" to that of the direct sound, and this although the distance travelled by the reflected sound is the greater. Tjmdall' lays it down as an essential condition of complete success in the more delicate experiments with these flames, *' thai a free way should be open for the transmission of the vibrations from the flame, hachuards, through the gaspipe which feeds it. The orifices of the stopcocks near the flame ought to be as wide as possible/* The recommendation is probably better justified than the reason given for it. Prof. Barrett" attributes the evil effect of a partially opened stopcock to the irregular flow and consequent ricochetting of the current of gas from side to side of the pipe. In some experiments of my own * the introduction of a glass nozzle into the supply pipe, making the flow of gas in the highest degree irregular, did not interfere, nor did other obstructions unless attended by hissing sounds. The prejudicial action of a partially opened stopcock was thus naturally attri- buted to the production of internal sounds of the kind to which the flame is sensitive, and this view of the matter was confirmed by some observations of the pressure of the gas in the neighbour- hood of the burner. " In the path of the gas there were inserted two stopcocks, one only a little way behind the manometer junction, the other separated from it by a loug length of indiik rubber tubing. When the first cock was fully open, flame was brought near the flaring-point by adjusti distant cock, the sensitiveness to external soud* ^ Proe. Roy, In»t, toI. xn. p. 192, ISSS; NiUur- * PhU, Mag. yol. zxxm. p. 99, 1867. * PhiL Mag. voL xzxm. p. 288« 1867. PERIODIC VIEW. I the manometer indicated a pressure of 10 inches (25-4 cm.) f water. But when the distant cock stood fully open and the ^ustment was effected at the other, high sensitiveneBS could not I obtained ; and the reason was obvious, because the flame red without external excitation while the pressure was still an (254 cm.) short of that which had been borne without inching in the former arrangement. On opening again the eighbouring cock to its full extent, and adjusting the distant ne until the pressure at the manometer measured 9 inches (22-9 cm.), the flame was found comparatively insensitive." The most direct and satisfactory evidence as to the manner of disintegration is of course that of actual observation. Using a jet of phosphorus smoke from a glass nozzle and a stroboscopic disc, I was able (in 1879) to see the sinuosities when the jet was disturbed by a fork of pitch 256 vibrating in its neighbourhood '. Moreover by placing the nozzle exactly in the plane of symmetry between the prongs of the tuning-lbrk, it could be verified that the disturbance required is motion transverse to the jet. In this position there was but little effect ; but the slightest displacement led to an early rupture of the jet. " In order to exalt the sensitiveness of jets to notes of mode- rate pitch, I found the use of resonators advantageous. These may be of Helmholtz's pattern ; but suitably selected wide-moutb bottles answer the purpose. What is essential is that the jet should issue from the nozzle in the region of rapid reciprocating motion at the mouth of the resonator, and in a transverse direction. " Good results were obtained at a pitch of 256. When two forks of about this pitch, and slightly out of tune with one another, were allowed to sound simultaneously, the evolutions of the smoke- jot in correspondence with the audible beats were very remarkable. By gradually raising the pressure at which the smoke is supplied, manner usual in these experiments, a high degree of sitiveness may be attained, either with a drawn-out glass or with the steatite pinhole burner used by Tyndall. In tees (even at pitch 256) the combination of jet and resoua- d almost as sensitive to sound as the ear itself. 'iour of the sensitive jet does not depend upon the i8>." uftice is merely to render the effects more 406 SENSITIVE JETS OF easily visible. I have repeated these observacioas without smo^f by simply causing air-jetH from the same nozzlea to irapioge Bp^H the flame ofa caudle placed at a suitable distance. Id such "^tB aa has been pointed out by Tynilall, the flame acts merely B£ uM indicator of the condition of the otherwise invisible jet. Exa 1 without a resimator the sensitiveness of such jets to hi»i]i(; I sounds may be taken advantage of to fonn a pretty experiment ' "The combination of jet, resonator, and flame kHows soincttme! a tendency to speak on its own account ; but I did not succeed in getting a well-tmslained sound. Such as it is, the effect prot)^ corresponds to one observed by Savart and Plateau with water-} breaking up under the operation of the capillary tension and, wb resolved into drops, impinging upon a solid obstacle, such as t bottom of a sink, in mechanical connection with the nozzle fn which the jet originally issues. In virtue of the connexion, s regular cycle in the mode of disintegration is able, as it were, pnipagate itself," "In the hope of being able to make better observatic upon the transformations of unstable jets, I next hod reooc coloured water issuing under water. In this form the experime ia more manageable than in the case of smoke-jets, which a difficult to light, and liable to be disturbed by the sliglit« draught. Permanganate of potash was preferred as a colunrii agent, and the colour may be discharged by mixing with tl general mass of liquid a little acid ferrous sulphate. The je Were usually projected downwards into a large beaker or I of glass, and were lighted &om behind through a piece of groui glass. " The notes of maximum sensitiveness of these liquid jets v, found to be far graver than for smoke-jets or for flames. Fori vibrating from 20 to 50 times per second appeared to produce tl maximum effect, to observe which it is only necessary to bring tl stalk of the fork into contact with the table supporting the app( ratus. The general behaviour of the jet could be observed witboc stroboscopic appliances by c:ins>-~~ -^c li((nid in ^^^jtttR ^ vibrate from side to aid*.' umK* if griivi colour proceeding from llu lu jconn and more sinuous, and -.i liit 4* k ance of a rope bent back war ■^I()wed the proceaa u£ il\aUoj ^^^^ m LIQUID IN LIQUID. 407 [uencies of vibrational distiubance from 1 or 2 per second up iabout 24 per second, using electro-raaguetic intemiptoi-s to i intermittent currents through an electro- magnet which acted nn a soft-iron armature attached to the nozzle. At each stage 9 pressure at which the jet is supplied should be adjusted so a8 Kgive the right degree of aensitivt-ness. If the pressure be too kat, the jet flares independently of the imposed vibration, and I transforrnatious become irregular: in the contrary case the lenomena, though usually observable, ai'e not so well marked as uitable adjustment is made. After a little practice it is Bible to interpret pretty well what is seen directly; but in ■der to have before the eye an image of what is really going on. we must have recourse to intermittent vision. The best results are obtained with two forks slightly out of tune, one of which is used to effect the disintegration of the jet, and the other (by means of perforated plates attached to its prongs) to give an intennittent view. The difference of frequencies should be about one per second. When the means of obtaining uniform rotation are at hand, a stroboscopic disk may be substituted for the second fork'. "The carrj-ing out of these observations, especially when it is desired to make a drawing, is difficult unless we can control the plane of the bendings. In order to see the phases properly it is necessary that the plane of bendings should be perpendicular to the line of vision : but with a symmetrical nozzle this would occur only by accident. The difficulty may be got over by slightly nicking the end of the drawn-out glass nozzle at two opposite points (Barrett). In this way the plane of bending is usually rendered determinate, being that which includes the nicks, so that by turning the nozzle round its axis the sinuosities of the jet may be properly presented to the eye, "Occasionally the jet appears to divide itself into two parts imperfectly connected by a sort of sheet. This seems to corre- apond to the duplication of flames and smoke-jets under powerful toroUB action, and to be due to what we may regard as the 1 waves taking altemately different courses." "It has already been noticed that the notes appropriate to • far graver than for air-jets from the same nozzles. « original paper (PAi7, Man. fl. xvil p. 188, 1884) drawingB by Mn I. See alto Proc. itoy. /m(. *aL uu. p. 361, 18B1, Im R«fO> VABYINO VISCOSITY. Moreover, the velocities suitable in the former case are much !i I the latter. This difference relates not, aa might perhi be at liKt supposed, to the greater density, but to the smaller viscosity of the water, measured of course kinematically. It is not dtfBcult to eve that the density, presumed to be the same for the jet and surrounding 6uid, is immaterial, except of course in ss.. far as a denser fluid requires a greater pressure to give it assigned velocity. The influence of fluid viscosity upou these phenomena is explained in a former paper on the Htabiiity or Instability of certain Fluid Motions'; and the laws of dynamical similarity with regard to fluid friction, laid down by Prof. Stokes', allow us to compare the behaviour of one fluid with another. The dimensions of the kinematic coefficient of viscosity are those of an area divided by a time. If we use the same nuzzle in both cases, we must keep the same standarti of length ; and thus the times must be taken invereely, and the velocities directly, as the co- efficients of viscosity. In passing from air to water the pitch and J velocity are to be reduced some ten times. But, in spite of the smaller velocity, the water-jet will require the greater pressure behind it, inasmuch as the densities differ in a ratio exceeding 100 ; 1." Guided by these considerations, I made experiments to ti] whether the jets would behave differently in warm (less viscous] water, and as to the effect of substituting for water a mixture ( alcohol and water in equal parts, a fluid known to be more visconi than either of its constituents. The effect of varying the viscosity was found to be very distinct. A jet which would not I a pressure of more than 1 inch (63 cm.) of water without flarinf when the liquid was water at a temperature under the boilinjf point required about 25 inches (63 cm.) pressure to make it flai when the alcoholic mixture was substituted. The importauct of viscosity in these phenomena wa.i thus abundantly established. The manner in which viscosity opei-ates is probably as followi At the root of the jet, just after it issues from the nozzle, there il a near approach to discontinuous motion, and a high degr« of instability. If a disturbance of sufficient intensity and i " Uath. Sue. Proc. Feb. 12, 18B0. See i 366. 9 Camli. Phil. Tram. 1850, " On the EQect of Internal FrictioD of Fluids ou tl Motion of PendalnniB," 9 G. 3ee also Hetmboltz, Ified. Ann. 6d. Tii. p. 337 (ls7t orBeprlat, vol. i. p. 991. *V0.] bell's experiment. 409 >%Vtable period have access, the regular luotion is lost and cannot t^lerwards be recovered. But the instability has a very short ■imf in which tu produce its effect. Under the influence of viscosity - lii' changes ol" velocity become more gradual, and the instability it-creases rapidly il' it does not disappear altogether. Thus if the ilisturbance be insufficient to cause disintegration during the fc>rief period of instability, the jet may behave very much as bhough it had not been disturbed at all, and may reach the full [jevelopement observed in long flames and smoke -jets. This temporary character of the instability is a second feature differ- entiating strongly these jets from those of Savart, in which capillarity has au unlimited time of action. When a flame is lighted at the burner, there are further complications of which it is difhcult to give an adequate explanation. The high temperature leads indeed to increased viscosity, and this tends to explain the higher pressure then admissible, and the graver notes which then become operative. But it is probable that the change due to ignition is of a still more fundamental nature. An ingenious method of observation, due to Mr C. Bell', may be applied so as to give valuable information with regard to the disintegration of jets ; but the results obtained by the author are not in harmony with the views of Mr Bell, who favours the symmetrical theory. In this method a second simitar nozzle faces directly the nozzle fi-oui which the air issues, and is con- nected with the ear of the observer by means of rubber tubing. Suitable means are provided whereby the position of the hearing nozzle may be adjusted with accuracy, both longitudinally and laterally. When the distance is properly chosen, small disturb- ances acting upon the jet are perceived upon a magnifled -'tcale. Thus a fork vibrating feebly and presented to the jet is loudly heard ; and that the Liffoct is due to the peculiar properties of ihu jet is proved at once by cutting off the supply of air, when • VQUnd becomes feeble, if not inaudible. Mr Bell proved that fficacy of the arraogemeut re'iuire.s a, anuill area in the ■ Utter be large enough to receive the whole f the jet, compaa-atively little is heard. »ir-jil I'lmui a well-regulated 'ipiuged Upon a similar 410 BIRD-CALLS. [376. hearing nozzle. It was excited by forks (c' or c") held in the neighbourhood. If the position of the fork was such that the plane of iti prongs was perpendicular to the jet, and. that the prolongation of the axis of the stalk intersected the delivery end of the nozde, the sound perceived was much less than when the fork was displaced laterally in its own plane so as to bring the noz^ nearer to one prong. This appears to prove that here again the effect is due, not to variation of pressure, but to transverse motiim, causing the jet to become sinuous. Confirmatory evidence may be drawn from observations upon the effect of slight movements of the hearing nozzle. When this is adjusted axially, but little is perceived of the fundamental tone of a fork presented laterally to the jet nozzle, but the octave tone is heard and often very strongly. When, however, the hearing nozzle is displaced laterally, the fundamental tone of the fork comes in loudly. 371. In that very convenient source of sounds of high pitch, the "bird-call," a stream of air issuing from a circular hole in a thin plate impinges centrically upon a similar hole in a parallel plate held at a little distance. The circumstances upon which the pitch depends have been investigated by Sondhauss^ but much remains obscure as regards the manner in which the vibrations are excited. According to Sondhauss. the pitch is comparatively inde- pendent of the size and shape of the plates, varying directly as the velocity (v) of the jet and inversely as the distance (d) between the plates. If we assume independence of other elements, and that the frequency (n) is a function only of «, rf, and 6 the diameter of the jet, it follows from dynamical similarity that n^vjd.fihld) (1), where / is an arbitrary function. Thus, if hjd be constant, Sondhauss' law must hold. From the very small dimensions employed it might fairly be argued that the action must be nearly independent of the velocity of sound, and therefore {v beinir ~" of the density of the gas; but the question viscosity may not be an elen ^ Pogg. Am EXTREME PITCH. 411 I geometrical slmilaiity is maintained (b proportional to d), I theoretical form, when viscosity is retained, is n = v/d.F{i'lvd) (2), King the kinematic coefficient of viscosity, of dimensions 2 npace and — 1 in time. But wlien we take a numerical example, Appears improbable that the degree of viscosity can play much [rt in determining the pitch. In c.o.s. measure (»=16 for air; 1 if the pressure propelling the jet be 1 cm. of mercury, v = (cm./sec.). Thus, if we take d = -l cm., we have vjvd = 0004, Rthat Fivjvd) could hai-dly differ much from jF(0). Bird-calls are very easily made. The first plate, of 1 or 2 cm. in diameter, is cemented, or soldered, to the end of a short supply lube. The second plate may conveniently be made triangular, the turned down corners being soldered to the first plate. For calls of medium pitch the holes may be made in tin plate, but when it is desired to attain a very high pitch thin brass, or sheet silver, is more suitable. The holes may then be as small as J mm. in diameter, and the distance between them as little ag 1 mm. In any case the edges of the holes should be sharp and clean '. In order to test a bird-call it should be connected with a well- regulated supply of wind and with a manometer by which the operative pressure can be measured with precision. When it is found to speak well, the pressure and corresponding wave- length should be recorded. If the tones are high or inaudible, a high-pressure sensitive flame is required, the wave-length being deduced from the interval between the positions in which a refiector must be held in order that the flame may shew the least disturbance (§ 370). There is no difficulty in obtaining wave-lengths (complete) as low as 1 cm., and with c:ire wave- lengths of (j cm. may be reached, corresponding to about 50,000 vibrations per second. lu experimenting upon minimum wave- lengths, the distance between the call and the flame ehould not exceed 50 cm., aud the fiame should be adjusted to the verge of flaring. In many cases a bird-call, which otherwise will not speak, may ^ made to do so by a reflecting plate held at a shoil distance in In practice the reflector is with advantage reduced to a A- M. Majrer haa eoofitnioted lieBulifullj Gaished bird.calls in whicli tlie ' ' ii adjaatAbU b; a screw motion. 412 ^OLTAN T0KE8. Htrip of metal, e.g. 1 era. wide ; and, when this assistanct; ui requind thu right distance ia an (even or odd) multiple of the half * length. In some cases the necesBary position of the strip ia tot sharpl}' detined, On the question whether the disturbance of the jet accom- panying the production of the sound is varicose or sinuous, i evidence may be derived from observations upon the nuuan which the sound radiates. Upon the latter view we might exftci that the sound would fall off, or even disappear alto^tber, in the axial direction, as happens, for example, in the case of the smnA radiated ^m a bell (§ 282). But, so far as I have been able i< observe, the sound emitted from a bird-call, speaking without tb< aid of a reflecting strip, is uniform through a wide angle ; and thi' fact may be regarded as telling strongly in favour of the view thai the disturbance is here symmetrical, or varicose, in characKi Other evidence tending in the same direction is afforded by the behaviour of resonating pipes made to speak with the aid of bird- calls. The pair of pei-fomted plates is moimted symmetrioally at one ond of a pipe 40 or 50 cm. long. The other end of the pipe li acoustically open, and a gentle stream of air is made to pass the bird-call, most easily with the aid of a very narrow tube ineierted into the open end and supplied from the mouth. By careful regu- lation of the force of the blast, the pipe may be made to speak in various harmonics, and the fact that it speaks nt all seems to aha^ that the issue of air through the bird-call is variable. ■ The manner of action is perhaps somewhat as follows, Whdl a symmetrical excrescence reaches the second plate, it is unable H pass the hole with freedom, and the disturbance is thrown bacn probably with the velocity of sound, to the Hrst plate, where 9 gives rise to a further disturbance, to grow in its turn during Clfl progress of the jet. But the elucidation of this and many kindrM phenomena remains still to be effected. I 372. ..^olian tones, as in the a'oliao harp, are generated whdfl wind plays upon a stretched wire capable of vibration at variod speeds, and their production also i" H^htless conDeo|MLiin(ii t]U instability of vortex sheuts. I' ntial, * '^| wire should partake in thi' vi' sp H has been investigated by Stm ^ i ' Wied. Ann. vol. T. p. Slli, laT- I .EOLIAN HARP. 413 I In Strouhal's experunents a vertical wire attached to a suitable ) was caused to revolve with iiuiform velocity about a parallel The pitch of the leolian tone generated by the relative rtion of the wire and of the air was found to be independent of i length and of the tension of the wire, but to vary with the meter (rf) and with the speed (w) of the relative motion. Vithin cert^n limits the relation between the frequency (n) and 3se data was expreasible by the centimetre and s ,( = ■185 v/rf (1). md being units. When the speed is such that the reolian tone coincides with one of the proper tones of wire, supported ao as to be capable of free independent vibration, the sound is greatly reinforced, and with this advantage Strouhal found it possible to extend the range of the observations. Under the more extreme conditions then practicable the observed pitch deviated sensibly from the value given by (I). He shewed further that with a given diameter and a given speed a rise of temperature was attended by a fall in pitch. Observations' upon a string, vibrating after the manner of the Kolian harp under the stimulus of a chimney draught, have shewn that, contrary to the opinion generally expressed, the vi- brations are effected in a plane perpendicular to the direction of the wind. According to (1) the distance travelled over by the wind during one complete vibration is about 6 times the diameter of the Bel If, as appears probable, the compressibility of the fluid may be ift out of account, we may regard n as a function of v, d, and v the kinematic coefficient of viscosity. In this case n is necessarily of the form ,> = vld.f(v/vd) (2), where / represents an arbitrary function ; and there is dynamical similarity, if f x nrf. In observations upon air at one temperature p i.t constant ; and, if d vary inversely as v, ndjv should be constant, a result fairly in harmony with the observations of Strouhat. Again, the temperature rises, v increases, and in order to accord with ration, we must suppose that the function /' diminishes with ig argument. 414 STROUHALS LAW. [37i Ad examination of the actual values in Strouhals experimeote shew that p/vd was always small ; and we are thus led to represent /by a few terms of Mac Laurin's series. If we take /{x) = o + 6a? + c«', we get (3). d d^ V(fi If the third term in (3) may be neglected, the relation between 71 and t; is linear. This law was formulated by Strouhal, and his diagrams shew that the coefficient b is negative, as is also required to express the observed effect of a rise of temperature. Further , dn ci^ (4). so that d . dnfdv is very nearly constant, a result also given by Strouhal on the basis of his measurements. On the whole it would appear that the phenomena are satis- factorily represented by (2) or (3), but a dynamical theory has yet to be given. It would also be of interest to extend the experi- ments to liquids. CHAPTER XXII. VIBRATIONS OF SOLID BODIES. 373. It is impossible in the present work to attempt anything approaching to a full consideration of the problems suggested by vibrating solid bodies ; and yet the simpler parts of the theory seem to demand our notice. We shall limit ourselves entirely to the case of isotropic matter. The general equations of equilibrium have already been stated in § 345. If p be the density, and a« = (/c + in)/^, b' = nlp (I), we have (a«-6»)^ + 6'v'a + -3r' = 0, etc (2), where X\ T\ Z' are the impressed forces reckoned per unit of iiiass. If from these we separate the reactions against accelei-ation, we obtain by D*Alembert*s principle £=<«'-^)£ + ^V'« + ^' (3), and two similar equations. In (8) S is the dilatation, related to a, y9, 7 according to B = da/dw{-dfildy'\'dyldz (4). If a, ^, 7, etc. be proportional to e*^S d^a/dt^^—p^a, and (3) becomes (a»-6»)^ + 6Va+i>'a + -3:'«0 (5). 