GC
30 /
A
UC-NRLF
SB E7D
ELEMENTARY THEORY
THE TIDES
T.K.ABBOTT.
LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
Class
ELEMENTARY THEORY
or
THE TIDES:
THE FUNDAMENTAL THEOREMS DEMONSTRA TED
WITHOUT MATHEMATICS,
AND THK
INFLUENCE ON THE LENGTH OF THE DAY DISCUSSED,
BY
T. K. ABBOTT, B.D., D.Lrrr.,
FELLOW OF TRINITY COLLEOK, DUBLIN.
UNIVERSITY SfcoND EDITION, REVISED.
LONGMANS, GREEN, AND CO
LONDON, NEW YORK, AND BOMBAY.
1901.
First Edition, 1888.
Second Edition, zgoz.
In Italian. By Prof. Edoardo de Ferrari (with additional
elucidations by the author], 8 pp. 67, Pistoia, iqoi.
Printed at THE DUBLIN UNIVERSITY PRESS, by PONSONBV & WELDRICK.
PREFACE,
THE substance of the following -pages has already ap-
peared, partly in the Philosophical Magazine, 1871, 1872,
and the Quarterly Journal of Mathematics, 1872, and
partly in Hermathena, 1882. Hitherto correct statements
about the Tides have been confined to treatises which
employ the resources of the higher mathematics. Other
works almost without exception* repeat such erroneous
statements as that the place of high water without
friction would be under the moon, and that high water
is retarded by friction. No apology then is needed for
the publication, in a more accessible form, of the present
Essay, in which the fundamental theorems are deduced
from elementary physical principles without the use of
mathematics, except for quantitative calculations. The
problem of the influence of the Tides on the length of
the day is discussed in a similar method.
* The only exception with which I am acquainted is Stubbs' edition of
Brinkley's Astronomy, in which the reasoning of this Essay is adopted.
IV PREFACE.
For the benefit of readers who may wish to see the
latter problem analytically treated, I have given in an
Appendix the substance of Sir Greorge Airy's investi-
gation.
In this second edition, I have introduced some correc-
tions and further elucidations, but have omitted the
Appendix. The subject of the first part of this Essay
has been treated with more detail by the late Dr. Samuel
Haughton in a paper entitled " Mathematical Principles
of Tidal Theory and Observation," published in Her-
mathena, 1879, p. 563. The present Essay has been
translated into Italian by Professor Edoardo de Ferrari,
of Empoli.
ELEMENTARY THEORY OF THE TIDES.
THE attractive force of the sun or moon on the solid mass
of the earth is the same as if the latter were concentrated
at the centre. But the attractive force, on movable par-
ticles at the surface, is greater than this on the side nearest
to the moon (or sun), and less on the opposite side. Thi&
excess and defect constitute the tide-producing force.
Confining ourselves to the moon :
The direction of the tide-producing force is always tangen-
tial, and towards the line joining the centres of the Earth
and moon.
The following (Newton's) construction represents this
force in direction and magnitude.
Fig. i.
I
Let xl (fig. ~^) be perpendicular to EH, and let Im = mn
= EL Then if ME represent the attractive force of
the moon at the centre E, xn will represent the whole
B
THE TIDES.
disturbing force in magnitude and direction. The proof
is as follows :
Let it be borne in mind that ME is about sixty times
the radius of the earth. Hence, if we consider Ml
the error cannot exceed = th part. For the
error is greatest when x coincides with D (fig. 2). But
in that case ML* = ME* + r* = (60r) 2 +r* = 3601 r*
,-. MD = rv/3601 = r ( 60 + -!-) nearly. Therefore
Again, Mn, Mm, Ml, ME, being arithmetical proportionals
with a difference varying from to ~th, may be regarded
as geometrical proportionals ; the greatest possible error
being about the same as before. Hence, Mn : Mloi MX : :
Ml* or MX* : ME*, i.e. as the moon's force at E : force at x ;
therefore, if MX represent the moon's force at x, Mn will
represent the force at E in magnitude and direction, and
the difference or disturbing force will be represented in
magnitude and direction by xti. In order to have a fixed
scale we must represent the force at the centre by ME.
On this scale xn is in the nearer hemisphere too small,
and in the more remote too large, in the proportion of
o -I
Mn to ME. This error is at most ^ths = -r^th. This will
bO 20
be considered by-and-by, but for the present it may be
overlooked.
The vertical component of xn is xh. The tangential
TIDAL FORCE.
or THE
UNIVERSITY
OF
actually exceeaea tne lorce ot gravity. And, as will be
seeu by-and-by, it is too minute to produce any indirect
effect.
The tide-producing force then always acts towards EM
(in the direction of the arrows in fig. 2) . From this we
can deduce theorems relating to the place of high and low
water, &c., without requiring to determine the magnitude
of the force, which will be hereafter taken into account.
At present we need only observe that it is very small
compared with gravity.
First, then, let us consider the case of water limited to
an equatorial canal. The moon being supposed in the
equator, we shall establish the following theorems :
i. If there were no friction it would be low water
under the moon, and high water in quadratures.
n. Friction accelerates the times of high and low water.
in. In addition to the oscillatory motion of the water
there is a constant current produced by the action of the
moon.
iv. The eifect of friction on this is to increase the
length of the day.
B2
THE TIDES.
ERRATA
Page 3, line 4 should read
= El sin MEx = Ex cos MEx sin MEx = Jr sin 2 Jf Ex,
MD -ME=^
ME
120 2 x 60 x 60
nearly.
Again, Mn, Mm, Ml, ME, being arithmetical proportionals
with a difference varying from to ~~th, may be regarded
as geometrical proportionals ; the greatest possible error
being about the same as before. Hence, Mn : Ml or MX : :
Ml 2 or Mx 2 : ME 2 , i.e. as the moon's force at E : force at x ;
therefore, if MX represent the moon's force at x, Mn will
represent the force at E in magnitude and direction, and
the difference or disturbing force will be represented in
magnitude and direction by xn. In order to have a fixed
scale we must represent the force at the centre by ME.
On this scale xn is in the nearer hemisphere too small,
and in the more remote too large, in the proportion of
O I
Mn to ME. This error is at most ^ths = rrr
bO 4,0
This will
be considered by-and-by, but for the present it may be
overlooked.
The vertical component of xn is xh. The tangential
TIDAL FORCE. 3
component of xn is equal and parallel to nh, the
perpendicular on the radius, and is, therefore, pro-
portional to Ik, which is one- third of nh. Note that
Ik = El sin HEi<= Ex cos If^sin MEfr= Jr sin 2 Mft
that is to say, the tangential force is proportional to sin 2
(angle from moon). The same reasoning applies to the
dotted letters in the further hemisphere.
The vertical component being in the same line as gravity
(either in the same or an opposite direction) cannot directly
produce any motion. In fact, it could not do so unless it
actually exceeded the force of gravity. And, as will be
seen by-and-by, it is too minute to produce any indirect
effect.
The tide-producing force then always acts towards EM
(in the direction of the arrows in fig. 2). From this we
can deduce theorems relating to the place of high and low
water, &c., without requiring to determine the magnitude
of the force, which will be hereafter taken into account.
At present we need only observe that it is very small
compared with gravity.
First, then, let us consider the case of water limited to
an equatorial canal. The moon being supposed in the
equator, we shall establish the following theorems :
i. If there were no friction it would be low water
under the moon, and high water in quadratures.
n. Friction accelerates the times of high and low water.
in. In addition to the oscillatory motion of the water
there is a constant current produced by the action of the
moon.
iv. The effect of friction on this is to increase the
length of the day.
B2
4 THE TIDES.
I. Without friction it would be loiv ivater under the moon,
and high icater in quadratures.
I suppose the moon to be fixed, and the earth rotating
in the direction ABCD, carrying the ocean with it. That
the ocean is so carried is a fact of experience.
Now, in the course of one lunar day every particle of
the ocean is subjected to precisely the same forces, acting
in the same order of succession and for the same periods,
being accelerated for about one quarter of a day, viz.,
while passing from B to C ; then retarded for a quarter,
from C to D, and so on. The variation in the amount of the
force does not concern us, being the same for every particle.
This being so, it is obvious that those particles will be
Fig. 2.
moving faster which have been for a longer time acted
on by an accelerating force, and the velocity will be a
maximum when the accelerating force has acted during
its full period, viz. through one quadrant. On the other
hand, those particles will be moving slower which have
been longer acted on by a retarding force, and the
absolute velocity will be a minimum when the retarding
force has acted during its full period, or through one
quadrant. The maximum velocity is therefore at A and
(7, the minimum at B and D.