416 PLANE WAVES [371 Differentiating equation (3) and its companions with leqwd to X, y, z, and adding, we obtain by (4) Similar equations may be obtained for the rokUions (compile § 239), defined by dy dz * dz dx * dx dy ^ ^ Thus, if we differentiate the third of equations (3) with respect to y, the second with respect to z, and subtract, d^^' L, 4 / , 1 dZ' - dV . ^-5- = 6»V'« + J^-i^ (8); and there are two similar equations relative to w'\ w'". It is to be observed that «•', «•", «•'" are subject to the relation dm/dx + dm''/dy'{'dv"/dz = (9). We will now consider briefly certain cases of the propagation of plane waves in the absence of impressed forces. In (6), if X\ y, Z vanish, and S be a function of x only, d?l\dff ^ a^d^Uda? (10), of which the solution is, as in § 245, S =/(a? - aO + ^(^ + at) (II). In this wave 8 = dajdx, while yS and 7 vanish ; so that the case is similar to that of the propagation of waves in a compressible fluid. It should be observed, however, that by (1) the velocity depends upon the constant of rigidity (n) as well as upon that of compressibility {ic). In the dilatational wave (11) the rotations w', «r", m'" vanish, as appears at once from their expressions in (7). We have now to consider a wave of transverse vibration for which S vanishes. If, for example, we suppose that a and ^ vanish and that 7 is a function of x (and t) only, we have 8 = 0, tsr' = tj'" = 0, 2m'' ^ ^ dfy/dx. The equation for w" is d'm'ldt^ = h^d^m^'lda^ (12), of the same form as (10) ; and the sa" ' ^ obtains for The transverse vibrat' velocity 6, a velbdV ->» 373.] LIMITED INITIAL DISTURBANCE. 417 The formation of stationary waves by auperposition of positive and negative progressive waves of like wave-leugth need not be dwelt upon. U k = iirjX, where \ is the wave-length, the super- position of the positive wave 7 = F cob k (bt — x) upon the negative wave 7 = r cos k (bt + x) gives 'f = 2Vcoskbt.co&kx (13), The second progressive wave may be the reflection of the first at a bounding surface impenetrable to energy. This may be either a free surface, or one at which 7 is prevented from varying. 374. The problem of the propagation in three dimensions of s. disturbance initially limited to a finite region of the solid was first successfully considered by Poisson, and the whole subject has been exhaustively treated by Stokes'. By (G), (8) § 373 the dila- tation and the rotations satisfy the equations d'S/d(> = a'V'8, d*mldf = b*V>-a (1), the solutions of which, applicable to the present purpose, have already been fully discussed in g 273, 274. It appears that distinct waves of dilatation aud distortion are propagated out- wards with different velocities, ao that at a sufficient distance from the source they become separated. If we consider what occurs at a distant point, we see that at first there is neither dilatation nor distortion. When the wave of dilatation arrives this effect commences, but there is no distortion. After a while the wave of dilatation passes, and there i.s an interval of no dilatation and no distortion. Then the wave of distortion arrives and for a time produces its effect, after which there is never again either dilatation or distortion. The complete discussion requires the expressions for the dis- placements in terms of h, w,, w„ «r,, for the derivation of which we have not space. From these it may be proved that before the arrival of the wave of dilatation and subsequently to the passage of the wave of distortion, the medium remains at rest. Between tJie two waves the medium is not absolutely undisturbed, although there is neither dilatation nor distortion. If the initial disturbances be of such a character that there is ^ wave of distortion, the whole disturbance is confined to the p of dilatation. »1 TLtoiy o( Diflriiction," Camh. PMl, Ttiim. Vo^.\i.■5.\,\Sl^a- 418 HARMONIC FORCE [37il 376. The subject of § 374 was the free propagation of mm resulting from a disturbance initially given. A problem al ktffc equally important is that of divergent waves ^ maintained \ij harmonic forces operative in the neighbourhood of a given centre. We may take first the case of a harmonic force of sadi a character as to generate waves of dilatation. By equaticm (6) § 373 we may suppose that at all points except the origin <( coordinates d«S/ctt« = o«V»S (1); or, if £ as a frmction of x, y, z depend upon r, or V(«' + y*+«^ only, and as a frmction of the time be proportional to e^, § 241, df^'^rTr^^'^-'' <«)' where h ^pjcu The solution of (2) is, as in § 277, « = — (3). In terms of real quantities ^__ il cos(p^ — Ar-f c) r in which A and e are arbitrary. (4), By transformation of (4) § 373, the relation between 5 and the radial displacement w may be shewn to be S = r-*d(i^w)/dr (5), or at a great distance from the origin simply S^dw/dr (6). Thus, when r is great, corresponding to (4) t(; = — r- ain{pt — hr + €) (7). In these purely dilatational waves the motion is parallel to the direction of propagation, and the symmetrical with respect to the origin. The theory of forced waves of dis from a centre is of stiU great- when the waves are due tA nuiiakMlN ^S|7S.] OPERATIVE AT A POINT. 419 a apace T at the origin. If we suppose in (8) § 373 that X', Y' nish, and that all the quantities are proportional to e*', we find W +A»«' +^b-^dZ-/di/ = (8), V'w"+f»r"-J6-'dZ7(tc = (9), V'nr'- + k-^-' =0 (10), I; l>eing written for p!b. These equations are solved a^ in § 277. We get w'" = 0, and Ldenoting the distance between the element at x, y, z near the 1 (0) and the point (P) under consideration, If we integrate ally with respect to y, we find "■ = -8^-//^'|('?)^'''^' <")• e integrated term vanishing in virtue of the condition that Z' is iiite only within the space T. Moreover, since the dimensions of T are supposed to be very small in comparison with the wave- length, d{r~'e~'^)jdy may be removed from under the integral sign. It will be convenient also to change the meaning of x, y, x, so that they shall now represent the coordinates of P relatively to 0. Thus, a Z' now stand for the mean value oi Z' throughout the space T, TZ- d (e-^\ y In like manner TZ' d ie-"^ X - -Si'W,W)-\ "»>■ and w"' = (14). In rirtue of the symraetry round the axis of z it suffices to ider points which lie in the plane ZX. Then or' vanishes, so ' rotation takes place about an axis perpendicular both to 'oropagation (r) and to that of the force (r). If Q ■ween these directions, the resultant rotation. 420 SMALL OBSTACLE. [37i If we confine our attention to points at^ a great distance, dui becomes simply ikTZ' 81X10 e-^ -^-OT---^ W The displacement, corresponding to (16), is perpendicular to r mi in the plane zr. Its value is given by or, if we restore the &ctor ^^, and reject the imaginaiy part of the solution, If Zi cos kbt denote the whole force applied at the origin, Z, = TZ\p (18). so that (17) may be written -2/.dr = -^-f22?M?Zir) (19^ The amplitude of the vibration radiated outwards is thus inverselj as the distance, and directly as the sine of the angle between the ray and the direction in which the force acts. In the latter direction itself there is no transverse vibration propagated. These expressions may be applied to find the secondary vibra- tion dispersed in various directions when plane waves impinge upon a small obstacle of density different from that of the rest of the solid. We may suppose that the plane waves are expressed by y = rcosk(bt-x) (20), and that they impinge at the origin upon an obstacle of volume T and density p\ The additional inertia of the solid at this place would be compensated by a force {p' — pY'i, or — (p — p)1^b^T coskbU acting throughout T\ and, if this force be actually applied, the primary waves would proceed without interruption. The secon- dary waves may thus be regarded as due to a force equal to the opposite of this, acting at parallel to Z, The whole amount of the force is given by ZiCosA*^ = (p'-p)ifc»6«rrcosifc6^ (21); so that by (19) the secondary displacement at a distant poiv (r, 6) is (p -p)ifc'rraing COS ib(^-r) 4rirp * T UNEAB SOURCE. 4isl ^e intensity of the scattered vibration is thus inversely as the mrth power of the wave-length (F being given), and aa the K{uare of the sine of the angle between the scattered ray and the rectiou of vibration in the primary waves. Thus, if the primary Biy be along a; and the secondary ray along £, there are no )ndary vibrations if (as above supposed) the primary vibrations B parallel to z ; but if the primary vibrations are parallel to y, fcere are secondary vibrations of full amplitude (flinff=l). and ! vibrations are themselves executed in a direction parallel 376. In I 375 we have examioed the effect of a periodic force )akbt, localized at the origin. We now proceed to consider case of a force uniformly distributed along an infinite line. Of this there are two principal sub-cases : the first where the *e, itself always parallel to z, is distributed along the axis of z, the second where the distribution is along the axis of y. In the first, with which we commence, the entire motion is in two dimensions, symmetrical with respect to OZ, and further ia auch that a and y9 vanish, while 7 is a function of (x' + y') only. If, as suffices, we limit ourselves to points situated along OX, sj', "" "' vanish, and we have only to find w". The simplest course to this end is by integration of the result given in (16) § 37-i. pTZ' will be replaced by Z„d£, the amount of the force distributed on rf; : r denotes the distance between P on OX and dz on OZ; 6 the angle between r and z. The rotation 0" about an axis parallel to 1/ and due to this element of the force is thus ikZudz ice"**' Swb'p r" In the integration x is constant, and r'' = x' + £', so that we have to consider K [ itr^dr j- le-^e-'^dh °'' }ix + h}.^{2j! + h).^/h ^''^• ..(1). ■ J Mr-. "we write r — x = h. > "On ihe Light from tfao fSkj, iu PoU'^' lU. pp. 107. 274, 1971 ; km aUo W gstioD of Ibe cuBO ohvie the obsUsI (rom the nmaiuder of Iti* meiUux 422 LINEAR SOURCK [37t From this integral a rigorous solution may be developed, kt, as in § 342, we may content ourselves with the limiting In assumed when kx is very great. Thus, as the equivalent of (^ we get x.^(2x)), VA x.^{2kx} ^^'' so that as the integral of (1) "^ 4w*V>/(2fcr)^ ^^^ From this 7 may be at once deduced. We have or, if we restore the time-factor, and omit the inoiaginary part of the solution, This corresponds to the force Z^ cos kht per unit of length of tiie axis of z. In virtue of the symmetry we may apply (6) to points not situated upon the axis of x, if we replace x by V'C^'+y')- That the value of 7 would be inversely as the square root of the distance irom the axis of z might have been anticipated from the principle of energy. The solution might also be investigated directly in terms of 7 without the aid of the rotations o. It now remains to consider the case in which the applied force, still parallel to ^, is distributed along OY, instead of along OZ, The point P, at which the effect is required, may be supposed to be situated in the plane ZX at a great distance R from and in such a direction that the angle ZOP is 0, In virtue of the two-dimensional character of the force, /8 = 0, while a, 7 are independent of y. Hence m\ m" vanish. But, although these component rotations vanish as regards the resultant effect, the action of a single element of the force Z^dy, situated at y, would be more complicated Into this, however, we need not enter, because, as before, the effect in reality depends only the elements in the neighbourhood of 0. Thus, in plac^ we may take ikZudff . sin e; S.] UNEAR OBSTACLE. 4^3 leing the distance between di/ and P, so that dylr = drly = dr/^(r--m. Wtitittg r - iJ = A, we get, as id (2), (3). (4), iTTfcV ^/(2/;fi)* *-°'' tad for the displacement, perpendicular to R, -''-■■'"-Wumrr"^ («> mce, corresponding to the force Z,i cos U>t per unit of length of {I axis of y, we have the displacement perpendicular to R at the W^w,"-*'"-^-^ ('»>■ 377. As in § -ITS, we may employ the results of § 376 to form expressions for the secondary waves dispersed from a small cylindrical obstacle, coincident with OZ and of density p', upon which primary parallel waves impinge. If the expression for the primary waves be (20) § 375, we have Zii = (p'-p)K^l^.Trd'.V 0), TTc' being the area of the cross section of the obstacle. Thus, if we denote ■>/(x'+y*) by r, we have from (6) §376 aa the expression of the secondary waves, k being replaced by its equivalent (2ir/X), In this case the lieeoondary waves are symmetrical, and their intensity varies in- iely as the distance and as the cube of the wave-length. The solution expressed by (10) § 376 shews that if primary raves ff = Bcosk(bt-x) (3) : upon the same small cylindrical obstacle, the displace- t perpendicular to the secondary ray, viz, r, will be 424 LINEAR OBSTACLE. [377. 6 denoting the angle between the direction of the primaiy lay (i) and the secondary ray (r). In this case the secondary disturbaooe vanishes in one direction, that is along a ray parallel to tk primary vibration. Returning to the first case, in which a and fi vanish through- out, while 7 is a function of x and y only, let us suppose tbat the material composing the cylindrical obstacle differs from ito surroundings in rigidity (n") as well as in density (p^ The conditions to be satisfied at the cylindrical surfisu^ are 7 (inside) = 7 (outside), n'drfldr (inside) = n dry/dr (outside). In the exterior space 7 satisfies the equation (§ 373) d^y/da^ + d^yjdf + ifc»7 = 0, where k = pjb ; and in the space interior to the cylinder 7 satisfies d^yjda^ + d»7/dy» + k'^y = 0, where k' =p/b' and V denotes the velocity of transverse vibrations in the material composing the cylinder. The investigation of the secondary waves thrown off by the obstacle when primary plane waves impinge upon it is then analogous to that of § 343, and the conclusion is that, corresponding to primary waves 7=rcos^(6<-a;) (5), the secondary waves thrown off by a small cylinder in a direction making an angle <? with x are given by which includes (2) as a particular case. 378. We now return to the fundamental problem, already partially treated in § 375, of the vibrations in an unlimited solid due to the application of a periodic force at the origin of coordi- nates. Equations (12), (13), (14) § 375 give the solution so fitf as to specify the values of the component rotations. If, as we shall ultimately suppose, the solid be incompressible, we have in addition £ s 0. On this basis the solution might be compki' but it may be more instructive to give an independent gafeion* ['8.] FORCE AT ONE POINT. 425 Since in the notation of § 373 X'=Y'=' 0, we have by (5) (a'-f)rf8/dc + i.'V'a + p»a = (1), (a'-6')dS/dy+i'V«y9+;,'jS = (2), ta'-6*)rfS/rf« + i'V'7 + ;)=T = -2' (3). Let us assume <i=-d?-^dxdz, = d'xldydz, y = d'xldi' + w (4), d accordingly S = d(V'x)lde + dw/d;s (5). The substitution of these values in (1) gives SO that (1) and (2) are satisfied if o»V'j^+p*;^ + (a'-6')Mj = (6). The same substitutions in (3) give or in virtue of (6) b'^'w + p'w+Z' = Q (7). By this equation w is determined, aud thence j^ by (6), In the notation of § 375, /.- = ;)/;., b=pja. Since Z" = at all points other than the origin, (7) becomes (S?' + k')w = (8), whence by (6) (V' + A') (V + yf) X = (9) is to be satisfied everywhere except at the origin. The solution of (9) is X-^^ + b'— (10), where A and B are constants. The corresponding values of w and S are by (6) and (5) «, = eA':^. S = -*'Bi(*-^') (11). To connect A and B with Z', we have from (7), as in g 37J> 4^6" , ///- -d.rfj,i-=,^i ^ 426 COMPLETE SOLUTION. [378. Again, by (6) § 373, V«S + A«S + a'^dZ'/dz « ; 80 that, as in § 375, 1 fffdZ^e^ Z^d(^ 4iroV jj dz r "^"'9'*' *na*pdz\ r J Thus, hy comparison with (11), "'^ ^-A;;^w-p — ;: — (i*> From the values of x ^^^ ^ ^^^^ ^1^7 determined oc. /9, 7 are found by simple differentiations, as indicated in (4). We have As the complete expressions are rather long, we will limit ourselves to the case of incompressibility (h^O). Thus, if we restore the time-factor (^f*) and throw away the imaginary part of the solution, we get o = — -^ [^(- 1 + j^) cos (p« - At) - ^ sin (p< - *r) - ^^ cos^J (17). the value of fi differing from that of a merely by the substitution of y for X, The value of il is given in (12), and Zicospt is the whole force operative at the origin at time t At a great distance from the origin (17), (18) reduce to _ Zi xz cos (pt — kr) n agreement with (Id'^^^l^. 378.] COMPARISON WITH HERTZ. 427 W. Konig' has remarked upon the iion -agreement of the complete solution (17), (18), firet given in a different form by 8tokes ', with the resulta of a somewhat similar investigation by Hertz', in which the terms involving cos pt, &in pt do not occur, and he seems disposed to regard Stokes' results as affected by error. But the fact is that the problems treated are essentially different, that of Hertz having no relation to elastic solids. The source of the discrepancy is in the first terms of (1) &c,, which are omitted by Hertz in his theory of the ether. But assuredly in a theory of elastic solids these terms must be retained. Even when the material is supposed to be incompressible, so that S vanishes, the retention is still necessarj', because, as was fully explained by Stokes in the memoir referred to, the factor (o*— 6') is infinite at the same time. If we suppose in (17), (18) that p and Ic are very small, and trace the limiting form, we obtain the solution of the statical problem of the deformation of an incompressible solid by a force localized at a point in its interior. 379. In § 373 we saw that in a uniform medium plane waves of transverse vibration « = 0. /3 = 0, y=Tcos(pt-kx) (1) may be propagated without limit. We will now suppose that on the positive side of the plane 3:= the medium changes, so that the density becomes p, instead of p, while the rigidity becomes ii, ^-instead of n. In the transmitted wave p remains the same, but k ^Us changed to k, , where M^ k,'/k' = npjn,p (2). ^KAHSuming, as will be verified presently, that no change of phase Bveed be allowed for, we may take as the expre&sions for the transmitted and reflected waves 7i = r, cos{/)( —kiJt), -y = r" cos (pt + kx) (3), that altogether the value of 7 in the first medium is = rcos(/)(-^-j) + r'cos(/)(+i-3-) (4), '^coB(;./-;',^) (5). i Worki. rol. n. p. Ma. 428 REFLECTION OF PLANE WAVES. [379. The coaditions to be satisfied at the interface (x =s 0), upon whiii no external force acts, are 7i = 7, fijdyjdx^ ndy/dx (6); so that r+r=r„ iiA;(r-r)=niJtir, (i\ If, as can plainly be done, F, Fi be determined in accordance with (7), the conditions are all satisfied. We have r ^ nk - n^fci V(np)-V(^pi) x«v r nk + fHk,^^(np)+^/{n,p,) ^ ^ r r V(np) + v(n,pO ^""^ by which the reflected and transmitted waves are determined The particular cases in which pi = p, or n, = /^, may be spedallf noted. When the incidence upon the plane separating the two bodies is oblique, the problem becomes more complicated, and divides itself into two parts according as the vibrations (always perpen- dicular to the incident ray) are executed in the plane of incidence, or in the perpendicular plane. Into these matters, which have been much discussed fix)m an optical point of view, we shall not enter. The method of investigation, due mainly to Green, is similar to that of § 270. A full account with the necessaiy references is given in Basset's Treatise on Physical Optics, Ch. xn. 380. The vibrations of solid bodies bounded by free surfaces which are plane, cylindrical, or spherical, can be investigated without great diflBculty, but the subject belongs rather to the Theory of Elasticity. For an infinite plate of constant thickness the functions of the coordinates required are merely circular and exponential*. The solution of the problem for an infinite cylinder^ depends upon Bessel's functions, and is of interest as giving a more complete view of the longitudinal and flexural vibrations of a thin rod. The case of the sphere is important as of a body limited in all directions. The symmetrical radial vibrations, purely tational in their character, were first investigated by PoiascA m ^ Proe, Land, Math. Soe. toL xvii. p. 4, 1885 ; vol. zx. p. 235. ^ s Pochhammer, CrelUf toL lzxxi. 1876 ; Chree, ^ lave'a Theory of Elottieity, ch, xtil. 380.] VIBRATIONS OF SOLID BODIfrS. 429 Clebsch '. The complete theory is due to Jaerisch * and especially to Lamb'. An exposition of it will be found iu Love's treatise already cited. The calculations of frequency are complicated by the exiateoce of two elastic constants k and n § 373, or (/ and ^ § 214. From the principle of § 88 we may infer, as Lamb has remarked, that the frequency increase.'* with any rise either of k or of n. for as appears from (1) § 345 either change increases the potential energy of a given deformation. 381', In the courae of this work we have had fi-equent occasion to notice the importance of the conclusions that may be arrived at by the method of dimensions. Now that we are in a position to draw illustrations from a greater variety of acoustical phenomena relating to the vibrations of both solids and fiuids, it will be convenient to resume the subject, and to develope somewhat in detail the principles upon which the method rests. ■ In the case of systems, such as bells or tuning-forks, formed of Httiiform isotropic material, and vibrating in virtue of elasticity, the Acoustical elements are the shape, the linear dimension c, the constants of elasticity (/ and *i (§ 149), and the density p. Hence, by the method of dimensions, the periodic time varies cfxteris paribus as the linear dimension, at least if the amplitude of vibra- tion be in the same proportiou ; and, if the law of isochronism be assumed, the last-named restriction may be dispensed with. In fact, since the dimensions of q and p are respectively [ML~^ r~'] and [ML~*], while /i. is a mere number, the only combination capable of representing a time is q~^ . p*.c. The argument which underlies this mathematical shorthand is t the following nature. Conceive two geometrically similar bodies, whose mechanical constitution at corresponding points is the , to execute similar movements in such a manner that the responding changes occupy times ' which are projwrlional to the ' Theorie dir Etailicim Ftiler Knrper, Leipzig, 1862, » CriUt, vol. (.iMTni. 1879, • FrtK. Lond. Mttli. Soe. vol, »in. p. 189, 1882. a appeared in the Ftiat Edilion as § 318. le oooeeption ol an alteration ot scale in upkce hat been raade ramiliar by s of map* and taodeta, but the correBpomltiig oouception for time Beference to the ease of a muHieal compoaitioQ perlonned si It the iaiaf iaaliou at the atudeat. 430 PRINCIPLE OF [881. linear dimensions — in the ratio, say, of 1 : n. Then, if the oat movement be possible as a consequence of the elastic forces, the other will be also. For the masses to be moved are as 1 : n*, the accelerations as 1 : nr^, and therefore the necessary forces aie as 1 : n' ; and, since the strains are the same, this is in Csict the ratio of the elastic forces due to them when refisTred to cone- sponding areas. If the elastic forces are competent to prodnee the supposed motion in the first case, they are also competent to produce the supposed motion in the second case. The dynamical similarity is disturbed by the operation of a force like gravity, proportional to the cubes, and not to the squares, of corresponding lines; but in cases where gravity is the sole motive power, dynamical similarity may be secured by a different relation between corresponding spaces and corresponding times. Thus if the ratio of corresponding spaces be 1 : n, and that of corresponding times be 1 : n^ the accelerations are in both cases the same, and may be the effects of forces in the ratio 1 : n' acting on masses which are in the same ratio. As examples coming under this head may be mentioned the common pendulum, sea-waves, whose velocity varies as the square root of the wave-length, and the whole theory of the comparison of ships and their models by which Froude predicted the behaviour of ships from experi- ments made on models of moderate dimensions. The same comparison that we have employed above for elastic solids applies also to aerial vibrations. The pressures in the cases to be compared are the same, and therefore when acting over areas in the ratio 1 : n', give forces in the same ratio. These forces operate on masses in the ratio 1 : n', and therefore produce accelerations in the ratio 1 : n~^ which is the ratio of the actual accelerations when both spaces and times are as 1 : n. Accordingly the periodic times of similar resonant cavities, filled with the same gas, are directly as the linear dimension — a very important law first formulated by Savart. Since the same method of comparison applies both to elastio solids and to elastic fluids, an extension may be made to systems ,^ into which both kinds of vibration enter. For example, tiie of a system compounded of a tuning-fork and of an air i" may be supposed to be altered without change in the m than that involved in taking the times in the r e linear dimenaiona. 381.] DYNAMICAL SIMILARITY. 431 Hitherto the alteration of scale has been supposed to be uniform in all dimensions, but there are cases, not coming under this head, to which the principle of dynamical similarity may be most usefully applied. Let us consider, for example, the flesural vibrations of a. system coniposed of a thin elastic lamina, plane or curved. By §§ 214, 21.i we see that the thickness of the lamina b, and the mechanical constants q and p, will occur only in the com- binations ql^ and bp, and thus a comparison may be made even although the alteration of thickness be not in the same proportion as for the other dimensions. If c be the linear dimension when the thickness is disregarded, the times muiit vary caiteris paribus as 5~* . p' . c" . b~\ For a given material, thickness, and shape, the times are therefore as the squares of the linear dimension. It must not be forgotten, however, that results auch as these, which involve a law whose truth is only approximate, stand on a different level from the more immediate consequences of the principle of similarity. CHAPTER XXIIL FACTS AND THEORIES OF AUDITION. 382. The subject of the present chapter has especial relatioD to the ear as the organ of hearing, but it can be considered only from the physical side. The discussion of anatomical or physio- logical questions would accord neither with the scope of this book nor with the qualifications of the author. Constant reference to the great work of Helmholtz is indispensable ^ Although, as we shall see, some of the positions taken by the author have be^ relinquished, perhaps too hastily, by subsequent writers, the im- portance of the observations and reasonings contained in it, as well as the charm with which they are expounded, ensure its long remaining the starting point of all discussions relating to sound sensations. 383. The range of pitch over which the ear is capable of perceiving sounds is very wide. Naturally neither limit is well defined. From his experiments Helmholtz concluded that the sensation of musical tone begins at about 30 vibrations per second, but that a determinate musical pitch is not perceived till about 40 vibrations are performed in a second. Preyer' believes that he heard pure tones as low as 15 per second, but it seems doubtful whether the octave was absolutely excluded. On a recent review of the evidence and in the light of some fresh experiments. Van Schaik' sees no reason for departing greatly from HelmholtifjB estimate, and fixes the limit at about 24 vibrations per seooodL 1 Tonempfindungen, 4th edition, 1877; Senaatiana of Ti" translated from the 4th German edition bj A. J. Ellis, this English edition, which is farther ftumished bor * notes. * PhyaiologUehe AbhanHimgen, Jf * Jreh. Nierl yoV. xxdl v* ^,U 384.] ESTIMATION OF PITCH. 433 On the upper side the discrepancies are still greater. Much «n doubt depends upon the intensity of the vibrations. In experi- ments with bird-calls (§ 371) nothing is heard above 10.000, Jilthough sensitive flames respond up to 50,000. Bni forks care- i'ully bowed, or metal bars struck with a hanimer, ajjpear to give rise to audible sounds of still higher frequencies. Preyer gives 20.000 as near the limit for normal ears. In the case of very high sounds there is little or no appreciation of pitch, so that for musical purposes nothing over 4000 need be considered. The next question is how accurately can we estimate pitch by the ear only ? The sounds are here supposed to be heard in succession, for (g 59) when two uniformly sustained notes are riouuded together there is no limit to the accuracy of comparison attainable by the method of beats. From a .series of elaborate experimeuts Preyer' concludes that at no part of the scale can '20 vibration per second be distinguished with certainty. The sensi- tivene&a varies vrith pitch. In the neighbourhood of 120, 4 vibration per second can be just distinguished ; at 500 about ".I vibration; and at 1000 about o vibration per second. In some cases where a difference of pitch was recognised, the observer could not decide which of the two sounds was the graver. 384. In determinations of the limits of pitch, or of the perceptible differences of pitch, the soimds are to be chosen of cii.uvenient intensity. But a further question remains behind as to the degree of inleasity at given pitch necessary for iiudibility. The earliest estimate of the amplitude of but ju,st audible sounds appears to be that of Toepler and Boltzmann'. It depends upon an ingenious application of v. Helmholtz's theory of the open organ- pipe (§ 313) to data relating to the maximum condensation within (he pipe, aa obtained by the authors experimentally (§ 322 d). They conclude that plane waves, of pitch 181, in which the ma ximum condensation (») is 6-5 x 10"*, are just audible. kit is evident that a superior limit to the amplitude of waves ID atidiblo sound may be derived fixim a knowledge of the ch must be expended in a given time in order to vft't ivurk was (jiven by A. .1. EUi» in the Procttdingt of the ^a, p. 1. 1877. 434 HINIUUM AMPLITUDE [3Bi | generate them and of the extent of surface over which the mm so generated are spread at the time of hearing. founded on these data will neceBsarily be too hig^h, both bean eound-waves must suffer some dissipation in their progren uit also because a part, and in some caaes a large part, of the enogj expended never takes the fono of sound-waves at all. In the first application of the method', the source of soimd was a whistle, mounted upon a Wolfe's bottle, in connection wilk which was a siphon manometer for the purpose of measuring tkt pressure of the wind. The apparatus was inflated from the \fap, and with a little practice there was no difficulty in maiDtainiDg t sufficiently constant blast of the requisite duration The moct suitable pressure was determined by preliminary trials, and vu measured by a column of water 9^ cm. high. The first point to be determined was the distance &om the source to which the sound remained clearly audible. The expeti- ment was tried upon a still winter's day and it was ascertained that the whistle could be heard without effort (in both directioiu) to a distance of 820 metres. The only remaining datum necessary for the calculation is the quwitity of air which passes through the whistle in a given time. This was determined by a laboratory experiment from which it appeared that the consumption was 196 cub. cents, per second. In working out the result it is most convenient to use con- sistently the c. G. s. system. On this system of measurement the pressure employed was 9^ x 981 dynes per sq. cent., aud therefore the work expended per second in generating the waves was 196 X 9i X 981 ergs'. Now (§ 245) the mechanical value of a series of progreoBiTe waves is the same as the kinetic energy <»f the whole mass of concerned, supposed to be moving with the maxinmm velocity (l of vibration; so that, if S denote the area of the wavt considered, a the velocity of sound, p the density of mechanical value of the waves passing in expressed by S.a.p.^if, in which the nutn< about 34100, and that of p about -OOIS. I tion S is the area of the sorfoce of a h^nii ' iV«e. iloy. Sot. ToL zztl p. f 384.] OF AUDIBLE SOUNDS. 435 82000 centimetres ; and thus, if the whole energy of the escaping air were converted into sound and there were no dissipation on the way, the value of r at a distance of 82000 centimetres would be given by the equation 2x196 x3^x981 whence = oou - 2Tr(82000)' x 34100 x 0013 ' « = '^ = 41xl0^. This result does not require a knowledge of the pitch of the sound. If the period be t, the relation between the maximum excursion a: and the maximum velocity v is x = vTJ2Tr. In the experiment under discussion the note was /"', with a frequency of about 2730. Hence ;00M. "^"2^x2730" = 81 X 10-»cm., or the amplitude of the aerial particles was less thau a ten- millionth of a centimetre. It was estimated that under favourable conditions an amplitude of 10~' cm. would still have been audible. It is an objection to the above method that when such large distances are concerned it is difficult to feel sure that the disturb- ing influence of atmospheric refraction is sufficiently excluded. Subsequently experiments were attempted with pipea of lower pitch which should be audible to a less distance, but theae were not successfiil, and ultimately recourse was had to tuning-forks. " A fork of known dimensions, vibrating with a known ampli- tude, may be regarded as a store of energy of which the amount may readily be calculated. This energy ia gradually consumed by internal friction and by generation of sound. When a resonator is employed the latter element is the more important, and in some 8 we may regard the dying down of the amplitude as auflSciently I &» by the eini-ssion of sound. Adopting this view for JT deduce the rate of emission of sonorous energy ^de of the fork at the moment in question e amplitude decreases. Thus if the 9 wnptitude of the fork, or e"" for :it time (, the rate at . ./(, or kE. The value ■ tr of d(.'cay, e.g. of the 436 MINIMUM AMPLITUDE [384. ments there is no difficulty in converting energy into sound upon a small scale, and thus in reducing the distance of audibility to such a figure as 30 metres. Under these circumstances the obm- vations are much more manageable than when the operaton are separated by half a mile, and there is no reason to fear distuil»Doe from atmospheric refraction. The fork is mounted upon a stand to which is also firmly attached the observing-microscope. Suitable points of light are obtained fit)m starch grains, and the line of light into which eadi point is extended by the vibration is determined with the aid d an eyepiece-micrometer. Each division of the micrometer-scak represents *001 centim. The resonator, when in use, is situated in the position of maximum effect, with its mouth under the fi'ee ends of the vibrating prongs. The course of an experiment was as follows : — In the first pbice the rates of dying down were observed, with and without the resonator, the stand being situated upon the ground in the middle of a lawn. The fork was set in vibration with a bow, and the time required for the double amplitude to fall to half its original value was determined. Thus in the case of a fork of frequency 256, the time during which the vibration fell from 20 micrometer-divisions to 10 micrometer-divisions was 16' without the resonator, and 9* when the resonator was in position. These times of halving were, as far as could be observed, independent of the initial amplitude. To determine the minimum audible, one observer (myself) took up a position 30 yards (27*4 metres) from the fork, and a second (Mr Gordon) communicated a large vibration to the fork. At the moment when the double amplitude measured 20 micrometer- divisions the second observer gave a signal, and immediately afterwards withdrew to a distance. The business of the first observer was to estimate for how many seconds after the signal the sound still remained audible. In the case referred to the time was 12*. When the distance was reduced to 15 yards (IS'7 metres), an initial double amplitude of 10 micrometer-divisions waB audible for almost exactly the same time. These estimates of audibility are not made without some difi* culty. There are usually 2 or 3 seconds during which the obsi is in doubt whether he hears or only imagines, and df individuals decide the question in opposite waya Tbf of course room fox a tc^ dA&^Ksctf^ ol \^»Karam^ ba& i OF AUDIBLE SOUNDS. 437 Tuded itself much. A given observer on a given day will often with himself aurpriaingly well, btit the accuracy thus (gested is, I think, illusory. Much depends upon freedom n disturbing noises. The wind in the trees or the twittering t birds embarrasses the observer, and interferes more or less with e accuracy of results. The equality of emission of sound in various horizontal direc- t was tested, but no difference could be found. The sound niea almost entirely from the resonator, and this may be expected ^ act as a simple source. When the time of audibility is regarded as known, it is easy to fieduce the amplitude of the vibration of the fork at the moment when the sound ceases to impress the observer. From this the rate of emission of sonorous enei^y and the amplitude of the aerial vibration as it reaches the observer are to be calculated. The first step in the calculation is the expression of the total energy of the fork as a function of the amplitude of vibration measured at the extremity of one of the prongs. This problem is considered in § 164. If / be the length, p the density, and w the _«ectional area of a rod damped at one end and free at the other, e kinotic energy T is connected with the displacement -ri at the e end by the equation (10) r= ^pita{d7]idty. i the moment of passage through the position of equilibrium "0 and dfjjdt has its maximum value, the whole enei^ being Inen kinetic. The maximum value of di/ldt is connected with the maximum value of 17 by the equation t if we now denote the double amplitude by 2)7, the whole ergy of the vibrating bar is jtpa>(7r'/T'.(2ij)', r the two bars composing the fork E^^ptol-n^l-r'.iiTir (A) e pml is the mass of each prong. The application of (A) to the S56-furk, %ibmting with a double bplitudo of 20 micrometer-divisiona, ia »< *" ' have l = l4,-0 cm., MK 1/t = 258. 438 MINIMUM AMPLITUDE [38t This is the whole energy of the fork when the actual doobk amplitude at the ends of the prongs is 050 oentim. As has already been shewn, the energy lost per second is kE, if the amplitude vary as er^. For the present purpose k must be regarded as made up of two parts, one k^ representing' the disspa- tion which occurs in the absence of the resonator, the other it, doe to the resonator. It is the latter part only which is effectife towards the production of sound. For when the resonator is out of use the fork is practically silent ; and, indeed, even if it were worth while to make a correction on account of the residual sound, its phase would only accidentally agree with that of the sound issuing from the resonator. The values of k^ and k are conveniently derived from the times, t^ and t, during which the amplitude falls to one half. Thus A = 21og.2./e, A:, = 21og.2./e,; so that Jfc, = 2 log. 2 . (1/t - l/t,) = 1-386 (l/t - l/t^y And the energy converted into sound per second is kJE. We may now apply these formulae to the case, already quoted, of the 256-fork, for which e = 9, e, = 16. Thus e,, the time which would be occupied in halving the amplitude were the dissipation due entirely to the resonator, is 20*6; and A:, = "0674. Accordingly, k^E^ 267 ergs per second, corresponding to a double amplitude represented by 20 micrometer- divisions. In the experiment quoted the duration of audibility was 12 seconds, during which the amplitude would fall in the ratio 2^ : 1, and the energy in the ratio 4^ : 1. Hence at the moment when the sound was just becoming inaudible the energy emitted as sound was 42*1 ergs per second ^ 1 It is of interest to compare with the energy-emission of a sooroe of lighit incandescent electric lamp of 200 candles absorbs about a hor8e-power« eigs per second. Of the total radiation only about y^ part acts the eye ; so that radiation of suitable quality consuming 5 x 10* corresponds to a candle-power. This is about 10* times that en* the fork in the experiment described above. At a dis^** metres, the stream of energy firom the ideal oanc'*' stream of energy jott audible to the ear. It ar zeqnired to influenoe the ^ye and the ear are < lOiMliifioii almdy dxawn \]V Toe|te •aadL^BtAtei P84.] OF AUDIBLE SOUNDS. 439 The question now remains, What is the corresponding ampli- tde or condensation in the progressive aeria! waves at 274 metres rom the source ? If we suppose, as in my former calculations, lat the ground reflects well, we are to treat the waves as bemi- pberical. On the whole this seems to be the best supposition to te, although the reflexion is doubtless imperfect. The area S fcvered at the distance of the observer is thus Stt x 2740' sq. tntim., and since <§ 245) S. tapi^=S.ipaV=421, ^= *2T trx 2740" X -00125 x 34100' ' fi = 60x10-*. lie condensation s is here reckoned in atmospheres } and the "■result shews that the ear is able to recognize the addition and subtraction of densities far less than those to be found in our highest vacua The amplitude of aerial vibration is given by asT/27r, where 1/t = 256. and is thus equal to 127 x 10"' cm. It is to be observed that the numbers thus obtained are still somewhat of the nature of superior limits, for they depend upon the assumption that all the dissipation due to the resonator repre- sents production of sound. This may not be strictly the case even with the moderate amplitudes here in question, but the uncertainty under this head is far less than in the case of resonators or organ- pipes caused to speak by wind. From the nature of the calculation by which the amplitude or condensation in the aeria) waves is deduced, a considerable loss of energy does not largely influence the final numbers. Similar experiments have been tried at various times with forks pf pitch 384 and 512. The results were not quite so accordant as Hum at first hoped might be the case, but they suflUce to fix with ^^■ime approximation the condensation necessary for audibility. The ^^Hnjmnlte are as follows: — 440 BINAURAL AUDITION. [i reserve ; so that the comparison must not be taken to prove mi more than that the condensation necessary for audibility varies slowly in the singly dashed octave*." Results of the same order of magnitude have been obt also by Wien', who used an entirely different method. 385. For most purposes of experiment and for many of ordinary life it makes but little difference whether we empkf one ear only, or both ; and yet there can be no doubt that we cut derive most important information fix>m the simultaneous use of the two ears. How this is effected still remains very obscure. Although the utmost precautions be taken to ensure separate action, it is certain that a sound led into one ear is capable d giving beats with a second sound of slightly different pitch led into the other ear. There is, of course, no approximation to sucK silence as would occur at the moment of antagonism were the two sounds conveyed to the same ear; but the beats are perfectly distinct, and remain so as the sounds die away so as to become single all but inaudible'. It is found, however, that combination tones (§ 391) are not produced under these conditions^ Some curious observations with the telephone are thus described by Prof. S. P. Thompson*. "Almost all persons who have experi- mented with the Bell telephone, when using a pair of instruments to receive the sound, one applied to each ear, have at some time or other noticed the apparent localization of the sounds of the telephone at the back of the head. Few, however, seemed to be aware that this was the result of either reversed order in the connection of the terminals of the instruments with the circuit, or reversed order in the polarity of the magnet of one of the receiving instruments. When the two vibrating discs execute similar vi- brations, both advancing or both receding at once, the sound is heard as usual in the ears ; but if the action of one instrument be reversed, so that when one disc advances the other recedes, and the vibrations have opposite phases, the sound apparently its place from the interior of the ear, and is heard as if pi from the back of the head, or, as I would rather say, from ^ Phil, Mag. Tol. jumu, p. 866, 1894. * Wied. Ann, vol. xxxyz. p. 884, 1889. * S. P. Thompson, Phil. Mag. toI. it. p. 874. ) * See also Dove, Pogg. Atm. vol. era, p. 6 * PHU Mtt9.^o\.^i>.«W^A«l^ tbe cwflbeHmn."..,.."! &miiged a Hughes's coicrciphoDe with two Us of a Falls's battery and tvo Bell telephones, ooe of them "*-^*Ting a cuiumatator uoder my cootroL Placing tbe telephones "■*!» my ears, I requested my a^istaut to tap on the wooden support ^^*^" tbe microphone. The result was deafening. I fclt as if simul- ■i-^^neous blows had been giren to the tjiupana of my eai*. But ***i reversing tbe current thD>ugh one telephone, I experienced a '^^■^aiisation only to be described as of some one tapping with a hammer *^»i the back- ufthe skull from the inaide" In oiu' estimation of the direction in which a sound comes to X:»s we are largely dependent upon the evidence afforded by bin- ^».ural audition. This is one of those familiar and instinctive *:»peration8 which often present peculiar difficulties to scientific ^Mialysis. A blindfold observer in the open air is usually able to indicate within a few degrees the direction of a sound, even though it be of short duration, such as a single vowel or a clap of the liands. The decision is made with con6dence and does not require » movement of the head. To obtain further evidence experiments were made with the approximately pure tones emitted from forks in assMiatiim with resonators; but in order to meet the objection that the first sound of the fork, especially when struck, might give a clue, and ao vitiate the experiment, two similar forks and resonators, of pitch 256, were provided. These were held by two assistants, between whom the observer stood midway. In each trial both forks were struck, and afterwards one only was held to its resonator. The results were perfectly clear. When the forks were to thi' right and to the left, the observer could distinguish them instinctively and without fail. But when he turned through a right angle, BO as to bring the forks to positions in front and behind him, no discrimination was possible, and an attempt to pronnunce wa« felt to be only guessing. That it should be impossible tn (HMtinguiwh whether a pure me comes from in front or from bi^hind 'n' iiiti llit^ilik- enough. 1 account of tbe symmetry the two ean would bi; atfectiid alike ^ to iJiii [HJHilion I original 442 HEAD AS AN OBSTACLE. P that the quality of a compound sound is liable to modification the external ear, which is differently presented in the two caso. The ready discrimination between right and left, even lAnj pure tones are concerned, is naturally attributed to the difeoll intensities with which the sound would be perceived by the t«| ears. But this explanation is not so complete as might be posed. It is true that very high sounds, such as a hiss^aiel heard vdth the averted ear; but when the pitch is moderate, <f 256 per second, the difference of intensity on the two sides dsa not seem very great. The experiment may easily be tried rongli^ by stopping one ear with the finger and turning round backiraidi and forwards while listening to a sound held steadily. Calcab* tion (§ 328) shews, moreover, that the human head, considered m an obstacle to the waves of sound, is scarcely big enough in relatioa to the wave length to throw a distinct shadow. As an ilhis- tration I have calculated the intensity of sound due to a distant source at various points on the sur£EU>e of a fixed spherical obstada The result depends upon the ratio (kc) between the circumfer^ooe of the sphere and the length of the wave. If we call the point upon the spherical surface nearest to the source the anterior pole, and the opposite point (where the shadow might be expected to be most intense) the posterior pole, the results on three suppositions as to the relative magnitudes of the sphere and wave length are as follows: — ke=2 kc=l kc=:i Anterior pole Posterior pole Equator •69 •32 •36 •50 •28 •24 •29 •26 •23 When for example the circumference of the sphere is but half the wave length, the intensity at the posterior pole is only about a tenth part less than at the anterior pole, while the intensity is least of all in a lateral direction. When kc is less than ^^ the difference of the intensities at the two poles is still less important, amounting to about 1 per cent when Ac = J. The case of the head and a pitch c' would correspond to ho ^ 'A about, so that the differences of intensity indicated by iheoacj decidedly small The explanation of the power of diserinui actually observed would be oasi^ox^ li \t ^v^Ke ^KMsible U ■Mecoui EXCEPTIONS TO OHM S LAW. soiipt taken of the different phases of the vibrations by which dtAhe two ears are attacked'. 