Secondly, it is clear that the tide will be rising where
PLACE OF HIGH WATER IN EQUATORIAL CANAL. 5
^each portion of water is moving faster than that just in
advance of it ; or, in other words, where water is flowing in
faster than it flows out. Where this process has gone on
for the maximum time the tide will be highest. On the
other hand, the tide will be falling where the water is
moving slower than that in advance of it or, in other
words, is flowing out faster than it flows in. Where this
has continued for the maximum time the tide is lowest.
Now consider any point s in the quadrant BC. The
water now passing s has been subject to an accelerating
force during the whole time since it passed B, longer
therefore than any particles behind it, as at r. It is there-
fore moving faster ; and as the water in the space r s is
thus flowing out faster than it flows in, the tide is falling.
This is the case through the whole quadrant BC.
At C the force changes and becomes a retarding force.
The particle at y has been subject to this retarding force
longer than one behind it, as at #, and is therefore moving
slower. Here, therefore, water is flowing in faster than it
flows out, and the tide is rising ; and this holds through
the quadrant CD. What is said of these quadrants holds
also of those opposite to them ; the tide is falling all
through DA and rising through AB. Hence it is highest
at B and D, lowest at A and C. Where will the tide be
falling fastest ? Clearly where the difference of velocity
between r and s is greatest, i.e. where the amount of force
to which the water at s has been subject since it passed r
is greatest in other words, where the force is at its
maximum, viz. at / (fig. 3), 45 from C. Similarly it
will be rising fastest at that point in the quadrant CD
where the force is greatest, viz. at ff, 45 from C.
6
THE TIDES.
On the whole, then, the water in the supposed equatorial
canal assumes the form of an ellipse ; and as it is the earth
that is rotating, this ellipse does not change its absolute
position except with the moon's motion ; only the water
accompanying the rotating earth moves fastest at A and
(7, and is there lowest; and slowest at B and D, and
is there highest. Relatively to the earth it is moving
westward from e to /and from g to h ; eastward from /to
g and from h to e. At A, J9, (7, D, the particles are in
their mean places ; at ,/, g, h, they are farthest from their
mean places, and change the direction of their relative
motion. This is represented in fig. 3, where the inside
BD
AC
Fig. 4.
arrows show the direction of the earth's motion ; the out-
side arrows that of the relative motion of the water. The
path of any one particle may be represented by fig. 4,
where the letter A indicates the position of the particle
when its mean place is at A in fig. 3.
FKICTION ACCELERATES HIGH WATER. 7
II. Friction accelerates the times of high and low water.
The theorem that the effect of friction is to accelerate
the time of high and low water admits of an equally
simple proof. As the water approaches (7, the tangential
force diminishes gradually to zero at C. Therefore it
must have been equal to the force of friction at some point
n (fig. 1), after which friction prevails and the velocity
diminishes. It is therefore low water at n. Approach-
ing D, the ocean is moving slower than the earth ; there-
fore here friction tends to accelerate it, while the retarding
force is decreasing to zero. The two forces, then, must be
equal at same point 0, after which the velocity again
increases. It is high water therefore at o.
It sounds paradoxical to say that friction " accelerates "
high water. The paradox is only apparent. Friction
checks the motion, so that the water stops rising or falling
sooner than it otherwise would ; and thus we may speak
of the phase of high or low water being accelerated.
The preceding proof assumes that the ocean is carried
round by the earth in its rotation. This amounts to
supposing that it has not assumed a position of equili-
brium.*
* It is important to observe that we are not entitled to assume that when
the tide is rising fastest the water is flowing in from both sides. This is
by no means evident. The rate of rise depends on the difference in velocity
between two successive parts of the ocean, and this may be greater when
the two velocities have the same sign than when they have different
signs. Taking into consideration the rotation of the earth, the assumption
amounts to this that the tide is rising fastest where the velocity of the
ocean is just equal to that of the earth. This is certainly not evident : in
fact it would not be true if the tangential force did not decrease at the same
rate on both sides of each of the four maxima. It ought not, therefore, to
be assumed, but deduced.