386. Passing on to another branch of our siibject, we have now to consider more closely the impression produced upon the ear by an arbitrary sequence of aerial pressures fluctuating about fi certain mean value. According to the literal statement of '.)hm'B law (§ 27) the ear is capable of hearing as separate tones ;i!l the simple vibrations into which the sequence of pressures may be analysed by Fourier's theorem, provided that the pitch of these components lies between certain limits. Components whose pitch lies outside the limits would be ignored. Moreover, within the limits of audibility the relative phases of the various components would be a matter of indifference. To the law stated in this extreme form there must obviously be exceptions. It is impossible to suppose that the ear would hear as separate tones simple components of extremely nearly the same frequency. Such components, it is well known, give rise to beats, and their relative phase is a material element in the question. Again, it will be evident that the corresponding tone will not be heard unless a vibration reaches a certain intensity. A finite intensity would be demanded, even if the vibration stood by itself; and we should expect that the intensity necessary for audibility would be greater in the presence of other vibraitions, especially perhaps when these correspond to harmonic undertones. It ivill be advisable to consider these necessary exceptions to the univer- sality of Ohm's law a little more in detail. The course of events, when the interval between two simple vibrations is gradually increased, has been specially studied by Busanquet'. As in ^ 30, 65a, if the components be coa 27r7ii(, coe 27rti,f, we have for the resultant. H^ u = cos 27rn,C + cos 2vn.it ^^1 ^ic(M7r(ti^ — nj)t.cosTr{n,+ n,)t. (1); shewing that the resultant of two simple vibrations of equal amplitudes and of frequencies Uj, n, can be represented mathe' inatically by a single vibration whose frequency is the mean, vi^ » Saturt. vol, xn. p. 33. 1876. Phil. .\l,ig. vol. in. p. 456. 1877; vol.v- I. )]. 420, 1881. 444 bosanquet's observations. [38t| ^ (n, + 71,), and whase amplitude varies according to the cosine bi.' involving a change of sign (§ 65a), with a frequency ^ (n, — n^). U single vibration is not simple. The question now arises under which of the two forms in (1) will the ear perceive the knibI According to the strict reading of Ohm's law the two tones ih aid n, would be perceived separately. We know that when f^ amiii are nearly enough equal this does not and could not hi^ipaL The second form then represents the phenomenon; and it indicttes beats, the tone ^ (t^ + iij) having an intensity which varies betwea and 4 with a frequency (n, — n,) equal to the difference of fre- quencies of the original tones. Mr Bosanquet found that ''(a) the critical interval at which two notes begin to be heard beside their beats, or resultant displacements, is about two commas, throughout that medium portion of the scale which is used in practical music; (13) this critical interval appears to be not exactly the same for &B ears/' But in both the cases examined the beats alone were heard with an internal of one comma, and the two notes were quite clear beside the beats with an interval of three commas. ''As the interval increases, the separate notes become more and more pro- minent, and the beats diminish in loudness and distinctness, till, by the time that a certain interval is reached, which is about a minor third in the middle of the scale, the beats practically dis- appear and the two notes alone survive." On the second question as to the strength in which a com- ponent simple vibration, of sufficiently distinct pitch, must be present in order to assert itself as a separate tone there is but little evidence, and that not very accordant. According to the experiments of Brandt and Helmholtz (§ 130) Young's law as to the absence in certain cases of particular components from the sound of a plucked string is verified. Observations of this kind are easily made with resonators; but for the present purpose the use of resonators is inadmissible, the question relating to the behaviour of the unassisted ear. On the other hand A. M. Mayer ^ found that sounds of consider- able intensity when heard by themselves were liable to be completa^JT obliterated by graver sounds of sufficient force. In some ments the graver note was from an open organ-pipe which a steadily, while the higher was that of a fork, excited y and then allowed to die down. The action of the fr* '- 38G.] MAVE31S OBSESIVATIOSS. 445 iiiiide ill terra ittent by moving the hand to and fro over the mouth it its resonance box. The results are thus described. "At first ■ viiTy time that the mouth of the box is open the sound of the ri.rk is distinctly heard and changes the quality of the note of the (ipen pipe. But as the vibrations of the fork run down in ampli- tude the sensations of its effect become less and less till they soon entirely vanish, and not the slightest change can be observed in the quality or intensity of the note of the organ-pipe, whether the resonance box of the fork be open or closed. Indeed at this stage uf the experiment the vibrations of the fork may be suddenly and totally stopped without the ear being able to detect the fact, But if instead of stopping the fork when it becomes inaudible we stop the sound of the orgaa-pipe, it is impossible not to feel surprised at the strong sound of the fork which the open pipe had smothered and had rendered powerless to affect the ear." But "no sound, even when very intense, can diminish or obliterate the sensation of a concurrent sound which is lower in pitch. This was proved by experiments similar to the last, but differing in having the more intense sound higher (instead of lower) in pitch. In this case, when the ear decides that the sound of the (lower and feebler) tuning-fork is just extinguished, it is generally discovered on stopping the higher sound that the /brk-, which should produce the lower sound, luis ceased ta vibrate. This surprising experiment must be made in order to be appre- ciated, I will only remark that very many similar experiments, ranging through four octaves, have been made, with consonant and di.ssonant intervals, and that scores of different hearers have mfirnied this discovery." These results, which are not difBcult to verify', involve a ioiis deduction from the universality of Ohm's law, and must ,ve an important bearing upon other unsettled questions relating to audition. It is to be observed that in Mayer's experiments the question is not merely whether a particular tone can be heaixl as such. The higher sound of feebler intensity is not heard gtalL The audibility of a 80ug^f^J[2||^|iy|^ted, is intluenced by sUte of Ihe .-ar as retfuduHtilUa*. .The effect ls K^sweiallv 446 FATIGUE OF EAR. [381 apparent with the very high notes of bird-calls (§ 371). "A bird-call was mounted in connection with a loaded gas-bag audi water-manometer, by which means the pressure could be kept constant for a considerable time. When the ear is placed at i moderate distance from the instrument, a disagreeable aound ii heard at first, but after a short interval, usually not more tina three or four seconds, fades away and disappears altogether. A very short intermission suflBces for at any rate a partial lecoveiy of the power of hearing. A pretty rapid passage of the hand, screening the ear for a fraction of a second, allows the sound to be heard again^" But although Ohm's law is subject to important limitations, it can hardly be disputed that the ear is capable of making a rough analysis of a compound vibration into its simple parts. The nature of the diflScuIty commonly met with has already been referred to (§§ 25, 26), but a few further remarks may here be made. In resolving compound notes a certain control over the attention is the principal requisite, and Helmholtz found that the advantage does not always lie with musically trained ears. Before a particular tone is listened for, it ought to be sounded so as to become fixed in the memory, but not too loudly, lest the sensitiveness of the ear be unduly impaired. As a rule the uneven component tones, twelfth, higher third, &c., are more easily recognised than the octaves. On the pianoforte, for example, let g be first gently given, and as soon as the key is released, let c be sounded strongly. The tone g' on which the attention should be kept rivetted throughout, may now be heard as part of the compound note c. A similar experiment may be made with the higher third «", and an acute ear may detect a slight fall in pitch. This is a consequence of the equal temperament tuning (§ 19), and shews clearly that the apparent prolongation of the tone is not due to imagination. In modem pianos the seventh and ninth component tones are often weak or altogether absent, but on the harmonium these tones may usually be heard. It is still better when the tone to be listened for is finfe obtained as a harmonic from the string c itself In the ^1 187.] HEARING OF OVERTONES. } twelfth, for example, strike the ke^ gently while the string is fctly touched at one-third of its length, and then after removal I the finger more stroQgly. The proper point may be cou- Iriently found by sliding the linger slowly along the string, jjle the key is continually struck. When a point of aliquot iBJon is reached, the corresponding harmonic rings out clearly ; ? the sound is feeble and muffied. In this way Helmholtz >ded in hearing the overtones of thin strings as far as the teenth. From this point they lie too close together lo be kniy distinguished. A further slight modification of this method is especially wmmended by Helmholtz. Instead of using the finger, the lal point is touched with a small camel's hair brush. This lowB the degree of damping to be varied at pleasure, and a idual transition to be made from the pure harmonic, free from I admixture of components which have not a node at the point inched, to the natural note of the string. But it is nith the assistance af resonators that overtones are easily heard in the first instance. For this purpose a "Resonator is chosen, tuned, say, to g' , and the ear is placed in communication with its cavity. When c is sounded, either on the piano or harmonium, or with the human voice, the tone g' may usually be heard very loud and distinct. Indeed on many pianofortes a tone g' may be heard as loudly from its harmonic undertones j; or c as from the string g itself. When an overtone has once been heard, the assistance of the resonator should be gradually withdrawn, which may be done either by removing it &om the ear, or putting it out of tunc by an obstacle (such as the iger) held near its mouth. 387. If it be admitted that the ear is capable of analysing a musical note into components, or partials, it follows almost of necessity that these more elementary sensations correspond to Himple vibrations. So long as we keep within the range of the principle of superposition, this is the kind of analysLs effected by mechanical appliances, such as resonators, and all the more patent facts go to prove that the ear resolves according to the same laws. Moreover, the d priori probabilities of the ease seem '~ the same direction. It is difficult to suppose ' 448 HELMHOLTZ'S THEORY. [S8?. produced directly by the impact of sonorous waves involTiig merely a variable fluid pressure. Helmholtz's theory of auditia is based upon the more natural supposition that the immediak effect of the waves is to set into ordinary mechanical vibnitNi certain internal vibrators^ and that nervous excitation follows as i secondary consequence. The modus operandi is conceived to be as follows. When a simple tone finds access to the ear, all the parts capable of motioii vibrate in synchronism ynth the source. If there be any part, approximately isolated, whose natural period nearly agrees with that of the sound, then the vibration of that part is tar moie intense than it would otherwise be. Practically this part of the system may be said to respond only to tones whose pitch lies within somewhat narrow limits. Now it is supposed that the auditory nerves are in communication with vibrating parts of the kind described, whose natural pitch ranges at small intervals between the limits of hearing in such a manner that when any part vibrates the corresponding nerve is excited and conveys the impression to the brain. In the case of a simple tone, one (or at most a relatively small number) of the whole series of nerves is excited, the excitation of the nerve being the proximate cause of the hearing of the tone. At this point the question presents itself whether more than one simple vibration may not have the power of exciting the same nerve ? A priori, this might well be the case ; for the vibrating parts might be susceptible of more than one mode of vibration, and therefore of more than one natural period. If we were to suppose that the natural periods of any vibrating part formed a harmonic scale, so that the same auditory nerve was excited by a tone and its octave, the supposition would certainly give a very ready explanation of the remarkable resemblance of octaves, and would tend to mitigate some of the difficulties which at present stand in the way of accepting Helmholtz's theory as a complete account of the facts of audition I As we shall see present!^, ^ The dram-skin and its attachments are here regarded as external to auditory mechanism. However important maj be the part thej pbur. rather to that of a hearing tube or of the disc of a meohaiiioal lili ^ A curious question suggests itself as to what would bftppf tions capable of exciting the same nerve deviated se&iili^ ^ harmonic scale. In this way ears naturally cooftind ^onoal relations may ea^ \>« Vbdaqsm^ 388-1 DEGREE OF DAMPING. 449 H«lmhoItz would admit, or rather asaert, that when the sounds are strong two originally simple vibrations, auch as c aud c, would excite to some extent the same nerve, but he regards this as depending upon a failure in the law of sujiL-qioaition, due to exeessive vibration. 388. It is evident that Helmholtz'a theory gives a verj' natural account of Ohm's law, as well as of the limitation to which it is subject when two simple vibration-s are in operation of nearly the same pitch. Some of the internal vibrators are then within the influence of both disturbing causes, and are accordingly excited in an intermittent manner, giving rise to beats, when the period is long, and to a sensation of roughness as the beats become too quick to be easily perceived separately. But when the interval between the two vibrations is increa.sed, a point is soon re-ached after which no internal vibrator is sensibly affected by both disturbing causes, so that from this point onwards the resulting sensation is free from beats or roughneases, or at least should be so according to the strict interpretation of the law. To this point we shall return later. The magnitude of the interval, ovei- which a single internal vibrator will respond 8en.>iibly. is an element of considerable importance in the theory. It has already been shewn (§ 49) that there is & relation between this interval and the number of free vibrations which can be executed by the vibrating body. Thus, if the interval between the natural and the forced vibration required to reduce the resonance to ^ of the maximum be a semitone, this implies that after 9.5 free vibrations the intensity would be reduced to -y^ of its original value, and ao on for other intervals. From a consideration of the effect of trills in music, Helmholtz concludes that the case of the ear corresponds somewhat to that r »bove specified, and he gives the accompanying table shewing the DUlereiice Intensity of , DifTerenoe Inteaaity ot otpiloh ot pitch rsBonanoe 0-0 IfjO 0-fi 7-2 01 74 0-7 5-4 ^.iva 41 0-8 4-2 U 0-9 3-3 L= ^r^c Wholfl tone 9-7 450 A. M. Mayer's experiments. [388. relation obtaining in this case between the difference of free and forced pitch and the intensity of resonance, measured by the square of the amplitude of vibration. Although according to Helmholtz's theory the sensation of dissonance is caused by intermittent excitation of those vibrating .parts which are within the range of two or more elements of the sound, it is not to be inferred that the number of beats is a sufficient measure of the dissonance. On the contrary it is found that if the number of beats be retained constant (e.g. 33 per second), the effect is more and more free from roughness as the sounds are made deeper, the intervals being correspondingly increased. The experiments of A. M. Mayer^ extend over a considerable range of pitch, and have been made by two methods. In the first method a sound, which would otherwise be a pure tone, is rendered intermittent, and the rate of intermittence is gradually raised to the point at which the effect upon the ear again becomes smooth. The results are shewn in the accompanjring table, in which the first column gives the pitch of the sound and the second the minimum number of intermittences per second required to eliminate the roughness. Pitch (n) Frequency of Intermittence (m) 64 1 231 128 36 256 62 320 73 384 88 512 108 640 126 768 143 1023 170 The theory of intermittent vibrations has already been given § 65 a. It is to be remembered that by the nature of the case an intermittent vibration cannot be simple. To a first approximatum it may be supposed to be equivalent to three simple vibratioiis cf frequencies, n — m, n, n + m, and t} experienced 1 Phil Mag. vol. max. ^. %^%, 1« 389.] SMALLEST CONSONANT INTERVAL. 451 the ear may be looked upon as due to the beats of these three tones. Mayer has experimented also upon the "smallest consonant intervals among simple tones/' i.e. upon the intervals at which the roughness due to beats just disappears, the plural being used since it is found that the necessary interval varies at different parts of the scale. Pitch Additional vibrations required («2 - Wi) Smallest consonant intervals in semitones 48 64 96 128 c 192 i^ 256 1^ 316 a.432 575 766 1707 2304 2560 2806 none none 41 615 38 4-50 48 3-86 58 3-53 68 3-34 85 312 107 2-95 130 2-70 210 200 245 1-76 256 1-64 266 1-54 Different observers agi-eed very closely as to the point at which roughness disappeared. According to the theory of intermittent sounds it is to be expected that for a given pitch m in the first set of experiments should be nearly the same as (/^ — n^) in the second, and this is pretty well verified by Mayer s numbers, at least over the middle region of the scale. 389. From the degree of damping above determined it follows that the natural pitch of the internal vibrators, which respond sensibly to a given simple sound, ranges over about a whole tone, and it may excite surprise that we are able to compare with such accuracy the pitch of musical sounds heard in succession. The explanation probably depends a good deal upon *ie symmetry of the effects on the two sides of the maximum. A ^arison with the capabilities of the eye in a similar case may 452 NUMBER OF IMPULSES NECESSARY [389. be instructive. In setting the cross wires of a telescope upon the centre of a symmetrical luminous band, e.g. an interference band, it is found that the error need not exceed j^ of the widtL A similarly accurate judgment as to the centre of the region excited by a given musical note would lead to an estimation of pitch accurate to about y^, agreeing well enough with the fiEU^ts to be explained. In the light of the same principle we may consider how &r the perception of pitch would be prejudiced by a limitation of the number of vibrations executed during the continuance of a sound. According to the estimate of Helmholtz already employed (§ 388) the internal vibrations, excited and then left to themselves, would remain sensible over about 10 periods. The number of impulses required to produce nearly the full eflfect is of this order of magnitude. If the number were increased beyond 20 or 30, there would be little further concenti*ation of effect in the neighbourhood of the maximum, and therefore little foundation for greater accuracy in the estimation of pitch. Experiments upon this subject have been made by SeebeckS Pfaundler', S. Exner', Auerbach*, and W. Kohlrausch*, those of the last being the most extensive. An arc of a circle carrying a limited number of teeth was attached to a pendulum, which could be let go under known conditions. In their passage the teeth struck against a card suitably held ; and the sound thus generated was compared with that of a monochoi-d. By varying the length in the usual manner the chord was tuned until the pitch was just perceptibly higher, or just perceptibly lower, than that proceeding from the card, and the interval between the two, called the characteristic interval, determined the precision ynth which the pitch could be estimated in the case of a given total number of vibrations. The best results were obtained only after considerable practice and in the entire absence of extraneous sounds. Sixteen teeth appeared to define the pitch with all precision attainable, the characteristic interval (on the number of experiments) being in this case •9922. El ' Pogg, Ann. vol. lhi. p. 417, 18^ '^ Wien. Ber. vol. lxxvi. p. 56L • PJliiger's ArehiVf vol. xm. p ^ Wied, Ann. voL vi. p. 691r • Wien. Atm. '^oV X. i^« V^ V 390.1 FOR ESTIMATION OF PITCH, ■fl teeth the characteristic interval was 9903, shewing that this small miraber of vibrations was capable of definiug the pitch to within one part in 200. But the most surprising results were those obtained with a very low number of teeth. For 3 teeth the characteristic interval was 9790, and tor 2 teeth 97l4i. The fact that pitch can be defined with ccinsiderable accumcy by »o small a setpjence of vibrations has sometimes been regarded as an objection to Helmholtz's theory of audition. I do not think that there is any gnxind for this opinion. So far as there is a difficulty, it is one that would tell equally against any other theory that could be proposed, It would seem that the delimitation of pitch in Kohlrauach's experiments may have been greatly favoured by the approximate discontinuity of the impulses. For it is to be remembered that the internal vibrators concerned arc not those only whose period ranges roumi the inten.'al between the taps, but also those whose periods aie approximately submultiples of this quantity. As regards the vibi-ators in the ixitave, the number of impulses is practically doubled, for the twelfth trebled, and so on, just as in optica the resolving power of a grating with a limited number of lines is increased in the spectra of the second and higher orders. Vibratiiius limited to a moderate number of periods are some- times generated by reflection of short sounds from railings or steps. At Tcrling there is a flight of about 20 steps which i-eturns an echo of a clap of the hands as a note resembling the chir]) of a sparrow. In all such cases the action is exactly analogous to that of a grating in optics. 390. When two sounds nearly in unison are compound, we have to cousidei' not only the beats of their first partials, or primes, but also the beats of the overtones. The beats of the ![i(ioneuts are twice, and those of the twelfth three times, 1- the simultaneous beats of the primes. In some cases, .1 1 , where the pitch is veiy low, mistakes may easily be 1 by overlooking the prime beats, which affect the ear but ■ctttVe beats be reckoned as though they were the the differt-no/ of pitch will be taken to be the I'll consonances other than the 454 CONSONANCES DEFINED [390. itself specially felt. For example, take the Fifth c — ff. The third partial of c and the second partial of g coincide at g\ If the int^n-al be tnie, there are no beats : but if it be slightly disturbed from the true ratio 3 : 2, the two previously coincident tones separate from one another and give rise to beats. The frequencr of the beats follows at once from the manner of their genesi& Thus if the lower note be disturbed from its original frequency by one vibration per second, its third partial is changed bj 3 vibrations per second, and 3 per second is accordingly the frequency of the beats. But if the upper note undergoes a disturbance of one Wbration per second, while the lower remains unaltered, the frequency of the beats is 2. This rule is evidently general. If the consonance be such that the hth partial of the lower note coincides with the kth partial of the upper note, then when the lower note is altered by one vibration per second, the frequency of the beats is A, and when the upper note is altered by the same quantity, the frequency of the beats is k, " We have stated that the beats heard are the beats of those partial tones of both compounds which nearly coincide. Now it is not always very easy on hearing a Fifth or an Octave which is slightly out of tune, to recognise clearly with the unassisted ear which part of the whole sound is beating. On listening we are apt to feel that the whole sound is alternately reinforced and weakened. Yet an ear accustomed to distinguish upper partial tones, after directing its attention upon the common upper partials concerned, will easily hear the strong beats of these particular tones, and recognise the continued and undisturbed sound of the primes. Strike the note (c), attend to its upper partial (g'\ and then strike a tempered Fifth {g) ; the beats of (^) will be clearly heard. To an unpractised ear the resonatoi-s already described will be of great assistance. Apply the resonator for (g'), and the above beats will be heard with great distinctness. If, on the other hand, a resonator, tuned to one of the primes (c) or (g), be employed, the beats are heard much less distinctly, because the continuous part of the tone is then reinforced'." Experiments of this kind are conveniently made on tlie harmonium. Small changes of pitch may be obtained by Qnbr partially (instead of fully) depressing the ke' is to flatten the note. The beats of the ^ « ■ ■*. BV BEATS OF UPPER PAHTIA.LS, easily heard whej] a (tempered) Fifth is sounded ; those of the t-Cjual teiuperament Third are somewhat rapid. The harnioiiiuni is also a suitable iustrumeDt for experimenti* illustrative of just intonation, A reed may be flattened by loading the free end of the tongue with a fragment of wax, and sharpened by a slight filing at the same place. It is easy, especially with the aid of resonators, to tune truly the choi-ds c' — e* — g'. f — a' — c". whose consonance will then contrast favour- ably with the unaltered tempered chcnl g' — 6' — d'. It is not consistent with the plan of this work to enter at length into questions of temperament and just intonation. Full particulars will be found in the English edition of HelmhoUz (with Ellis's notes) and in Mr Bciaanquet's treatise. According to Helmholtz's theory it is mainly the boats of the upper partiala which determine the ordinary consonant intervals, any departure from which is made evident by the beats of the previously coincident overtones. But even when the notes are truly tuned, the varione conaonaiic&'i differ io degree, iin account of the disturbances which may arise from overtones which approach one another too nearly. The unison, octave, twelfth, double octave, etc.. may be regarded a« absolute consonances, the second component intro- ducing no new element but merely reinforcing a part of the other. The remaining consonant intervals, such as the Fifth and the Major Third, are in a manner disturbed by their ueighbom-hood to other consonant intervals. In the case of the truly tuned Fifth, for example, with frequencies represented by 3 and 2. there is indeed coincidence between the second partial of the higher note and the third partial of the graver note, but the partials which define the Fourth, of pitch 3x3 = 9 and 4x2 = 8, are within a whole Tone of one another and accordingly near enough to produce disturbance. In like manner the Major Third may bo regarded an disturbed by itt neighbourhood to the Fourth, and so 1 in the case of other intervals, ! im[)ortaucti of these disturbances, and consequently thv ■ in which the various intervals stand in respect to their <n.sonance, varies with the quality of the sounds. As tuple where overtones are present tn considerable strength. 456 DIFFERENCE TONES [390. intervals on the violinf and has exhibited the results in the form of a curve *. 391. The principle of superposition (§ 83), assumed in ordinary acoustical discussions, depends for its validity upon the assumption that the vibrations concerned are infinitely small, or at any rate similar in their character to infinitely small vibrationi, and it is only upon this supposition that Ohm's law fimk immediate application. One apparent exception to the law has long been known. This is the combination-tone discovered hj Sorge and Tartini in the last century. If two notes, at ^e inter^'al for example of a Major Third, be sounded together strongly, there is heard a grave sound in addition to the two others. In the case specified, where the primary sounds, or generators, as they may conveniently be called, are represented by the numbers 4 and 5, the combination-tone is represented by 1, being thus two octaves below the graver generator. In the above example the new tone has the period of the cycle of the generating tones ; but Helmholtz found that this rule fidls in many cases. The following table' exhibits his results as obtained by means of tuning-forks: Relative Frequency Combination — _ _ _ - — Generators tone Generators Combination tone b £' B 2 :3 f b' B 3:4 1 b d' B.i 4 :5 ^ d' f B., 5 : 6 1 f as' B., 6 :7 1 b g' es 3 :5 2 d' as' B 5 : 7 2 d' b' f 5 :8 3 In the last three cases the tone* **^*wd period of the complete cycle, bi ences of the frequencies of the which was found to apply in c question are called diflference-tt 1 Sensationt of Tor ksdi. SOMETIMES EXTERNAL TO EAR. 457 According to Heltnholtz it is necessary to the distinct audibility I of corobinntion-toncs that the generators be strong. We shall see . prefiently that this statement has been contested. "They are most easily heard when the two generating tones arc less than nn octave apart, because in that case the differential is deeper than either of the two generating tones. To hear it at first, choose two tones which can be held with great force for some time, and form a justly intoned hai-monic interval. First sound the low tone and I then the high one. On properly directing attention, a weaker low tone will be heard at the moment that the higher note is struck ; this is the required combinational tone. For particular instm- mentfi, a.^ the harmonium, the combinational tones can be made more audible by properly tuned resonators. In this case the tones are generated in the air contained within the iuntrument. But in fer cases where they are generated solely within the ear, the >nator8 are of little or no use'." On the strength of some observations by Bosauquet and Preyer, doubts have been expressed as to the correctness of Helmholtz'a statement that combination- tunes may exist outside the eai-, and strangely enough they have been adopted by Ellis. The question has an important bearing Upon the theory of conibinalion-tone.-^ ; and it has recently been examined by Rlicker and Edser'. who used apparatus entirely independent of the ear. They conclude that " Helmholtz was correct in stating that the sireu produces two objective notes the frequencies of which are respectively equal to the sum and difference of the frequencies of the fundamentals." My own observations have been made upon the harmonium, and leave me at a loss to understand how two opinions are possible. The mator is held with its mouth as near as may be to the reeds h sound the generating notes, and is put in and out of tune Ae difference- tone by slight movements of the finger. When I (tming is good, the difference-tone swells out with considerable ih, but a slight miutuning (probably of the order of a le) reduces it almost to silence. In some cases, e.g. when rval between the generatora is a (tempered) Fifth, the tone is heard to beat. 'verrntion proves that in some cases there exist two of nearly the same pitch. Helmholtz finds the Mom V roHt, p. lU. 458 BEATS OF DIFFEREN'CE TONE. [391. explaimtion of this in the compound nature of the sounds. Thus in thf case of the Fifth, represented by the numbers 2 and 3, we have not only the primes to consider, but the overtones 2x2, 3x2, etc., 2x3, 3x3, etc. Accordingly the difference-tone 1 may bo derived from 2x2 = 4 and 3, as well as from 3 and i, and sinco the octave partial is usually strong, the one sourcje may be as important as the other. But if we substitute the Major Third (5 : 4) for the Fifth, we do not get a second difference-tone 1 witil we come to the fourth partial (16) of the graver note and the third (15) of the higher, and these would usually be too feeble to prodace much effect. As regards the frecjuency of the beats, let us I'etum to the case of the Fifth, supposing it to be so disturbed that the frequencies are 200 and 301. The difference tone due to the primes is 301 — 200 = 101, and that due to the octave partial is 2x200-301=99; and these difference-tones sounding together will give beats with frequency 2. This, it will be observed, is the same number of beats as is due to the common overtone, viz. 2 x 301 — 3 x 200 ; but while the latter beats are those of the tone 600, the beats of the combination- tone are at pitch 100. 392. According to the views of the older theorists Chladni, Lagrange, Young, etc., the explanation of the difference-tone presentinl no particular difficulty. As the generators separate in pitch, the boats quicken and at last become too i-apid for apprecia- tion as such, passing into a difference-tone, whose fi^uency is continuous with the frequency of the beats. This view of the matter, which has commended itself to many writers, was rejected by Helmholtz, as inconsistent with Ohm's law ; and that physicist liius elaborated an alternative theor}% according to which the failure is not in Ohm's law, but in the principle of superposition. Helmholtz s calculation of the effect of a want of sjTiimetry in the forces of restitution, when the vibrations of a system cannot be rt»giinled as infinitely small, has already been given (§ 68). It appears that in addition to the terms in pt, qt, corresponding to the generating forces, there nmst be added other terms of the second order in 2pt, 2qty (p + q) t, (p — q) t, the last of which repto sents the difference-tone. This explanation depends, as Her ^92.] SUMMATION TONE. has remarked, uiwu the aaauined failure at syninietry. If, as in § 67. wt< suppose a force of restitution proportiuoal partlj' to the first power and partly to the cube of the displacement, we do not obtain a term in (p — q)t, but in place of it terms of the iidrd 'irder involving (2/) — g)(, (2y — ;>)(, etc, This objection, however, IS of little pi-actical importance, because the failure of symmetrj- atmost always occui^i. It may suffice to instance the all important case of auriat vibrations. Whether we are considering progressive waves advancing from a source, or the stationary vibmtions of a resonator, there is an essential want of symmetry between conden- sation and rarefaction, and the formation in some degree of octaves and combination -tones is a mathematical necesiiity. The production of external, or objective, combination-tones demands the coexistence of the generators at a place where they are strong'. This will usually occur only when the generating sounds are closely associated, as in the polyphonic siren and in the harmonium. In these coses the conditions are especial!)- tavourahle. because the limited mass of air included within the instniment is necessarily strongly affected by both tones. When the generating sources are two organ-pipes, even though they stand pretty neai- together, the difference- tone is not appreciably strengthened by a resonator, from which we may infer that but little of it exists externally to the ear. We have as yet said nothing about the summation-tone, corre- sponding to the term in (p -f q) t. The existence of this tone was deduced by Uelmholtz theoretically ; and he afterwards succeeded in hearing it, not only from the siren and harmonium, where it exists objectively and is reinforced by resonatore, but also from tuning-forks and oi;gan-pipes. Helnihtiltz narrates also an experi- ment in which he caused a membrane to vibrate in respouse to the summation- tone, and similar experiments have recently been carried out with success by Rilcker and Eds«r (I. c). Xevertheless, it must be admitted that summation-tones are extremely ditbcult to hear. Hermann (1. c.) asserts that he can I neither hew theui himself nor find any one able to do so ; and he B this diSiculty as a serious objection to Helmholtz's theory, Mrding to which the summation and the difference tone should e about equally strong. ites lit cuodcuHtion (| !lii4| lor loiuKln just aadible mkke it hiiflily |t UiM Uu) jinnai^U of »uf uixwUiuii oould fail to appljr \a aouuds of tluit 460 U£LMHOLTZS VIEWS [391 An objection of another kind has been raised by Konig'. He remarks that even if a tone exist of the pitch of the sumwatioD- tone, it may in reality be a difference-tone, derived from the upper paitials of the generators. As a matter of arithmetic this aiga- ment cannot be disputed ; for if p and q be commensurable, it will always be possible to find integers h and it, such that p-^q^hp-kq. But this explanation is plausible only when h and k are tmaU integers. It seems to me that the comparative diflScuIty with which summation-tones are heard is in great measure, if not altogether, explained by the observations of Mayer (§ 386). These tones are of necessity higher in pitch than their generators, and are accord- ingly liable to be overwhelmed and rendered inaudible. On the other hand the difference-tone, being usually graver, and often much graver, than either of its generators, is able to make itself felt in spite of them. And even as regards difference-tones, it had already been remarked by Helmholtz that they become more difficult to hear when they cease to constitute the gravest element of the sound by reason of the interval between the generators exceeding an octave. 393. In the numerous cases where differential tones are audible which are not reinforced by resonators, it is necessary in order U> carry out Helm hoi tz*s theory to suppose that they have their origin in the vibrating parts of the outer ear, such as the drum-skin and its attachments. Helmholtz considers that the structure of these parts is so unsymmetrical that there is nothing forced in such a supposition. But it is evident that this explana- tion is admissible only when the generating sounds are loud, ie. powerful as they reach the ear. Now, the opponents of Helmholtz s views, represented by Hermann, maintain that this condition is not at all necessary to the perception of difference-tones. Here we have an issue as to facts, the satisfactory resolution of which demands better experiments, preferably of a quantitative nature, than any yet executed. My own experience tends rather to support the view of Helmholtz that loud generators are neoesauy. On several occasions stopped organ-pipes d''\ ^'\ were blown wx 393.] AND CKITICIRMS THEREON. 461 a steady wind, and were 30 timed that the difference-tone gave slow beats with an electrically maintained fork, of pitch 128, mounted in association with a ^ei^uDato^ of the same pitch. When the ear was brought up close to the mouthu oi' the pipes, the difference- tone was so loud as to require all the force of the fork in order to get the most distinct beats. These beats could be made so slow as to allow the momentary disappeai'anee of the grave sound, when the intensities were rightly adjusted, to be observed with some precision. In this state of things the two tones of pitch 128, one the difference-tone and the other derived from the fork, weje of ecjual strength as they reached the observer; but as the ear was withdrawn so as to enfeeble both sounds by distance, it seemed that the combination- tone fell off more quickly than the ordinary tone from the foi^k. It might be possible to execute an experiment of this kind which should prove decisively whether the combination-tone is really an effect of the second order, or not. In default of decisive experiments we must endeavour to balance the a priori probabilities of the case. According to the views of the older theorists, adopted by KSnig, Hermann, and other critics of Helmholtz, the beats of the generatorK, with their alternations of swellings and pauses, pass into the differentia! tone of like frequency, without any such failure of superposition as is invoked by Helmholtz. The critics go further, and maintain that the ear is capjible of recognising as a tone any periodicity within certain limits of frequency'. Plausible as this doctrine it^ from certain points of view, a ^r examination will, I think, shew that it is encumbered with ifficullies. Among these is the ambiguity, referred to in § 12. as to what exactly is meant by period. A periodicity with frequency 128 is also periodicity with frequency 64. Is the latter tone to be beard as well as the former ? So far as theory is concerned, such questions are satisfactorily answered by Ohm's law. Experiment may compel us to abandon this law, but it is well to remember that there is nothing to take its place. Again, by consideration of particular cases it is not difficult to prove that the general doctrine above formulated cannot be true. Take the example above tioned in which two organ-pipes gave a difference- tone of 128. There is periodicity with frequency 128, and the ceriai k»ei ffici 462 DIFFERENCE TONES. [5 corresponding tone is heard*. So far, so good. But experii proves also that it is only necessar}' to superpose upon this anc tone of frequency 128, obtained from a fork, in order to neul the combination-tone and reduce it to silence. The periodicit] 128 remains, if anything in a more marked manner than bel but the corresponding tone is not heard. I think it is often overlooked in discussions upon this subj^ that a difference-tone is not a mere sensation, but involve vibrattan of definite amplitude and phase. The question at arises, how is the phase determined ? It would seem natural suppose that the maximum swell of the beats corresponds to or other extreme elongation in the difference-tone, but upon principles under discussion there seems to be no ground for selection between the alternatives. Again, how is the amplituc determined ? The tone certainly vanishes with either of tl generators. From this it would seem to follow that its amplitudej must be proportional to the product of the amplitudes of the generators, exactly as in Helmholtz's theory. If so, we come back to difference-tones of the second order, and their asserted easy audibility from feeble generators is no more an objection to one theory than to another. An observation, of great interest in itself, and with a possible bearing upon our present subject, has been made by Konig and Mayer*. Experimenting both with forks and bird-calls, they have found that audible difference-tones may arise from generators whose pitch is so high that they are separately inaudible. Perhaps an interpretation might be given in more than one way, but the passage of an inaudible beat into an audible diflference-tone seems to be more easily explicable upon the basis of Helmholtz's theory. Upon the whole this theory seems to afford the best ex- planation of the facts thus far considered, but it presupposes a more ready departure from superposition of vibrations within the ear than would have been expected. 394, In § 390 we saw that in the case of ordinary compound sounds, containing upper partials fairly developed, the recognised consonant intervals are distinguished from neighbouring intervals ^ In Btrictness, the periodicity is incomplete, unless p and q are i" iP-9)' < Mayer, Rep. Brit. Am. p» hl%^ \%^. IM-] BEATS OF IMPERFECT C0N5>0NANCF:S. MM\ well marked pheni>meQa. i^f which there wius no iliAicuIty in ing a satistactory acci.«iint. We have now to consider the difficult subject of coiiS«.»naiice among pun.' tones; anil we have to encDunter com?iderable difterences of opinion, not only tto theoretical exphiuatiims, but as to matters t»f observation. ire, as elsewhere, it will be convenient to begin with a statement tf Helmholtz s views' ticc<»rding to which, in a won!, the beats SisQch mistuned consonances are due to combination-tones. "If combinational tones were not taken into account, two pimple tones, as th4>se of tuning-forks, or stt»pped organ-pipes, poald not produce beats unless they were very nearly of the ■une pitch, and such beats are stn)ng when their interval is B minor or major second, but weak for a ThinI, and then only Roognisable in the lower psirts of the scale, and they gnidually diinimsh in distinctness as the interval incretises, without shewing any special differences for the hannonic intervals themselves. Fur My larger interval between two simple tones there would In: absolutely no beats at all, if thert* were no upptT partial oi- combinational tones, and hence the consonant intervals... would be in no way distinguished frr>ni adjacent intervals; there would in fact be no distinction at all V>«rtw<:«;n wide consonant intervals and cibsolutely dissonant inten'al.*^. Now such wider interval-? b«;twf:»rn siniple tones are known to produce beats, although verj* i/:';^h woakfrr than thosti hith(;rto considered, so that even for -r-'jch v^:-«r»r •.fi'.re is a diffen.^nci; be- tween consonances and di^v.-riaTiC^-r. a!:?*'^'igh it is very much iiore imperfect than for c^ijujy.j;,'; y,:.*^':" Experiments upt^n this »^ubi«r'.••. <..•* \.ir,':'i\i to execuW sat.is- fact<)rily. In the first pW*^ i: i- .'j'.': *->^-y y, ^.-cure simple tones. A-s sources recourse is usually hti^ v. ►-/. j>r>-'i organ-pipes or to tuning-forks, but nmch pfv':a, -.::•?. ,^ •..•..."r<3. From the fre(^ ends i>f the \'ibrating ppjijgs '»f u {■••■t: r:^*/.-. v . J:»-/>rjf.K may usually be heard^ Again, if a fori- *jt e!t :•.•.•>■--. ^rv-r the manner oi' musicians with its stalk pn-^M.-'^ ii^;; m-' <. vvsvria'.ing board, the octiive is loud and often p!»-'i'»j;;. ';»,•• '.'•..• •>--•. way i^^ to hold I . *.• ■ .-.- * A(icrib«rd bv hiu to Uallsir<^fi ^^'^ * SenMitioiu of Totir, p. lifj^. ' Kueu^'e experiments Hliew tfiii' i:.« . • -^ >. *. .. <.«/>* v^*;!:: the prongs m*' thin. Wied. Ann. vol. xiv. p. iilc i^y * The ynme tone mKy e\«/. •ii«j»>/^<» ^ . s,.. ■. ^, •• * .'^v* r^toral position 464 HELMHOLTZ'S VIEWS. [391 thu frtH.' ends of the i)n)ngs over a suitably tuned resonator. Bttl even then we cannot be sure that a loud sound thus obtained is absolutely free from the octave partial. In the case of the octave the differential tone ak-eady con- sidered suffices. " If the lower note makes 100 vibrations per Mi*cond, while the imperfect octave makes 201, the first differentiil tone makes 201 — 100 = 101, and hence nearly coincides with the lower note of 100 vibrations, producing one beat for each 100 vibrations. There is no difficulty in hearing these beats, and hence it is easily possible to distinguish imperfect octaves from jK*rfect ones, even for simple tones, by the beats produced by the former.'* The frequency of the beats is the same as if it were due to overtones; but there is one important difference between the two CAses noted by Ellis though scarcely, if at all, referred to by Hehuholtz. In the latter the beats would affect the octave tone, whereas according to the above theory the beats will belong to the lower tone. Bosanquet, Konig and others are agreed that in this resjKJct the theor}' is verified. Again, if the beats were due to combination-tones, they must tend to disappear Jis the sounds die away. The experiment is very easily tried with forks, and acconling to my experience the facts are in hannonv. When the sounds are much reduced, ihr niistuuing fails to make itself apparent. *' For the Fifth, the first order of differential tones no longer suffices. Take an imjx'rfect Fifth with the ratio 200 : 301 ; then the differential tone of the first order is 101, which is too far from either primary to generate beiits. But it forms an imperfect Octjive with the tone 200, and, as just seen, in such a case beats ensue. Here they are pnxluced by the differential tone 99 arising from the tone 101 and the tone 200, and this tone 99 makes two beats in a second with the tone 101. These beats then serve to distinguish the imperfect from the justly intoned Fifth, even in the case of two simple tones. The number of these beats is also exactly the siime as if they were the beats due to will depress the centre of inertia, the stalk being immovable, bat if the proogi an closest above, the contrary result may ensue. There must be some intermedUlt conntruction for which the centre of inertia will remain at rest during the Tibralta. In this case the sound from a resonance board is of the seoond ovder, and is destitute of the prime tone. P4.] DIFFERENTIAL TONES OF SECOND ORDER. 