8 THE TIDES.
It is a priori an admissible suggestion that the ocean is
in a state of equilibrium under the moon's action, i.e. that
it is absolutely at rest (relatively to the moon), while the
earth rotates. But this would imply an apparent move-
ment of the whole body of water with a velocity equal and
opposite to that of the earth's rotation, i.e. at the equator
there would be an apparent current of about 1000 miles
per hour. As this does not correspond to the fact, the
hypothesis is practically inadmissible ; but when friction
is considered it appears theoretically inadmissible also.
For in this case friction would continually act in the
same direction, and its effect would be to make the east-
ward forces preponderate ; so that although the ocean
should be supposed at rest at first, it would ultimately be
dragged round by the earth. The actual form of the
earth, moreover, in which the equatorial ocean is inter-
rupted by continents, would render this equilibrium of the
ocean impossible.
III. There is a constant current westward produced by
the moon's disturbing force.
This occurs from two causes. First, the water in the
supposed equatorial canal has now taken the form of an
ellipse ; and, in consequence of friction, the places of
greatest elevation are not at B and _D, but somewhere
in the quadrants BA, CD. Now, the moon's tangential
force, hn, is, cceteris paribus, proportional to Ex the distance
of the particles attracted from the centre of the earth. It
follows that it is greater in the quadrants BA y CD,
than in the other two ; but in the former the force is
CONSTANT WESTWARD CURRENT. 9
retarding ; in the latter it is accelerating ; therefore the
retarding force exceeds the accelerating, and produces a
permanent westward motion.
Secondly, the water having reached its mean place at n,
and passed it with its greatest eastward velocity, it is, when
it reaches (7, eastward of its mean place, i.e. it is nearer to
g. On the whole way before reaching g it is nearer to
that point than if there were no friction ; but on passing g
it begins to move westward ; but its eastward excursion
having been shortened by friction, it begins this motion to
the west of where it would otherwise be. At o it again
arrives at its mean place, which, without friction, it would
not reach until D. Thus, in the whole quadrant CD, the
particles are nearer to g than if friction had not operated.
But the tangential force is greater the nearer the particles
are to g, being proportional to sin 2 (angle from moon)
= cos 2 (angle from/ or g) ; hence the force in the quadrant
CD, which is a retarding force, is increased. After passing
its mean place at 0, the water going westward is, on arriv-
ing at Z), west of its mean place ; and until it reaches h
it continues to be west of the place which it would have
occupied had friction not operated, i.e. friction withdraws
it from h. At h its westward excursion is stopped, and it
begins to return eastward. But now from h to A. it is east-
ward of the place due to it without friction. Thus through-
out this quadrant the particles are brought farther from h
by friction. But here the force is accelerating. Therefore
the force in the accelerating quadrants is diminished, while
that in the retarding quadrants is increased, and hence
again a balance of retarding force, and therefore a current
westward. Or thus : Without friction, the quadrant
10 THE TIDES.
fg, throughout which the water is moving faster than
the earth, has its middle point at C ; and the following
quadrant in which it is slower has its middle point at D.
These quadrants are, therefore, equally divided between the
accelerating and the retarding quadrants. With friction,
the middle points being displaced to n and o respectively,
the water is moving faster than the earth through more than
half the quadrant BC, and slower through more than half
the quadrant CD ; and similarly in the opposite quadrants.
But BC, DE are the accelerating quadrants, and CD, AB
the retarding quadrants. Therefore the water is exposed
for a longer time to the retarding than to the accelerating
force.
We have here, therefore, a vera causa which may pos-
sibly be effective in retarding the earth's rotation. An
attempt will presently be made to estimate the maximum
amount of this effect.
On the Quantitative Valuation of the Tidal Disturbance.
To determine more precisely the magnitude of the dis-
turbing force :
The moon's attraction at x : force of gravity : :
moon's mass earth's mass 1
- -, or nearly = : 81J ; there-
fore the whole attraction of the moon (represented in fig. 1
by ME] = i . But on the same scale the greatest
o
tide-producing force is represented by-r (the greatest value
HEIGHT OF TIDE. 11
of Ik being - r), i.e. by -^-. (This we shall call H. ) The
greatest tangential force then is
9 9 1
40x81ix60 2 11,736,000 365,000'
nearly. The tangential force at any given point is
= - Hsiu 2o) (<o being the angle from the moon) . Neglect-
ing the effect of pressure, the effect of the moon's action
through one quadrant : the effect of this maximum con-
tinued for the same period : : 1 : JTT (this appears from the
construction in fig. 5).