465 upper partial tones. But to observe these beats the twg mnry tonea must be loud, and thti ear must not be distracted r any extraneous noise. Under favourable circumstances, how- ler, they are not difficult to hear," It is important to he clear as to the order of magnitude of various differential tones concerned. If the primary tones, I frequencies represented by p and q, have amplitudes e and jectively, quantities of the first order, then (§ C8) the first ference and summation tones have frequencies corresponding Zp, 2g, p + q, p-q. \ are of the second order in e and /. A complete treatment I the second differential tones requires the retention of another 1 /3u' (§ 67) in the expression of the force of restitution. From this will arise terms of the third order in e and,/* with frequencies corresponding to 3p. 2p±?, p±2j. 3s;' Rnd there are in addition other terms of the same frequencies and order of magnitude, independent of $, arising from the full development to the third order of ctu'. In the case of the disturbed Fifth above taken, the beats are between the tone 2g — /» = 99, which is of the thinl order of magnitude, andp — (/ = 10I of the x>nd order. The exposition, quoted from lielmholtz, refers to ! terms last mentioned, which are independent of ff. The beats of a disturbed Fourth or major Third depend upon difference- ton e« of a still higher order of magnitude, and according to Helmholtz's observations they are scarcely, if at all, audible, even when the primary tones are strong. This is no more than would have been expected; the difficulty is rather to understand hnw the beats of the disturbed Fifth are perceptible and those of the disturbed Octave so easy to hear. ^^B When more than two simple tones are sounded together, ^^nah conditions arise. " We have seen that Octaves are precisely ^^Btited even for simple tonea by the beats of the first differential ^^Kie with the lower primary. Xow sa^^XMe that an Octave has ^Ken tuned p«'rfectly, and that then « ||^^iJmuBJntefpoeed ^) act as a Fiah. Th.-n if th« Fifth is flaH*""''^H»fl| «Dsae from the first differential tone. fine Tl 466 CHORD OF THREE NOTES. [394. Let the tones forming the perfect Octave have the pitch numbers 200 and 400, and let that of the imperfect Fifth be 301. The differential tones are 400-301= 99 301 - 200 = 101 Number of beats 2. These beats of the Fifth which lies between two Octaves are much more audible than those of the Fifth alone without its Octave. The latter depend on the weak differential tones of the second order, the former on those of the first order. Henee Scheibler some time ago laid down the rule for tuning tuning- forks, first to tune two of them as a perfect Octave, and then to sound them both at once with the Fifth, in order to tune the latter. If Fifth and Octave are both perfect, they also give together the perfect Fourth. The case is similar, when two simple tones have been tuned to a perfect Fifth, and we interpose a new tone between them to act as a major Third. Let the perfect Fifth have the pitch numbers 400 and 600. On intercalating the impure major Third with the pitch number 501 in lieu of 500, the differential tones are 600-501= 99 500 - 400 = 101 Number of beats 2. » 396. In Helmholtz's theory of imperfect consonances the cycles heard are regarded as risings and fallings of intensity of one or more of the constituents of the sound, whether these be present from the first, or be generated by transformation, to use Bosanquet's phrase, in the transmitting mechanism of the ear. According to Ohm s law, such changes of intensity are the only thing that could be heard, for the relative phases of the constitu- ents (supposed to be suflSciently removed from one another in pitch) are asserted to be matters of indifference. This question of independence of phase-relation was examined by Helmholtz in connection with his researches upon vowel sounds (§ 397). Various forks, electrically driven from one interrupter (§ 64), could be made to sound the prime tone, octave, twelfth etc., of a compoimd note, and the intensities and phases of the constituents could \>e coii\>To\\e4 \>^ ^^^^^ Titfs^&s»a(a£A& in the 3'.)5.] QUESTION OF PHASE. 467 ( natural) pitch of the forks and associated resonatore. According Im Helmholtz's observations changes of phase were without liistinct effect Upon the quality of the compound sound. It is evident, however, that the i|uestion of the effect, if any, upon the ear of a change in the phase relationship of the various components of a sound can be more advantageously examined by the method of slightly mistuned consonances. If, for esample, an Octave interval between two pure tones be a very little imperfect, the effect upon the ear at any particular moment will be that of a true interval with a certain relation of phases, but after a short time, the phase relationship will change, and will pass in turn through every possible value. The audibility of the cycle is accordingly a criterion for the question whether or not the ear appreciates phase relationship ; and the results recorded by Helmholtz himself, and easily to be repeated, shew that in a certain sense the answer must be in the affirmative. Otherwise slow beaty of an imperfect Octave would not be heard. The explanation by means of CO nibiuatiOQ -tones does not alter the fact that the ear appreciates the phase relationship of two originally simple tones, at any rate when they are moderately loud'- According to the observations of Lord Kelvin' the "beats of imperfect harmonies," other than the Octave and Fifth, are not so difficult to hear as Helmholtz supposed. The tuning-forks employed were mounted upon box resonators, and it might indeed be argued that the sounds conveyed down the stalks were not thoroughly purged from Octave partials. But this consideration would hardly affect the result in some of the cases mentioned. It appeared that the beats on approxtmatioDS to each of the harmonies 2 : 3. -S : 4, 4 : 5. 5 : 6. 6 : 7. 7 : 8, 1 : 3, 3 : 5 could be distinctly heard, and that they all " fiilfil the condition of having the whole period of the imperfection, and not any sub-multiple of it, for their period," the sauae rule as would apply were the beats due to nearly coincident overtones. As regards the necessity for loud notes, Kelvin found that the beats of an imperfectly tuned chord 3 : 4 : .5 were some- times the very la.'^t sound heard, as the vibrations of the forks died down, when the intensities of the three notes chanced at t^ 1 to be suitably proportioned. 468 konig's observations, [395. The last observation is certainly difficult to reconcile with a theory which ascribes the beats to combination-tones. But on the other side it may be remarked that the relatively easy audibility of the beats from a disturbed Octave and from a disturbed chord of three notes (3:4: 5), which would depend upon the first differ- ential tone, is in good accord with that theory^ and (so fiir as appears) is not explained by any other. 396. But the observations most difficult of recondliaticm with the theory of Helmholtz are those recorded by Konig', who finds tones, described as beat-tones, not included among the combination-tones; and these observations, coming from so skilful and so well equipped an investigator, must carry great weight. The principal conclusions are thus summarised by Ellis". "If two simple tones of either very slightly or greatly different pitches, called generators, be sounded together, then the upper pitch number necessarily lies between two multiples of the lower pitch number, one smaller and the other greater, and the differences between these multiples of the pitch number of the lower generator and the pitch number of the upper generator give two numbers which either determine the frequency of the two sets of beats which may be heard or the pitch of the beat-notes which may be heard in their place. The frequency arising from the lower multiple of the lower generator is called the frequency of the lower beat or lower beat- note, that arising from the higher multiple is called the frequency of the higher beat or beat-note, without at all implying that one set of beats should be greater or less than the other, or that one beat-note should be sharper or flatter than the other. They are in reality sometimes one way and sometimes the other. Both sets of beats, or both beat-notes, are not usually heard at the same time. If we divide the intervals examined into groups (1) from 1 : 1 to 1 : 2, (2) from 1 : 2 to 1 : 3, (3) from 1 : 3 to 1 : 4, (4) from 1 : 4 to 1 : .5, and so on, the lower beats and beat-tones extend over little more than the lower half of each group, and the upper beats and beat-tones over little more than the upper hal£ For a short distance in the middle of each period both sets of beats, or both beat-notes are audible, and these beat-notes beat with each ^ Pogg. Ann, voL CLvn. p. 177, 187& * SeiuoHoiu of Tone, ^. lbl9A. ^OG.'] KONia's OBSERVATrONS. 469 other, forming secondary beats, or are replaced by new or secondary bt;at -notes." In certain cases the beat-notes coincide with the differential tone, but Kiinig considers that the existence of combinational tones has not been proved with certainty. It ia to be observed that in these experiments the generating tones were as simple as Konig could make them : but the possibility remains that overtones, not iiudible except through their beats, may have arisen within the ejir by transformation. This is the view favoured by Bosanquet, who has also made independent observations with results leas difB- ciill uf accommodation to Helmholtz's views. It will be seen that Kiinig adopts in its entirety the opinion that beats, when quick enough, pass into tones. Some objections to this idea have already been jKiinted out; and the question must be regarded as still an open one. Experiments upon these subjects have hitherto been of a merely qualitative character. The diffi- culties of going further are doubtless considerable ; but I am di?p«aed tu thiok that whnl is moat wanted at the present time is a better reckoning of the intensities of the various tones dealt with and obscn'ed. If, for example, it could be shewn that the intensity of a beat-tone is proportional to that of the generators, it would become clear that something more than combination -tones necessary to explain the effects. Eonig has also examined the question of the dependence of lity upon phase relation, using a special siren of his own con- struction'. His conclusion is that while quality is mainly deter- mined by the number and relative intensity of the harmonic tones, still the influence of phase is not to be neglected. A variation of phase produces such differences as are met with in different instruments of the same class, or in various voices singing the same vowel. A ready appreciation of such minor differences re- quires a series of notes, u{Kin which a melody can be executed, and they may escape observatiim when only a single note is available. To me it appears that these results are in harmony with the view A would ascribe the departure from Ohm's law, involved in any >gnition of phase relations, to secondary causes. 397. The dependence of the quality of musical sounds of given , pitch upon the proportions in which the various partial tones as?' ^j0oe ^loali Ton 470 WILLIS* EXPERIMENTS [397. present has been investigated by Helmholtz in the case of seven! musical instruments. Further observations upon wind instru- ments will be found in a paper by Blaikley^ But the most interesting, and the most disputed, application of the theory is to the vowel sounds of human speech. The acoustical treatment of this subject may be considered to date from a remarkable memoir by Willis*. His experiments were conducted by means of the free reed, invented by Kratzen- stein (1780) and subsequently by Grenie, which imitates with fisur accunicy the operation of the larjmx. Having first repeated success- fully Kempelen s experiment of the production of vowel sounds by shading in various degrees the mouth of a funnel-shaped cavity in association with the reed, he passed on to examine the effect of various lengths of cylindrical tube, the mounting being similar to that adopted in organ-pipes. The results shewed that the vowel ()uality depended upon the length of the tube. From these and other experiments he concluded that cavities yielding (when sounded in- dependently) an identical note " will impart the same vowel quality to a given reed, or indeed to any reed, provided the note of the reed be flatter than that of the cavity." Willis proceeds (p. 243) : " A few theoretical considerations will shew that some such effects iis we have seen, might perhaps have been expected. According to Euler, if a single pulsation be excited at the bottom of a tube closed at one end, it will travel to the mouth of this tube with the velocity of sound. Here an echo of the pulsation will be formed which will run back again, be reflected from the bottom of the tube, and again present itself at the mouth where a new echo will be produced, and so on in succession till the motion is destroyed by friction and imperfect reflect ion.... The effect therefore will be a propagation from the mouth of the tube of a succession of ecjuidistant pulsations alternately condensed and rarefied, at intervals corresponding to the time required for the pulse to travel down the tube and back again ; that is to say, a short burst of the musical note corresponding to a stopped pipe of the length in question, will be produced. Let us now endeavour to apply this result of Euler's to the case before us, of a vibrating reed, applied to a pipe of any length, 1 Phil Mag. toL vi. p. 119, 1878. > On the Vowel SonndB, and on Beed Organ-pipes. Comb. PkiL Ttam, tqL m* ^ S81, 1899. .] UPON VOWEL SOUNDS. 471 examine the nature of the series of pulsations that ought e produced by such a system upon this theory. [ The vibrating tongue of the reed will generate a series of ■Isations of equal force, at equal intervals of time, but alternately idenscd and rarefied, which we may call the primary pulsations; ! other hand each of these will be followed by a aeries secondary pulsations of decreasing Btrength, but also at equal ; from their respective primaries, the interval between iing, as we have seen, regulated by the length of the iched pipe." i further on (p. 247) : " Experiment shews ub that the series r effects produced are characterized and distinguished from each other by that quality we call the vowel, and it shews us more, it shews us not only that the pitch of the sound produced is always that of the reed or primary pulse, but that the vowel produced is always identical for the same value of s [the period of the secondary pulses]. Thus, in the example just adduced, g" is peculiar to the vowel A" [as in Paw, Nought'^ ; when this is repeated 512 times in a second, the pitch of the sound is n, and the vowel ia A' : if by means of another reed applied to the same pipe it were repeated 340 limes in a second, the pitch would be f, but the vowel still ^'. Hoiice it would appear that the ear in losing the consciousness of the pitch of s, is yet able to identify it by this vowel quality." From the importance of his results and from the fact that the early volumes of the Cambridge Ti-ansactions are not everywhere accessible, I have thought it desirable to let Willis speak for himself. It will be seen that so far as general principles are concerned, he left little to be effected by his successors. Some- what later in the same memoir (p. 249) he gives an account of a special experiment undertaken as a test of his theory. "Having shewn the probability that a given vowel is merely the rapid repetition of its peculiar note, it should follow that if we can produce this rapid repetition in any other way, we may expect to hear vowels. Robison and others had shtiwn that a quill held against a revolving toothed wheel, would produce a musical note by the rapid equidistaiit repetition of the snaps of the quiti upon the teeth. For the quill I substituted a piece of watch-spring pressed lightly against the teeth of the wheel, so that I'ach snaD became tbe musical note of the spring. The spring being ' I in a pair of mnoew. bo a» to ■ ' 472 HELMHOLTZ'S VIEWS. [397. alteration in length of the vibrating portion. This system evidently produces a compound sound similar to that of the pipe and reed, and an alteration in the length of the spring ought therefore to produce the same effect as that of the pipe. In effect the sound produced retains the same pitch as long as the wheel revolves uniformly, but puts on in succession all the vowel qualities, as the effective length of the spring is altered, and thst with considerable distinctness, when due allowance is made for the harsh and disagreeable quality of the sound itself." In his presentation of vowel theory Helmholtz, following Wheatstone^ puts the matter a little differently. The aerial vibrations constituting natural or artificial rowels are, when a uniform regime has been attained (§§ 4f8, 66, 322 k), truly periodic, and the period is that of the reed. According to Fourier's theorem they are susceptible of analysis into simple vibrations, whose periods are accurately submultiples of the reed period. The effect of an associated resonator can only be to modify the intensity and phase of the several components, whose periods are already prescribed. If the note of the resonating cavity — the mouth-tone — coincide with one of the partial tones of the voice- or larynx-note, the effect must be to exalt in a special degree the intensity of that tone ; and whether there be coincidence or not, those partial tones whose pitch approximates to that of the mouth-tone will be favoured. This view of the action of a resonator is of course perfectly correct ; but at first sight it may appear essentially different from, or even inconsistent with, the account of the matter given by Willis. For example, according to the latter the mouth-tone may be, and generally will be, inharmonic as regards the larynx-tone. In order to understand this matter we must bear in mind two things which are often imperfectly appreciated. The first is the distinction between forced and free vibrations. Although the natural vibrations of the oral cavity may be inharmonic, the forced vibrations can include only harmonic partials of the larynx note. And again, it is important to remember the definition of simple vibrations, according to which no vibrations can be simple that are not permanently maintained without variation of amplitude or phase. The secondary vibrations of Willis, whidi > London and Wettmintter Review, Oct. 18S7 ; IVheaUtone't SeiMt{fie London, 1879, p. &4B. B down after a few periods, are not simple. When the complete sceesion of them is resolved by Fourier's theorem, it is repre- , not by one simple vibration, but by a large or infinite mber of such. theae considerations it will be seen that both ways regarding the subject are legitimate and not inconsistent with B another. When the relative pitch of the mouth-tone is low, r, for example, the partial of the larynx note most reinforced e second or the third, the analysis by Fourier's series is the k>pcr treatment. But when the pitch of the mouth-tone is high, i each succession of vibrations occupies only a small fraction of complete period, we may agree with Hermann that the resolution by Fourier's series is unnatural, and that we may do better to concentrate our attention upon the actual form of the curve by which the complete vibration is expressed. More especially shall we be inclined to take this course if we entertain doubts as to the applicability of Ohm's law to partials of high order. Siuce the publication of Helmholtz'a treatise the question has been much discussed whether a given vowel is characterized by the prominence of partials of given order (the relative pitch theorj-). T by the prominence of partials of given pitch (the fixed pitch theory), and every possible conclusion has been advocated. We have seen that Willis decided the question, without even expressly formulating it, in favour of the fixed pitch theory. Helmholtz himself, if not very explicitly, appeared to hold the same opinion, perhaps more on a priori grounds than as the result of experiment. If indeed, as has usually been assumed by Hvriters on phonetics, a particular vowel quality is associated with i. given oral configuration, the question is scarcely an open one. nbscquently under Helmholtz's superintendence the matter was rthei' examined by Auerbach', who along with other methods Btployed a direct analysis of the various vowels by means ^ resonators associated with the ear. His conclusion on the lestion under discussion was the intermediate one that both icteristics were concerned. The analysi:« shewed also that in 8 the first, or fundamental tone, was the strongest element I the sound. A few years later Edison's beautiful invention of the pht'no- 474 EXPERIMENTS WITH PHONOGRAPH. [397. graph stimulated anew inquiry upon this subject by apparently affording easy means of making an experimentufn cruets. If vowels were characterized by fixed pitch, they should undergo alteration with the speed of the machine; but if on the other hand the relative pitch theory were the true one, the vowel quality should be preserved and only the pitch of the note be altered. But, owing probably to the imperfection of the earlier instruments, the results arrived at by various observers were still discrepant. The balance of evidence inclined perhaps in favour of the fixed pitch theory ^ Jenkin and Ewing' analysed the impressions actually made upon the recording cylinder, and their results led them to take an intermediate view, similar to that of Auerbach. It is clear, they say, "that the quality of a vowel sound does not depend either on the absolute pitch of reinforce- ment of the constituent tones alone, or on the simple grouping of relative partials independently of pitch. Before the constituents of a vowel can be assigned, the pitch of the prime must be given ; and, on the other hand, the pitch of the most strongly reinforced partial is not alone sufficient to allow us to name the vowel." With the improved phonographs of recent years the question can be attacked with greater advantage, and observations have been made by McKendrickand others, but still with variable results. Especially to be noted are the extensive researches of Hermann published in Pjlilgers Archiv, Hermann pronounces unequivocally in favour of the fixed pitch characteristic as at any rate by far the more important, and |his experiments apparently justify this conclusion. He finds that the vowels sounded by the phonograph are markedly altered when the speed is varied. Hermann 8 general view, to which he was led independently, is identical with that of Willis. ** The vowel character consists in a mouth-tone of amplitude variable in the period of the larynx tone*." The propriety of this point of view may perhaps be considered to be established, but Hermann somewhat exaggerates the ditference between it and that of Helmholtz. His examination of the automatically recorded curves was effected in more than one way. In the case of the vowel A * the 1 Graham Bell, Ann, Joum. of Otology, toL i. July, 1879. ' Edin, Tram, toI. xxvm. p. 745, 1878. * PJl&g. Arch, yol. zlto. p. 851, 1890. * The vowel rigna wte <A cwawft V^ ^Oaft ^gnaawBfitA yf mxwiriiiir'' 397.] HERMANNS EXPERIMENTS. 475 amplitudes of the various partials, as given by the Fourier analysis, are set forth in the annexed table, from which it appeal's that the favoured partial lies throughout between ^ and ^. Vowel A. Note G A H c d e fl8 g a h c' d' 1 2 3 Ordinal namber of partial. 4 5 6 7 8 9 10 •12 ^37 d* <f« •42 g* •11 a« •12 •13 30 33 ci8* e* < g* •10 a* •09 •08 ci8' •06 H •09 fi8' •22 37 46 •lO h' di8« JB8« <a« •16 •11 c •19 -64 ^38 16 c* e* g* <ai8* •09 •10 ds •29 62 -08 -18 d« fi82 a« <c5 •06 •13 •66 28 24 ^07 e« gi8« h« <d« » •30 cis* •61 07 •ll •ll 68* ai8* ci8* <e* •11 g •39 d« •66 21 •ll •OB g2 h2 d' <f5 •71 •18 18 -09 a* ci8' e' — — •74 fi8« •17 13 h« di8» •41 C3 •64 g* •40 •ll c3 e* — •71 d* •31 a* •26 ds The analysis of the curves into their Fourier components involves a great deal of computation, and Hermann is of opinion that the principal result, the pitch of the vowel characteristic, can be obtained as accurately and far more simply by direct measure- ment on the diagram of the wave-lengths of the intermittent vibrations. The application of this method to the curves for A before used gave 476 VOWEL CHARACTERISTIC. [397. Vowel A. L 1 ChanoterisUe tone Note 1 G 98 mm. 18-5 mm. Freqiien<7 Note 2-4 756 1 > fis* (740) A 110 16-3 2-5 717 >f (698-5) >f (698-5) H 123-5 14-9 2-6 708 c 130-8 13-6 2-55 698 f d 146-8 11-6 2-4 710 >f* (698-5) e 164-8 10-9 2-3 781 <g» (784) fis 185 9-8 2-5 725 <fi8« (744) g 196 9-1 2-5 714 > P (698-5) a 220 8-2 2-5 714 > f (698-5) h 246-9 7-3 2-6 693 < f* (698-5) c' 261-7 6-8 1 ? d' 293-7 6-2 ? ? Here L is the double period of the complete vibration and I the double period of the vowel characteristic. It appears plainly that I preserves a nearly constant value when L varies over a considerable range. A general comparison of his results with those obtained by other methods has been given by Hermann, from which it will be seen that much remains to be done before the perplexities involving this subject can be removed. Some of the discrepancies Characteristic Mouth-tones aocordin^ \ to tone from Vowel A graphical records Hermann • Donders Uelmholtz Kdnig Anerbach e— gis' b' V b* P E h» 0* cis' f,,b» b* a'( i) I d*-g* f U* b* f d« e« d' b' b' a' U c«~^« f f b f that have been encountered may probably have their origin in real differences of pronunciation to which only experts in phcmolicr' are su£Sciently alive ^ Again, the question of double roKMP has to be considered, for the known shape of the cavities oo E 397.] 18 THE PRIME TONE SENSIBLE? 477 renders it not unlikely that the complete characterization of a vowel is of a multiple nature (§ 310). It should be mentioned that in Lloyd's view the double characteristic is essential, and that the identity of a vowel depends not upon the absolute pitch of one or more resonances, but upon the relative pitch of two or more. In this way he explains the diflSculty arising from the fact that the articulation for a given vowet appears to bo the same for an infant and for a grown man, although on account of the great difference in the size of the resonating cavities the absolute pitch must vary widely. It would not be consistent with the plan of this work to go further into details with regard to particular vowels ; but one rcnmrkable discrepancy between the results of Hermann and Auerbach must be alluded to. The measurements by the former of graphical records shew in all cases a nearly complete absence of the first, or fundamental, tone from the general sound, which Auerbach on the contrary, using resonators, found this tone le moat prominent of all. Hermann, while admitting that the ! is heard, regards it as developed within the ear after manner of combination-tones (§ 393). I have endeavoured to repeat some of Auerbach's observations, and I find that for all the principal vowels (except perhaps A) the fundamental tone is loudly reinforced, the contrast being very marked as the resonator is pnt in and out of tune by a movement of the finger over its mouth. This must be taken to prove that the tone in question does exist externally to the ear, as indeed from the r in which the sound is produced could hardly fail to be the and the contrary evidence from the records must be ilained in some other way. An important branch of the subject is the artificial imitation f vowel sounds. The actual s^Tilhesis by putting together in feitable strengths the various partials was effected by Helm- loltz '. For this purpose he used tuning-forks and resonators, the ■rks being all driven from a single intc-mipter (^ 63, 64). These Kriments are difficult, and do not appear to have been n [nhottz was satisfied with the reproduction )ugh in others the imitation was incomplete. . lults were attained when organ-pipes ' ( forks. 478 ARTIFICIAL VOWELS. [397. Vowel sounds have been successfully imitated by Preece and Stroh\ who employed an apparatus upon the principle of the phonograph, in which the motion of the membrane was contioUed by specially shaped teeth, cut upon the circumference of a re- volving wheel They found that the vowel quality miderw^t important changes as the speed of rotation was altered. For artificial vowels, illustrative of his special views, Hermann recommends the poljrphonic siren (§ 11). If when the series of 12 holes is in operation and a suitable velocity has been attained, the series of 18 holes be put alternately into and out of action, the difference-tone (6) is heard with great loudness and it assumes diBtinctfy the character of an 0. At a greater speed the vowel is Ao, and at a still higher speed an unmistakable A. With the use of double resonators, suitably proportioned, Lloyd has successfully imitated some of the whispered vowels. In the account here given of the vowel question it has onlj been possible to touch upon a few of the more general aspects of it. The reader who wishes to form a judgment upon controverted points and to pursue the subject into detail must consult the original writings of recent workers, among whom may be specially mentioned Hermann, Pipping, and Lloyd. The field is an attractive one ; but those who would work in it need to be well equipped, both on the physical and on the phonetic side. * Proe. Roy. Soc, voL xxyui. p. 868, 1879. 479 NOTE TO § 86 \ It may be observed that the motion of any point belonging to a system of n degrees of freedom, which executes a harmonic motion, i& in general linear. For, if a;, ^, 2 be the space coordinates of the point,, we have x = X cos nty y=T cos nty z = Z cos rU^ where X, F, Z are certain constants ; so that at all times x\y\z = X: Y :Z, If there be more than one mode of the frequency in question, the coordinates are not necessarily in the same phase. The most general values of a;, y^ z, subject to the given periodicity, are then x^ Xi cos fU + X2 sin nt, y= Ti cos nt + F, sin rU, z= Zi cos 7U+ Z2 sin n/, equations which indicate elliptic motion in the plane x{Y,z, - z, r.) + y (z,x, - jr,z,) + z (X, r,- i^i, = 0. > This note appears now for the first time. APPENDIX TO CHAPTER Y\ ON THE VIBRATIONS OF COMPOUND 8T8TBM8 WHEN THE AMPLITUDB ARE NOT INFINITELY SMALL. In ^ 67, 68 we have found second approximations for the vibrations of systems of one degree of freedom, both in the case where the vibrations are free and where they are due to the imposition of given forces acting from without. It is now proposed to extend the investi- gation to cases where there is more than one degree of freedom. In the absence of dissipative and of impressed forces, everything may be expressed (§ 80) by means of the functions T and F. In the case of infinitely small motion in the neighbourhood of the configuration of equilibrium, T and V reduce themselves to quadratic functions of the velocities and displacements with constant coefficients, and by a suitable choice of coordinates the terms involving products of the several coordinates may be made to disappear (§ 87). Even though we intend to include terms of higher order, we may still avail ourselves of this simplification, choosing as coordinates those which have the property of reducing the terms of the second order to sums of squares. Thus we may write T= ii4i,«^i'+ Uaa*,' -I- ... + ^ij<^i<^+ -4u<^*8 + (1), in which ilj,, A^^.,, are functions of <^,, ^,... including constant terms O], a^. ..., while il,,, il,,, ... are functions of the same variables without constant tenns : V=^c,i>,^ + ^c,i>,^^...-\- K, + ^4+ (2), where F,, F4, ... denote the parts of V which are of degree 3, 4, ... in <f>i, if>2f ... For the first approximation, applicable to infinitely small vibratioiis, we have i!n = ai, il„ = aj, ... iii, = 0, -4^ = 0..., r, = 0, r4=0, ♦•*: ^ This aippendix ftppean now for the fiisl tinMu VIBRATIONS OP THE SECOND ORDER. 481 Kthat {§ 87) Liigrange'e equntiunB are + e,^ = 0, a,i, + e,^ = 0, A:c (3), 1 which the coordinates are separated. The Bolution relative to ^ may be taken to be *, = //iCo8n(, *, = 0, *i = 0, ...&c (4), where (r,-n'a, = (5). Similar solutions exist relative to the other coordinates. The second approximation, to which we now proceed, is to be founded on (4), (5) ; and thus <fi,„ <^. am to be rej^arded as small quantities relatively to •)>,. ^^i For the coefficients in (1) we write m (6). " «idm(2) r..,,«,' + ,,*/^+ (7); BO that for a further approximation t'?'/''*i = (Oi + '>ii*i)*i + <h<f>,'k + <h,<f'ji> + ■■■. + a5<^il^, + IL,<jl,4h+ "l^l*! +Oj*i'^. + ■■■■ ./y/d*! = la„^,' + «,^,.^ + a,.;i,^ + ... Thus as the equation (§80) for tfi,, terms of the order <fi,' being retained, we get (a, + a„^)«, + K'^' + <i*. + 3Y.*.' = {8)- To this order of approximation the coordinate ^ is separated from the others, and the solution proceeds as iu the case of but one degree of freedom (S 67). We have from (+) ^_ *j *i = - «'-ffi' cos' «{ - - i»'ff,' ( 1 + cos 2m(). ^^h ^*= n'/Ti'sin'rti = Jn'//,' (1 - coB2ni), ^^^Hhat (8) becx>mee ^m a,^ + C,^ + (- in>a„ + gy.) //," + (- Jn'a,. + Sy,) i/,' eo« 2«( = ■T (9). ] 11>9 solution of (9) may be expressed in the form ^1 = Z^„ + ff, cos »( + .ff, cos 2Tii + (10), n oomimrison gives «,ff.-(K",.-liy,)«,-. (t,-.,'a,)i/,.0, 482 VIBRATIONS OF THE SECOND OBDEB. Thus to a second approximation and the value of n is the same, Le. ^(<h/^X '^ ^ ^^^ ^^^ aj^roxi- mation. We have now to express the corresponding values of ^, ^.... From (6) and Lagrange's equation becomes, terms of order ^' being retained, o,^ + C,^^ + a,^^ + (a, - iou) ^* + y,<^« = 0, or on substitution from (4) in the small terms <h4^ + <5i^ + (- i^'ttis + ht) ^\ + (- »*'«« + i*»*«w + ht) -^1* cos 2n< =0 (12). Accordingly, if «^ = Aro + ^i00sn< + Z,cos2n^ + (13X we find on comparison with (12) ^^o=(K«i,-Jy.)^i* (1^), (c,-n»a,)iri = (15), (c.-4n«a,)JS:, = (nX-K«ii~ira)^i* O^)- Thus A\ = 0, and the introduction of the values of JTq and K^ from (14), (16) into (13) gives the complete value of ^ to the second approxima- tion. The values of ^, ^4, &g« are obtained in a similar manner, and thus we find to a second approximation the complete expression for those vibrations of a system of any number of degrees of freedom which to a first approximation are expressed by (4). The principal results of the second approximation are (i) that the motion remains periodic with frequency unaltered, (ii) that terms, constant and proportional to cos2n<, are added to the value of that coordinate which is finite in the first approximation, as well as to those which in the first approximation are zero. We now proceed to a third approximation ; but for brevity we will confine ourselves to the case (a) where there are but two ^ of freedom, and {fi) where the kinetic energy is oompletd.T a sum of squares of the velocities with constant 0^ include the vibrations of a particle moTinr ^he neighbourhood of a plao^ ot eo^^iilibH VIBRATIONS OF THE SECOND ORDER. 483 We have where l'. = >i*i' + ri*!**^-^ /*!*>' + (1^). r..8,0,* + S.*,'«.+ (18); so that Lagrange's eqoations are o,*, + c,*, + 3t.*,»+2t,^<^+4B,^.' = (19), fl5^+c,^+ y,«,* + 2y^^+ 8,*,' = (20). As before, we are to take for the iirat approximation ^-ff, cosn(, <^ = (21). For the solution of (19), (30) we may write 4^ = ff^ + II,coant + ffjcaaint + H,co9Snt + (22), ^ = jrj +^iCoaB(+^,coa2ni + J", cos3Mi+ (23). Ill (22), (23) If^, H,, A\, K, are quantities of the Beeond order in tf „ whose values have already been given, while X,, H„ f, are of the third order. Retaining t«rms of the third nrder, we have 0,'= Jtf,' + (3//„tf, + /T.//,) cos n( + Jff.' COS 3«i + ff,//, COS 371^ *,*, = (H^ E„ + J^i K^) 008 n( + Jff.f , C08 3ji(, ^' = iff,'cosrU + iZ/,'tMB3n(. Substituting these valuer in the xmall terms of (19), (20), and from (32), (23) in the two tirst terms, we get the following 8 equations, correct to the third order, c,ff,+ »7,//,' = (24), c, - «'a, + 3y, (2ff, + ff.) + 27, (jT, + jr.) + 3fi, tf,» - 0...{25), (c-4n'a,)fl, + Jy,Z/,' = (26), (e,-&n'a,)H,+ Sy,Btff,+y,H,K, + S,fl,' = 0...(27); c,A\+^y,B,'=Q (28), (c, - n'a,) A', + y,I/, {211, + //,) + y'H, (^K, t «,) + ja,ff,' = 0...{29), {c,-in'a,)S,+ Jy,tf," = ...(30), (c, - 9»'.m A'. + y,ff,//, + y«,A', + iS,//,' = 0...t31). se (24), (26), (28), (30) give immediately the values of S,, U„ , which are the same nd to tlie second order of approximation, 1 the substitution of these values in (27), (29), (31) determines ta i.',, A', as quantities of the third order. Tiie remaining equation 8 to detenuine n. We find as correct to this order + 38,...<38). 484 VIBRATIONS OF THE SECONB ORDER. If y,»0, this result will be found to harmonize with (9) § 67, whcntkl differences of notation are allowed for, and the first apprazimatioiitoi| is substituted in the small terms. The vibration above determined is that founded upon (21)asfirt' approximation. The other mode, in which approximatelj ^ = 0, en be investigated in like manner. If r be an even function both of ^ and ^, yi, y„ y\ h^ vanish, vk the third approximation is expressed by Indeed under this condition ^ vanishes to any order of approxi- mation. These examples may suffice to elucidate the process of approximatioB. An examination of its nature in the general case shews that the following conclusions hold good however far the approximation may be carried. (a) The solution obtained by this process is periodic, and the frequency is an even function of the amplitude of the principal term (//i). (6) The Fourier series expressive of each coordinate contains cosines only, without sines, of the multiples of nt. Thus the whole system comes to rest at the same moment of time, e.g. ^ = 0, and then retraces its course. (c) The coefficient of cos ml in the series for any coordinate is of the rth order (at least) in the amplitude (U^ of the principal term. For example, the series of the third approximation, in which higher powers of H^ than H^ are neglected, stop at cos 37i/. (d) There are as many types of solution as degrees of freedom ; but, it need hardly be said, the various solutions are not superposable. One important reservation has yet to be made. It has been assumed that all the factors, such as (c, — 4n'a,) in (30), are finite, that is, that no coincidence occurs between a harmonic of the actual frequency and the natural frequency of some other mode of infinitesimal vibration. Otherwise, some of the coefficients, originally assumed to be subordinate, e.g. K^ in (30), become infinite, and the approximalioii breaks down. We are thus precluded from obtaining a aolutiai some of the cases where we should mop^ As an example of th*" vibrations in one < VIBRATIONS OF THE SECOND ORDER. 485 Iptained in. a cyliDdrical tube with stopped ends. The equation to lie isfied throughout, (4) § 249, is of the form \'U) (le (W Ind the procedure suggested by the general theory is t y = a--»-y„ + w, cosjii +y,co8 2ni + ..., = Ii„ '^II„> a2x ■ ffia sin 3a! + . ■ /?B Bin 3ai + . , hth In tiie (irat approximation y = x+ H„ sin x cob nt. But when we proceed to a second approximatit (4«'-4)ffn--i"'^ii'. 11 with n equal to 1, so that the method breaks down. 2j'cos2iie in the value of y, originally supposed to be si with an intinite coefficient. The tenu ibordinate, It is possible that we have here on explanation of the difficulty of using long narrow pipes to speak in their gravest mode. The behaviour of a system vibrating under the action of an impressed force may be treated in a very similar manner. Taking, for example, the case of two degrees of freedom already considered in respect of its free ^nbrations, let us suppose that the impreeaed *,-JF,co8;>f, ♦, = (33), In that the solution to a eft-h ..(31). & tirjt approximation is Kth substitution of /> for n equations (22), (23) are still applicable, miting equations (24) to (31), except that in (25) the left-hand memlier is to be multiplied by H, and that on the right E, is to be substituted for zero. This equation now serves to determine £f,, iuBtead of, an before, to determine it, s evident that in this way a truly periodic solution can always kbuilt up. The period is that of the force, and the phases are such % the entire qratem comes to rest at the moment when the force is k maximum (jiositive or negative). After this tlie previous course I in the caw of free vibrations, each seriee of cosines i whan the aign of f i» rereraed. NOTE TO §273\ A MBTHOD of obtaining Poisaon's solution (8) given by Liouville^is worthy of notice. If r be the polar radius vector measured from any point O, and the general differential equation be integrated over the volume included between spherical surfaces of radii r and r + dr, we find on ta-ansformft- tion of the second integral by Green's theorem in which X = jj^dUr^ that is to say is proportional to the mean value of ^ reckoned over the spherical surface of radius r. Equation (a) may be regarded as an extension of (1) § 279 ; it may also be proved from the expression (5) § 241 for V*^ in terms of the ordinary polar co-ordinates r, By o>. The general solution of (a) is rX = x(«<+^) + ^(«<-^) (fi\ where \ and B are arbitrary functions ; but, as in § 279, if the pole be not a source, \ {at) + B (at) = 0, so that rX = x(«< + *")-x(«^-*") (r)- It appears from (y) that at 0, when r = 0, X = 2^' {cU\ which is therefore also the value of 4ir<^ at at time t. Again from (y) ^ ,, , d(r\) d{r\) so that 2x'(r) = [^.-i^J^^^^^. or in the notation of § 273 By writing cU in place of r in (8) 4ir(^, which agrees with (8) § 273. ' This note appeared ii * Lionville, torn. L p. 1, APPENDIX A. (§ 307'.) CORRECTION FOR OPEN END. The problem of detern lining tlie correction for the open end of a be is one of considerable difficulty, even wlien there is aii infinite age. It is proved in the text {§ 3QT} thiit the correction a is greater w It, and less than (S/3<r) R. The latter value is obtained by Iculating the energy of the motion on the supposition that the velocity .rallel to the axis is constant over the plane of the mouth, and inparing this enet^ with the square of the total current. The actual iJocity, no douht, increases from the centre outwards, becoming infinite the sharp edge ; and the assumption of a constant value is a some- tat violent one. Nevertheless the value of a so calculated turns out i be not greatly in excess of the truth. It is evident that we lould be justified in expecting a very good result, if we assume an ial velocity of the form 1 t/tr'/.ff' + ^V/^, denotinf; the distance of the point considered from the centre of the mth, and then determine ^ and fi' so as to make the whole energy a nimum. The ener/.'y so calculated, though necessarily in excess, must B a very good approximation to the truth. In carrying out this plan we have two distinct problems to deal with, the determination of the motion (1) outside, and (3) inside the cylinder. The former, being the easier, we will take first. The conditions are that ^ vanish at infinity, and that when x = 0, d^/dce vanish, except over the area of the circle r-H, where d^!<lx= 1 + ^r'Z-fi' + /iV/^ (1). r these circumstances we know (g 278} that 1 .Cd4>da *— 2;;JJ^7 ■* f denotes the distance of the point where ^ is to be estimated t element of area dir. Now Ittii »p|iendii an*'aied in tbo Orat edition. -■(2), 488 CORRECTION FOR OPEN SND. if P represent the potential on itself of a diK of radius R^ whose density = 1 + /if«/i2* + /i'^/^. The value of P is to be calculated bj the method employed in the ten (§ 307) for a uniform density. At the edge of the diaoy when cut dovi to radius a, we have the potential ^ . 20fka^ 356/4V and thus »wS^ f. U 5 , 314 , 214 , 89 _) on effecting the integration. This quantity divided by w gives twice the kinetic energy of the motion defined by (1). The total current = J*2«rrfr(l+^J + ,i'~) = iriP(l+i^ + J^') (5). We have next to consider the problem of determining the moticm of an incompressible fluid within a rigid cylinder under the conditions that the axial velocity shall be uniform when as = - oo , and when x = shall be of the form d4>ldx = 1 + fif^/R" + fi'f^/R*. It will conduce to clearness if we separate from ^, that part of it which corresponds to a uniform flow. Thus, if we take if/ will correspond to a motion which vanishes when x is numerically great. When x=^0, cAA/cfo: = M(r'-i) + ft'(r*-i) (6), if for the sake of brevity we put jR = 1. Now ijf may be expanded in the series ^ = 2ap«^^.(pr) (7), where p denotes a root of the equation Jo'(P) = (8)'. Each term of thiiS series satisfies the condition of giving no radial ^ l%e nomerioal values of the roots are apr |»i= 8-881705, p^-- IT4. Bcity, when r - I ; and no motion of any kind, when aj- — so . s to determme the ooeiEcients a^ ao aa to satisfy (6), when x • = to I" = 1, we must have iKnce mnltiplyiug by J,(pr) rdr and integrating from to 1, P«r [J. iP)]' = ^ I '■''^•^0 {yr) {(. {r= - i) * ^' (r- - J)!. ? term on the left, except one, vanishing by the property of the For the rigbt-hand side i rdrJ,(pr)^0, ['/rfrJ„(pr) = (i-p)/,{,-); The velocity-potential ^ of the whole motion is thus the summation extending to all the admissible values of p. We have □ow to Snd the energy of motion of so much of the fluid as is included between a; = 0, and x-~l, where / is so great that the velocity is there sensibly constant. y Green's theorem ByC la(kinetic energy) =f^^2)n-A- (x^O)- j <f, tf* 2wrrfr (x -I)- at the second term ia t/(1 + Jfi + if(')'. n calculating the first term, we must remember that if />, and Pf 490 CORRECTION FOR OPEN £ND. Thus = 16irS{/i + 2M'(l-|.)}V*- 8 Accordingly, on restoring E, 2 (kinetic energy) = wE^l (1 + J/* + ifi*)* To this must be added the energy of the motion on the positiye aide of X = 0. On the whole 2 kinetic energy _ I 16 ^ J o ' /i ^ M * s (current)' "i^"^^J?(l + i,x + J^')' ^ j'* "^ ->* V p/J ^" ■" Sir* A (1 + i/* + if^T Hence, if a be the correction to the length, SwalSB =-. [1 + liM + HI/ + (6» Sp- + A) m' + {24,r(Sp--8Sp-') + |||}Mfi' + {24,r (Sp- - 16S/>-' + 64Sp-») + ^} ^^-(l + Jm + JmO*- By numerical calculation from the values of p Sj5-» = -00128266 ; S;?-» - 82;?-'= -00061255, 2/?-' - 162p-' + 642jt?-' = 00030351, and thus 3ira/8i? = [1 + -9333333^1 + -5980951 ft' + -2622728 ^t' + -363223 fif/ + -1307634 fi^-5-(l + ift + JmT •0666667^1 + 0685716^1^- 1227 28^'- -029890Atft^- -0196523/4^ (11). The fraction on the right is the ratio of two quadratic functions of fly fjLj and our object is to determine its maximum value. In general if S and S' be two quadratic functions, the maximum and minimum values of « = S-r-S" are given by the cubic equation - A2-» + ez'^ - 0'«-^ + A' = 0, where ^S' = a/i' + bfi^ + c + 2//x' + 2gfi + 2hfifi, S' = a'fjL^ + 6'/" + C+ 2/V + 2g'fx + 2AV/, A = a6c-H2M-a/'^6<^-cA«=<^'-^>^^"'^>-^' = (ftc -/«) a' + (ca - ^ 6' + (oft - A«) c' + 2(^A-a/)/'+2(V-8i CORRECTION FOR OPEN END. 491 and &, A', are derived from nnd A by interchanging the accented and unacct^nted letters. In the present case, since .S" is a product of linear factoi's, A' = ; and since the two factors are the same, 0' = 0, so that s = A -;■ simply, ^substituting the numerical values, and effecting the calculationa, we tind z - -0289864, which is the maximom value of the fraction consistent with real values of /i anil /i'. The corresponding value of o is -82422 ft than which the true correction cannot be greater. If weiisBumefi' = 0, the greatest value of : then possible is '0343(>3, which gives a =-828146^'. On the other hand if we put /i = 0, the maximum value of z comes out -027653, whence a = -825353 R. It would appear from this result that the variable part uf the normal velocity at the mouth ia better represented by a term varying as r*, than by one varying as r*. The value a = -8242 R is probably pretty close to the truth. If the normal velocity be assumed constant, o - -848826 R ; if of the form 1 + /ir*, a = -82815 R, when /i is suitably determined ; and when the form 1 + fit' + fit*, containing another arbitrary constant, is made the foundation of the calculation, we get a = -8242 R. The true value of a is probably about -83 R. In the case of ^ - 0, the minimum energy corresponds to ;i' - 1-103, so that d^ldx=\ + l-103r'/^. On this supposition the normal velocity of the edge (r - R) would bo about double of that near the centre. I Notee on Benset'a fuaclinnn. Phil. Mag. Not. 1873. INDEX OF AUTHORS. Aiiy.i. 49; n. 127 Ango, 1.8 AristoUe, i. 182 Anerbaoh, n. 452, 478, 476, 477 Barrett, n. 401, 402, 404, 407 Barry, n. 402 BasMt, I. 482 ; n. 18, 841, 875, 876, 428 Beets, n. 870 BeU, C, n. 868, 409 Bell, Graham, x. 470, 471 ; n. 474 Bernard, x. 847 Bernoulli, i. 188, 181, 255 ; il 114, 201 Bertrand, i. 100 Betti, I. 157 Bidone, ii. 855, 857 BiUet, n. 119 Blaikley, ii. 115, 208, 219, 235 Boltzmann, ii. 228, 438 Boole, n. 87 Bosanquet, n. 17, 47, 201, 202, 216, 448, 444, 455, 457, 464, 466 Bosscha, IX. 48, 283 Boorget, i. 829, 880, 846, 347 Boutey, n. 115 Boys, II. 45 Brandt, i. 192 ; ii. 444 Brayais, i. 2 ; ii. 48 Bryan, x. 887 Baff, II. 356 Buijs Ballot, n. 155 Burton, i. 78; ii. 88, 207, 276 Gagniard de la Tour, x. 5 Csaohy, l 255, 417; n. ^ Cayaill^-Goll, n. 219 Chladni, i. 254, 861, 862, 867, 371, 31 879, 381, 887; n. 61, 227, 846, 458 Gbree, x. 252; ii. 272, 428 Christiansen, i. 168 Chrystal, i. 445, 449 Clarke, x. 60, 87 Clebsch, I. 297, 845, 858 ; n. 429 Clement, n. 21, 22 CoUadon, i. 8; n. 817 Crum Brown, x. 72 D*Alembert, 1. 170, 225 Davis, n. 225 De la Biye, ii. 230 De Morgan, ii. 23 Derham, n. 187 Deschanel, x. 7 Desor, n. 137 Desormes, n. 21, 22 Donders, n. 476 Donkin, i. 196, 209, 212, 253, 285, 28< 297, 298, 299, 353 Doppler, II. 155, 156 Dove, X. 446; ii. 440 Duhamel, n. 70, 114, 297 Duhem, i. 150 Dulong, n. 61 Dvof4k, n. 41, 48, 60, 88S» Eamshaw, xi. 86, ^ Edison, z. 474- Edser, n. Ellis, n. L ^^^^1 INDEX OF AUTHOHS. 493 M Her. r. 170. 161,356; ii. 51, 201 ^^^1 nrett, I. 7; u. 130 32 ^^^H ring/n. *7* ^^^H ner, a. 463 SeUia. L 24, 36, 08. 99, 100, 104, lOS^^ 109, 139, 253, 353. 363, 402. 483, 434. 1 i»d«y, t. Ba. ase ; n. 326. 227. 3i(J 436, 443, 466, 476, 478; ii. 7. 10, 11. 1 akoet, I. 387 13. 16, 23, 43, 44. 106, 129. 137. 389. 1 rmat. u. 126, 133 397. 813, 841. 345, 360. 370. 376. 400. 1 rraris, r. 473 467 irrers. ii. 936, 34B Kitohlioll. 1. 29C. 345. 863. 368, 362, 863. 1 irbes, n. 336 869, 370. 371; ii, 99. 148. 216. 319. nirier, r. 34, 26, 118, 303, 303 330, 321. 333. S34. 336. 337. 333 KBOel, ». 82. liy. 130. 131 Koenig, R.. t 86, 383 ; ri. 48, 62, 16o. rondB. 1. 213, 475. 477 ; n, 430 460. 461, 463, 463. 464, 467. 468, 476 Koenig, W.. n. 44. 46. 427 tnw. I. 410. 41G •imuD, 8., L SGS KolkieV, n. 193 DldiDgham, n. 47 Hnndt, i. 168; ii. 30, 46. 47, 67. 68. 59. ovi. II. 403 60, 74, 75, 836 ra;. n. 399 ►Men, I. 352; il 10,86,144,348 La Conr, r. 67 -OwenMll, i. 378 ' Qripon, n. 30a, 223 438; n.4, 5, 6.51, 201.468 Onthrie. lu 4S. 800 liBmb. 1. 157. 305. 417. 418. 419, 4H3. 463, 463 ; n. 18, 206. 376. 391, 429 HillWtem, n. 463 Lapla<». I. 109, 148; n. 19. 30, 86, 330 Huidel, L 9 Lam6, t. 315. 318. 363 HBDsen. I. S21, 329 Leeonte. n. 401, 402 403. Lenard. Ii. 871 L 483, 464, 467, 469. 470 Le Boux. ti. 48 ■Salne. it. 390, 397 Leslie, ii. 226, 230, 244, 246 ' ^htlmholtz, t. 13. 13. U, 16. G2, 69. 68. Liouville, L 317, 319. 221. 223, 294; i ■ 154. 1E7. 190, laa, 197. 198. 309 313. n. 486 ■ 231. 434, 435; it. S, 11, 86, 94. 108, H 143, 146, 146. 164. 175. 177, 187 106, Liworiui, u. 187 ^ 301,208,316,220.221,234,236 352, LiBssjom. 1. 29. 31. 33, 34. G2, 356, 285; 319. 376, 406, ch. iziii pauim n. 350 Henry. I. 434, 438. 440: 11.129.135 138 Lloyd, u. 476. 477, 478 Hennum. n. 458. 459, 460, 461, 474, Lodge, A.. I. 860 476, 476. 477. 478 Lodge. 0., 1. 483, 449. 456 ^Bu«alMl, J.. 1. 14S: n. 65. 310. 246 Lommal. 1.330; 11.143.167.399 ^■farU, t. 433, 4T4; □. 437 Lore. I. 806. 403, 407, 409. 415. 483; ^B^t^ n. 438 Mach. u. 110. 155 *61. 1 MB«na8, n, 356, 8.^7, 364. 371 Mmliii-Ti, ,. ,H3. 370 11 J1I9 . 11. 348 lt:0. 374,433, 435. 437. 1I'>U. 494 INDEX OF AUTHOBa Mayer, n. 22 BtUyer, A. M., l 71, 88; n. 80, 41, 56, 118, 156, 411, 444, 450, 451, 460, 462 MoKendriok, n. 474 MoLeod, X. 60, 87 McMahon, x. 829 ; xi. 267, 298 Melde, i. 81, 197, 848, 887; n. 220 Meraexme, x. 8, 181, 182, 188 MicheU, x. 214, 805, 844 Moll, I. 2; XI. 47 NewtoD, n. 18, 19 Oberbeok, x. 462 Ohm, X. 12, 16, 18 Oppel, n. 72 Page, XI. 225 P6an(m,L 296 Perry, n. 250 Petzval, XI. 155 Pfaondler, n. 452 Pipping, IX. 478 Plateau, x. 84; n. 860, 868, 864, 868, 871, 875, 402, 406 PockelB, I. 848, 846 Pockhammer, x. 252, 257 ; n. 428 Poisson, I. 255, 264, 845, 858, 862, 869 ; n. 21, 84, 85, 88, 41, 57, 64, 85, 97, 107, 249, 815, 417, 429, 486 Poncelet, u. 850 Preeoe, x. 473, 474; xi. 478 Preston, n. 80 Preyer, ii. 482, 488, 457 Salmon, x. 125 Savart, x. 258, 846, 847, 849, 882, 3M, 868, 869 ; n. 61, 201, 815, 860, MH, 868, 864, 868, 870, 871, 402, 40S,I» Savart, N., n. 77 Sohaifgotioh, n. S38 Scheibler, x. 60, 61, 68, 88, 183 ; xl 4tt Sohneebeli, n. 290, 886 Scbaik, van, n. 221, 482 Schnster, xi. 119 Schwartz, x. 150 ScoU Buasell, ix. 158, 860 Seebeok, i. 184, 285, 284, 285, 286, 29S. 300,801; n. 225, 452 Seebeok, A., ix. 826 Sellmeier, x. 168 Sidgwick, Mrs, n. 407 Simpson, x. 898 Smith, n. 220 Somof, X. 109 Sondhanss, n. 88, 175, 187, 188, 20S, 220, 228, 227, 228, 280, 281, 410 Soret, n. 142 Stampfer, n. 48 Stokes, X. 128, 129, 185, 803, 329, 475; n. 6, 12, 24, 25, 26, 28, 85, 36, 99. 101, 108, 110, 132, 237, 239, 240, 243, 246, 284, 804, 806, 315, 320, 341, 406. 417, 427 Stone, E. J., ii. 48 Stone, W. H., n. 235 Strehlke, i. 255, 277, 285, 362, 363, 375 Stroh, n. 478 Strouhal, n. 412, 413, 414 Storm, I. 217, 219, 221, 294 Qoincke, ii. 65, 210 Bankine, i. 477; n. 23 Regnaolt, x. 2; ix. 23, 47, 48 Beynolds, i. 476, 477 ; n. 129, 185, 188 Biccati, i. 255 Bidont, II. 402 Biemann, i. 181, 811, 828, 842 ; u. 88, 39,40 Biess, II. 283 Bijke, II. 232, 233 Bowland, n. 96 Bonth, I. 109, 120, 122, 123, 125, 189, 141, 142 Btloker, u. 457, 459 Tait, I. 24, 25, 72, 98, 104, 108, 109, 139, 253, 352, 402, 473, 475 ; n. 10, 44, 106, 289, 318 Taylor Brook, 1. 181 Terquem, l 253 Thompson, S. P., n. 440 Thomson, James, ii. 130 Thomson, J. J., i. 488, 442, 448^ Thomson, W., tee Kelvin Todhunter, i. 296; n. 286, Tdpler, 1.35; 11.110,228^ Treyelyan, n. 224, 225 Tyndall, 1. 18 ; xi. 60 147, 225, 228, 8] 405^406 INDEX OF AUTHORS. Verfet, a. 138 TioUe. II. 48 Voigt, I. 373 Volkamnn, u. 19 Watenton. ii. 30 Webb, n. S73 Weber, i. 377 Weber. H.,l. *S7 WeTtbeim, 1.353; i 495 377, 383,161, 4S3; WhefttBtone, i. Jl. 228, 472 Wieo, n. 440 WiUis, II. 470,471, 473 Yoong, I. 7. 88, 181. 182, 187, 191. 331, 385, 239, 241. 377 ; u. 72. 8fi, 168 INDEX OF SUBJECTS. t> »t Abnormal disperaion, i. 168 Absolute pitch, l 85, 183 Aeolian barp, i. 212; n. 418 „ tones, n. 412 Aerial vibrations, general equations of, n. 97 ,, in reotangnlar ohamber, n. 70 , , maintained by Heat, n. 226 Air, Velocity of Sound in, i. 2 „ viseosity of, n. 818 Amplitude, 1. 19 „ of but just audible sounds, ii. 488 Ampton bell, i. 892 Analogy of fluid motions with heat and electricity, n. 18 Analysis of sounds by ear, i. 14, 191 ; n. ch. xxiu Aperture, conductivity of elliptic, n. 177 Approximately simple systems, i. 113 Approximation, second, i. 76, 78 ; n. 480 Arbitrary initial disturbance, il 98, 417 Atmospheric refraction, n. 130 Attenuation by distance, i. 3 Attractions due to fluid motion, ii. 43 Audibility, amplitude necessary for, n. 433 „ of one sound in presence of another, ii. 444 Audition, binormal. Facts and Theories of, ch. xxiu It Balance, induction, of Hughes, i. 446 Barrett's observations upon sensitive flames, ii. 401 Bar, loaded, i. 249 Bars, circular, i. 804 „ expression for V, i. 267 „ Foarier*8 solution, i. 90% »f ti If Bars, groups of waves, z. 801 „ initial conditions, x. 960 „ lateral vibrations of^ z. 855 „ loaded, i. 288 „ longitudinal vibrations of, i. %i „ normal functions in various oa 1.272 „ permanent tension, z. 296 „ positions of nodes, i. 887 „ variable density, z. 894 Beat-notes, n. 468 Beats, I. 22; u. 444 due to overtones, i. 26 ; n. 4fr of bells, I. 889 of chords, u. 465 „ of fifth, n. 464 „ of fourth, u. 465 ,, of octave, u. 464 of third, n. 465 of upper partials, ii. 453 slow versuM quick, i. 61 Bell emits no sound along axis, n. 11 „ sounded in hydrogen, n. 839 Bell's experiments, n. 409 „ „ on liquid jets, n. 8 Bells, beats of, i. 389 „ Belgian, i. 393 „ church, I. 391 false octaves, i. 394 nodal meridians, i. 389 „ observations upon glass, z. 390 Bending, potential energy of, z. 8ff7« M 363, 411, 426 . Bertrand's theorem, 1. 100 :jdi Bessel's functions, n. 91, 1( 364,897 imaginac roots ot >t ti fi »» II II II II II \\ \^ ^^I^J^I^H INDEX OF SUB-IECTS. 497 Crispations, investigated by Faraday, Binl^Bll, 11. HI, 410 11.346 Currents, free, in cylinder, i. 461 Bow, nctioii of. 1. U12 induced electrical, I. 43(i Bojle'B Uw, 11. ly initial, I. 430 CyUndsr, vibrations within a cloeed. ii. 300 Cyliaders. liquid, under oapiUnry force, Cuble rormnla of LorJ Kslnu, i. 41(0 11. 352 Capillarity, ii. 3*8 Cylindrical obstacle, it. 3011 Capillnty tension determined hj ripples, shell, conditions of inciten- 11. 31G sion, 1. 399 Chamler, lectangulai, ii. IQtl „ shell, effect of friction, t. 3SB Character of Sounds, i. 13; li. 470 1.387 Chladni's figuros, i. 368 shell, potential and kinetic Chords, beats of, tl. 465 energies, I. 385 shell vibrating in two dimen. Clement and Duaonues' eiperimenH. It. sions, 1. H84 •il shell, tangential vibrationa, i. CloodB. aeoustic. u. 136 3BS CwxiEtanoe o( smalt motions, i. 105 ColMonnotdrops. 11.369 D'AJembcrt'B solution of differential Comma, i. 10 equation. I. 226 Damping of vibrations within the ear IM ,, ,, random vibrations. 1, 96 theory, n. 449 Conditiona. initial, i. 127 Conducting screen, 1. 160 DeiiHity, string of variable, l. 216 Conductivity of apertures, ri. 173, 175 DiBrtenee-tonea, ii. 456 twoks. 11. 181 HelmholtE's theory of, approximately cylindri- 11. 466, 460 cal tnbes. II. IM o( seoond order, :i. 464 order of magnitude, ii. Conjogalc property, i. 127, 263, 36« 4IIG Conical ahetl. i. 3^9 „ tube. II. 113, 114 oar. It. 457 Diffraction. Ti. 139 463 spectra, u. HO defined by beats of upper partials. n. 4M II. 416, IIS ConstrainlJi. seTeral, i. 138 Disc, shadowofcircomr.il. 148 Cotitinuily. eqnaliou of. il. 3 .. Btroboacopic. 1. 36; 11.407 1 ■.luvcyarioe of Souud ly Wire*, i. 3 ,. tospended, forces upon, ii. 44 oonnal.(. 107 function, t. 102 ^Dm r"~a .. for viaooua flnid. ^B, 11. 315 ^^^h !■■ I:n<:a. I. 4a, 102. 130 498 INDEX OF SUBJECTS. Diverging waves, n. 123, 239 Dominant, i. 8 Drums, i. 848 Doppler*8 principle, n. 154 Driving point, 1. 158 Drops, collisions of, ii. 869 electrified, ii. 874 vibrations of, il 871 Dynamical similarity, n. 429 n tf »» tt Echoes, harmonic, n. 152 Elastic solid, potential of strain, n. 313 Elasticity, comparison of notations, i. 358 Electrical system, i. 126 „ vibrations, i. 438 Electric fork, i. 65 Electricity, conducting screen, i. 460 conductors in parallel, i. 441 Edison's transmitter, i. 474 effect upon a small fountain, 11.369 free currents in cylinder, i . 461 Heaviside*8 theory of wires, 1.467 Hughes* apparatus, i. 458 induced currents, i. 436 induction balance. Wheat- stone's bridge, i. 449 inductometer, i. 457 initial currents, i. 439 resistance of wires to alter- nating currents, i. 464 telephone, i. 471 transmitter, i. 470 Electrified drops, ii. 874 Electromagnet and leyden, i. 484 „ forced vibrations, i. 435 Elliptic aperture, conductivity of, n. 177 „ comparison with circular, n. 179 Enclosure, vibrations in two dimensions within a circular, n. 297 Enclosure, vibrations of a gas contained within a spherical, ii. 264 End, correction for open, ii. 487 Energy emitted from vibrating spherical surface, u. 252 kinetic, i. 96 law of, verified in reflection, n. 85 of spherical waves, ii. 112 „ when confined in a conical tube, u. »» «» ♦• ft ♦» »» ti ft ff *f »t t» Energy, potential, i. 93 of eftndensaticii, ilIj «, bending, i. 25$ tr&nnDUSBion of eneigy in plni aerinl waTos, n. 16 Equal roots of determinantal e^aitan, z. 109 Tempenunent, z. 11 Equations, Lagnuig^'s, z. 100 Equilibriom theozy, z. 188 Fabrics, interferenoeof partial zvtelHM from, zz. 811 „ passage of sound throni^ ilSII Faraday's investigations on criqiati«ii, n. 846 Fatigue of ear, n. 446 Fermat*s principle of least time, n. 196 Fifth, I. 8 „ beats of, ii. 464 Flame, reflection of sound from, n. 83 „ sensitive for diffraction ezperi- mente, ii. 141 Flames, sensitive, ii. 400 singing, ii. 227 ff »t Sondhauss' experiments upon, n. 227 Fluid, perfect, ii. 1 Fog signals, ii. 137 Force applied at a single point, 1. 134 „ at one point of elastic solid, n. 425 Forced electrical vibrations, i. 435 vibration, i. 46, 68, 145 „ of string, r. 192 Fork for intermittent illumination, l 84 „ electric, i. 65 „ ideal, i. 58 „ opposing action of two prongs, n. 806 Forks for experiments on interference, n. 117 „ tuning-, I. 59 Fountain, disturbed by electrieilj,; Fourier's solution for tions of ban „ Theorem« ' Fourth, z. 8 If Frsp ft tf ^^^^^^H^^H INDEX OF stiarECTs. 499 Intinitiea ooouning wbon n + KU = U, II. refraoted waves, ii. ^ 398 nones, i[. 118 friction fluid. II. 313 Instability, i. 73, 143 FuDotiaus. uormftl. 1. 118 of eUctriQed drops, r. a74 .. jets. n. 860 „ vortex motion, It, 378 lulBUsily, m«ftn, i. 39 •J7 „ ., „ free vtbralion, r. 138 Oeuanliied CootdiDatsB. i. 91 Intarferenee, r. SO vibration8,i.71.166;ii.410 Onting cireulBT. n. Ii3 Interrupter, fork, i. 66, 45S Interval, emallest eonsouant, ii. 4S) retrftotiOD. n. 78 InlorvaU, i. 7, 8 .. theorera, Helniholti-o citenBion Invursiuu of luterials, i. « of, 11. 144 Oroii|>s of waves, i. HOI UrrosMIicteriiifi. 1. 1[)4 Jet inlerrupler, i. 456; n. 368 Jets. Uell's oxperimeutB, u. 3ti8 Harmonia curve, i. 31 eoboM, 11. 153 Kale. L 8 .. iuBlabilityof,duotovorticitj.ii.380 „ wave leagtli of luaximimi instabi- H«rmonie«. beaW of imperfBot, ii. 407 lity, II. BGl Harmonium, absolute pilch by, I. 8R ., under electrical inflnenoe, u. S69 Harp. ..^liaa. 1. 2Vi ; ii. 413 ., used tu Olid tlie tension ol reoeully Head as m obstacle, n. 413 formed luifaota, ii. 369 Heat, analogy with flnid motion, ii. 13 ,, varioose or ainuousV, ii. 403 ., vibrations about ■ circular (jgnre, ii. 357 means of, ii. 234 HeaU, Bpeciflo, n. 30 Kaleidophone, i. »'J UntviMde'a tbeoi? ot electrioal projinMa- lion in irirea, i. 467 Kettle -dcnms, i. 344 Hrfmholts'fl ntension of arean'i. iheo. Key-nolc. i. 8 rem. II. lit Kmetic energy, I. ilf. repiprocftt theoieiii. u. U!> Hooke's law, i. 171 ot sound in narrow tubes, ii. ais Hnjgens' prinoiplc, ii. 1111 Hagbea' apparatus, t. 463 1. 83 UydrogED, bell sounded in, ii. 3Jt9 Kuiidf. tube, ii. 17, .'.7, an aamo.. ti. 337 Lasraniiu'ii ciiuatiunB, i, 100 r-„,..,l«M, 1.9(1 Ihrorom in fluid motion, n. 6 number nwM"Bai7 to dvSnt.' Laplaae's correction tn velocity of Sound, Biloh. 11. 453 It. 10, 20 ■ flaU, n. !1 ' 44). Lnlcral inertia of bar*, i, 331 vibrations of ba»>, i. 36S 500 INDEX OF SUBJECTS. Lealia's eiperiment of bell Kinek in hyixogta, n. 339 LeTden Mid «l«etniiiuipiet, t. 4S4 Lionrills'i theorem, i. 333 Liquid oylinderuidOkpilUTy force, ii. 862 Liaujoai' Figora, i, 38 „ plwnomenciD, c. 349 Lottd CMTJed by atriDg, i. 68 Loaded ipting, i. 67 LongitndinBl Vibratioiu, i. 343 LondncM of SoimdB, i. IS Low note* from flunes, u. 33S HaiDtenftnce of fteriftt vibratioiu by baat, 11.33$ „ Tibntions, i. 79, 81 Htaa, effect of inoreue in, i. Ill Helde'i experiment, i. 81 Hembrmne*, boaodar; ui ftppioiim»te circle, i. S87 ., Boarget'a obMmtioDB od, 1.847 drculu, 1. 316 elliptical bonnduy, i. 348 foroed vibntioQi, i, 349 form of mBiimom period, I. 341 loaded, I. 331 Dodal flgnres of, t. SSI potential energy, i. 307 reolangnlar, i. 807 triangular, t. 817 Ueraenne's laws tor vibration u( string!, I. 183 MtoroMwpe, vibration, i. 84 Modulation, 1. 10 Moiature, effect of, on Telooity of Sound, 11.30 Motional foroea, i. 104 MotioQH, ooeiiEtenoe of nnall, i. 105 Maltiple souroeB, ii. 249 Multiply-oonneoted Bpaaee, n. 11 Mnsical lounds, i. 4 Narrow tubee, propagations of soond in, II. 319 Nodal lines foroireular membrane,!. 381 „ „ „ rectangular membrane, 1.814 „ „ of aquare platea, i. S74 „ meridiana of bells, I. 889, SSI ffodMsndliOopa, n. U.TI, 4*n Node* of vlbr»ting abri Nonnal ooonliiwtea, i. „ fnnotionB, i. IJ for Uluial ribcaliM I , oompMiiaoB <rf (elartidql, I 490 Note* and Ntnses, i. 4 „ Tones, i. 18 Obataele, eylindritml. n. SO „ in eUatio aoUd, x ■pherioal, i. S7S Octave, Beata of, n. 4U corre^Kinda to 3 : 1, i. 7, 9 Ohm'a law, ezoepttona to, ii. 443 One degree of freedom, i. 4t Open end, oonditian for, ti. 63, 196 .. „ oorreotion for, n. 4S7 ., „ experiments upon 301 Order, vibrations of the m Organ-inpee, ii. 316 „ inflnenoe of wind in dii- maintenanoe of vibratiaD. OrertoueB, i. 13 absolate pitch by. i. 88 bert way of hnriog, n. 446 Pendnlons vibration, t. 19 Period, 1. 19 „ calculation of, i. 44 Periodie vibration, i. 6 Period* of tree vtbiationa, r. lOB ,. Iateta!vilirationofbaT».i,inJ „ torreotiingul&rmoDibrane,l^B] ., stationary io value, i. 109 Permanent type, waves ol, n. 3 Peraiatanoea, theorem n Phase, I. 19 iflaenoe qo* HMi^^^^^^^H I Nil EX OF 8UB.IECTS. 501 1-Llch, I. i. VA Beaotion at dririnK point, i, 158 „ absolute. 1. 86 „ eetimalionof.il. 433 Kcciprocal relation, i. 93, !*o. 08, 150 ., high, bird-oallflqf. 11. 411 theorem, n. 145 P „ number of impulses neccasar? fur HectanBiilnr chamber, ii. 70, 166 membrane, i. 307 H •' '-aiKO or BudibQiCy, ii. 433 Plate of air, n. 74 ^B „ related to Frequency, t. 6 Reed inatmrnents, ii. 234 ^B „ standard. I, 9 ,. interrupter, t. 467 ^^hne waves of aurial vibration, ii. 16 ^F]7 tcflcotion of, ir. 427 11,78 ^Koeau'a theory of jeU, II. S6t ., &om a oorriigated smface, it. ^Hhle plane, t. 401 8!) ^H.„ vibrating oirciilar, rtactiou of air plate of air of Hnite ^f npon. 11. lf>'i IhiokncBs, a. 87 ^Plates, oireular. i. 359 porons waU, it. 830 .. clamped odKB, i. 3(17 ,. curved BurfHces, ii. 126 ., comparison with olservalinn. i. „ strata of voijing tempe- 3(12 rature, n. 83 .. conditions for free edge. :. 357 .. wall, u. 77 ,. curved, i. 395 of waves at a Junction of two ., gravaal mode of square, i, 379 strings, I. -234 ., Kirohhofl's theory, i. 3G3. 370 „ w»veBiQela8tiasolid,il.427 ., nodal lino* by ajmmetry. i. H81 tQtBl. n. 84 „ oscillation of nodes, t. 3GS l(e/rootion, atmospheric, n. 180 ., potential energy of bendlDjj.i. 353 by wind, II. 133, 13,^ ., rectaugiilar. i. 371 ., theory of a special case, i. 373 heats, 11. 23 „ vibrations of. i. 352 Point, most ^onural motion of a. o( n ,. forces of, I. 137 systom executing simple vibrations. n- 470 of wirea to alternating our- PoisBon-siutei^l. II, 3H. 41 renla, i. 464 ,, Hulutioii fur Btbitrnry initial cases, 1. 5(1 roroui walls, II. 328 rutt^iitiiil c;iii.T(^j, 1. 112. Sfi3 multiple, a. 1K9 ^.n.'.i,ling, 1,356 Bcsonator, ii. 447 1 ;. -:■■. . |.Mi.. ■■! 11. a, 14 anddoublBSource. 11. 314 ^Jj|jjr||j||^|VK^^«i cloeetosoiiree. ri. 311 ^^■HHK|ta dun-inx. of. II. 318 ^^^^^^^^^^hfefeB ^^^^^^^^^^^^^^^^^B n- 203 ^^^^^^^^^^^^^^H" I.I'd vibmUon of. n- 195 ^^^^^^^^^^^^^^^^^^k - of energy from, n. 198 ■ HI or more, 11.215 ^^^^^^^^^^^^^^^^^m ^bn«Uir« and forks, i, H6 ^^^^^^^^^^^^^^^^^M ^^^H ^^^^^1»7 502 INDEX OF SUBJECTS. lieflonaton, repulsion of, ii. 42 „ theory of, ii. 170 Riemann's equations, n. 89 Bijke's Sound, n. 282 Ring, vibrations of, i. 888 Rings, cironlar, vibrations of, i. 804 Ripples, used for determination of capil- lary tension, ii. 846 Roots of determinantal equation, i. 189 Routh's theorems, 1. 140 Sand, movements of, i. 868 Savart's observations upon jets, u. 868, 871 Second approximation, i. 76, 78 ; ii. 480 „ order, phenomena of, ii. 41 Secondaiy drouit, influence of, i. 160, 487 „ waves, due to variation of medium, ii. 150 Self-induction, i. 160, 487, 434 Sensitive flames, ii. 400 „ jets of liquid in liquid, n. 406 Shadow caused by sphere, ii. 255 ,, of circular disc, ii. 148 Shadows, ii. 119 Shell, cylindrical, i. 384 „ effect of rotation, i. 887 „ observations by Fenker, i. 887 „ potential and kinetic energies, i. 385 tangential vibrations, i. 388 Shells, I. 395 „ conditions of inextension, i. 398 „ conical, i. 399 ,, cylindrical, poti^ntial energy, i. 403 „ ,, extcnnional vibra- tions, I. 407 ,, potential energy of bending, i. 411 „ fiexural and cxtensional vibra- tions, I. 396 „ normal inextensional modes, i. 401 „ spherical, i. 401, 417, 420 Signals, fog, ii. 135 Silence, points of, due to interference, II. 116 Similarity, dynamical, ii. 410, 413, 429 Singing flames, ii. 29*7 Smoke jets, aenn tt ft »t >• tt tt tt Smoke jets, periodie view ol, n. 405 Solid bodies, vibrstioiie of, u. 415 elaaUe plAne waves, n. 416 limited initial dlBtarbaiiee, a fl? imali obataole in, n. 420 Sondhanes' obaomttione upon biri-cdi. II. 410 Sonometer, 1. 188 Sound, movementB of, i. 868 Sooroe, linear, n. 4S1 of harmonio tjrpe, n. 105 of sound, direction oi; n. 411 Souroes, moltiple, n. 849 „ simple and donble, zl 146 Sparks for intermittent iliamination, l9( Speaking trumpet, n. 113, 138 „ tabes, I. 8 Specific heats, ii. 20 Sphere, commnnication of motion to ur from Tibrating, xx. 828 obstructing, on which pliae waves impinge, n. S7S „ pressure npon, n. 279 Spherical enclosure, gas contained with- in a, n. 864 waves, energy propagated, n. 112 harmonics, table of zonal, n. 251 sheet of gas, n. 285 „ transition to two dimen- sions, n. 296 waves, n. 109 Spring, I. 67 Standard of pitch, i. 9, 60 Standing waves on running water, ii. 350 „ jets of liquid in liquid, ii. 406 Statical theorems, i. 92, 95 Steel, velocity of sound in a wire of, l 245 Steps, reflection from, ii. 453 Stokes, investigation of communication of vibration from sounding body to a gas, ii. 289 on effect of radiation on propa- gation of Sound, n. 84 theorem. ?. 128 Stop-cocI i^orbiiig tive : Strean Striatii »t »t »t «t If tf II INDEX Of aUIUECTS. unlysis ofsnunJs hj l!io ear, Torapenturc, effect of. o 1.200 It aljsolnlfilj Axed. noitelMd, J. 304 (orced vibrfttioM o(. i. 192 iiDpGrfect fleiibilily, i. 339 m*«B Doiiceatrat«d in eqnidiBtaot pnints, I. ITi uodeH DDder applied force, i. 223 iiorniNl Modes, i. IS5 of pjuioforte. t. lUl „ variable denRitj', i, 115, '215 partml Diffenntiai Eqnalioa, i. 177 propagation of waves along, t. 334 reQoolion at a jnnotioD, i. 336 Secbeck'a obsetVBtioDU, t. 1S4 ,. atretched on spliiirica] aarlaoe. I. 313 „ tone* form a mnaioal note, I. IBl ., tranBTerie Tibrationi of. i. 170 .. lalaeii of T and V, i. 178 „ Tibrationn started by plucking, i. 1M8 ,, ., ..Bblow.l.lSS ,. violin, t. SOW ., with loud, I. 53 „ tvo MtMhcd miuseji, i. IGS BtroboBoopio diso, i. BS ; ii. 407 Stronhal'a obiiervationB upuii ii<(ilian loiiee, u. 413 Sturm's llieoremit, i. 317 8nbaidonoe, rattw »f, I. 138 SDmmation-tone. ti. 4S9 finpBTpoaitioii, prineiple of, i. 49 Snppl; tube, influence of, in llamcn, II. 33!) Syrt'ii, I. ii^ II. 468 „ fur deteiminJns pitoh. i. Tdepbone oxperimcDt i influence on velocity of wand, It. 39 Tenaion, capillary, determined by me- thod of ripples, n. 34G Telling beUB, i. 393 Theory, Belmholtz'e, of audition, ii. 448 Thiri, I. 8 „ major, bcata of, ii. 4Bu Time, principle of least, n. 13fi Tone correepondi to nmple vibration. . 17; .447 Tones and Notes, i. 13 „ pure from forks, t. S9; i Tonic, I. 8 Tonometer, Scbcibler'e, i, C3 Toraional Tibrations ol bars, i. xoa .-. Transformation to sums of winares, t. 106 Transition, gradual, of deniity, t. 335 Transverse vibtatiuns in elastlu solids, II. 4ie Trevdyao's rocker, n. 224 TriftnguUr membrane, i. 317 Trumpet, speaking, it. 113, 138 Tnbe. DDlimited, containing simple source, ii. ISH Tubes, branched, ,, Kundfs, 1 ,, lectangnli variable suctiof ,, vibrations in. i Tuning by beats, [. 23 Twelfth (3 ; 1), i. 7 Two degrees of freedoi Tyndall's bigb presxai II. 101 Type, chanRB of, it 3J 47; I oandoctlug Variable section, tubes uf, ii. (IT Vehicle necesMry. i. 1 I ourreni audible, Telodty and condensatian, relation be- I. 473 twceu, ii. 15. 3fi plak. 1.367 ,. in Air. i. 2 (•« Electridly). theory of, .. independent of Intensity and I. 471 Pitch. 1. 3 LAnperatnent, I. 10; II, 44G minimam, of waves on water, oqaal, t. 10 n. M5 imlore, eSaot of, in Uterlng via- „ ■9W^ " ■INDEX OF SUBJECTS. ^^^"^""^ Water, surface waves on, u. 344 npon. II. 47 ,, waves on running, n. 350 in water, n. 30 Waves, aerial, diverging in two dimfn- sions, 11. 304 «. 19, 2(J New ton 'b calculation, 11. 416 11. 18 ., diverging, n. 133 -potential, II. 4, 8, 15 of permanent type, lu 32 Velocitien.Bjstem star ted with given, I. !»H ,. on water, IE. 344 Vibration, (orccd. i. 63 plane, energy half potential »nd Vibrations, forced and free, i, 49 half kinetic, ii. 17 of the second ordir, ii, 480 Violin string, I. 309 31 Viscosity, analogy with elastic Btraiu, of aerial vibration. It. U n, 313 defined, II. 313 11. 416 ., positive and negative, t. 227 of nir, 11. 313 „ progressive, i, 47S . „ „ subject to dompiQfi 1. 232 in, 11. 316. 332 „ secondoiy, doe to variation of „ threads of, 11. 376 medium, u. 150 „ ,, timsverBB vibrations in. ,, sphenoal, ii. 109 n. 317 „ standing, on rnnniag water, it Vortei motion and afneitive jets, n. 376 350 Vortices in Knndt's lubes, ii. 840 Vorticity, oBse of stability, ii. 384 two trains croBsing obliquelfr general equation for stratified. n. 78 11. -iBS „ layers of uniform, n. 386 kaloidophone, i. 32 Vowel A, Hermann's results, n. 47fi, 476 Wheel, phonic, i. 67 VoweU. arliflcUl, ii. 471. 477, 47B Wliiapering galleries, ii. 127 Whistle, steam, n. 233 474 Whistling by the mouth, ii. 224 „ pitcli of eharacteristic, two theo- Wind, retraction by, n, 132, 135 ries, 11. 473 Windows, how affected by explosioiiij It. Ill „ question of double resonance. Wires, coovcjaoce of sound by, i. 3, 361' II. 477 „ electriuoi ouirantH in, t. 464 „ WheaUtone and Eelmholtz's, 11. 472 Young's modulus, t. 348 ries, 11. 470 strings. 1. 187 Wall, porous, ii. 328 „ refleiion from filed, li. 77, 108 Zones of Huygens or Freanel. a. tlS ^k 141 J^ "*"" ^__™™ ^^^^ • * ■ J t i ■:':-i i; i, ;^ ■■ :*: r'- . '■•■ IT-' u ij L.I ^ *. »r