The number of seconds in the mean lunar day being
89432, the velocity generated in one-fourth of this time is
22358 1 ,
= ^ feet nearly.
365000 x TT 26
This is the difference between the greatest eastward
and the greatest westward velocity ; therefore the greatest
eastward velocity is , and the greatest westward velocity
0/4
is also 1.
As the same amount of water passes through a given
section in a given time, the increase in height : total
depth of the sea : : relative westward velocity of the
water : earth's velocity of rotation (relatively to the
moon). The last is about 1470 feet per second. Hence
depth of sea 1 depth of sea
the rue of the hde --- x-= --
For a sea of three miles in depth this would give for the
22
lunar tide 2*4 inches. The solar tide is about -^ of this.
DO
OF THE
IYER8ITY
or
12 THE TIDES.
The following is a geometrical construction for the velocity
and height at any place:
Round the radius OB describe a circle. Since the angle
at BcO is right, Be is equal to the perpendicular from CE,
i.e. to xl in fig. 1, and cp equal to Ik ; so that the tangential
disturbing force at a is proportional to the perpendicular cp.
M:
1'ig. 5.
If aa f be the space passed over in the rotation of the earth in
one second, the force acting on the water may be supposed
unchanged while it passes from a to a' ; and its effect during
that interval (i.e. in this quadrant, the retardation) will
also be proportional to cp or its double cf y and to the
time : that is, to aa', or the angle at 0, aOa'. Calling H
the moon's greatest tide-producing force, r the earth's
radius, and r the angular velocity = - : :
seconds in lunar day
2?r ,. , ,. Hcf aOd XT , ,
= rjTTYqs, t* 16 retardation = - x . Now the angle at
oy4o/c v T
= the angle at/, being in the same segment; and this
angle multiplied by cf= the small perpendicular cd, or pp' 9
CONSTRUCTION FOR VELOCITY AND HEIGHT. 13
which is parallel and equal to it. Therefore the whole
retardation since leaving B is proportional to the sum of
all the abscissae pp' that is, to Bp'. It is H - .
T ZTT
This represents the defect from the greatest eastward velo-
city ; and after passing its mean value at the middle point
s it represents a velocity which, relatively to the earth, is
westerly. The velocity of the current relatively to the
earth is represented hy ps.
We shall now show that the height of the tide at '
ahove its lowest point is also proportional to Bp'.
If at any point in the supposed canal a thin section be
taken, the quantity of water entering this section in a given
time is proportional to the product of the depth and the
velocity. If the water flows in a little more rapidly than
it flows out, it is clear that the increase in the quantity
contained in the section, and therefore the increase in depth,
will be proportional to the difference between these two
i i j xu /diff. of vel. x depth \
velocities and to the whole depth ~ - }.
\ length of section /
This holds as long as the change is small compared with
the whole depth. If this be supposed uniform throughout
the canal, the increase in it (that is, in the height of the
tide) at a' is therefore proportional to the retardation ;
and since the tide began to rise at B, where the velocity
began to diminish, it follows that Bp' is also proportional
to the height of the tide at a above its lowest point.
It is easy to deduce from this construction the cor-
responding formulae. For, if OB = r, we have
And since sB is proportional to the mean height, the defect
14 THE TIDES.
from this height is proportional to ps, and therefore to
08 2w.
The effect of pressure with such a tide will be extremely
small. As it operates to send the water away from its
position of greatest elevation, it will so far assist the moon's
force without changing the place of high water.
To estimate the effect of friction, the moon's force being
= - R sin 2o>, the velocity v, undisturbed by friction
= Fcos 2o> (V being the greatest velocity). If the dis-
placement caused by friction is very small, we may take
this value of v as a sufficient approximation. The friction,
assumed proportional to velocity = Vf cos 2o>, and the
velocity begins to be checked when Vf cos 2w = - ZTsin 2&>
or, tan 2, . - . _ /= _ 700 0/nearl y .
_tl O^
When the displacement (which we shall call 8) is sensible,
we shall have as follows : since the velocity is periodic and
is a maximum, positive or negative, when w = 8 ; and is = 0,
when w- 8 = 45, 135, we may assume fl=Fcos2(w-8).
Then friction = fV cos 2 (w - 8) and the net amount of
force = - H sin 2w -fV cos 2 (co - 8). Hence
H cos 2o> /Fsin 2 ( w - 8)
~2T~ ~^7~
By hypothesis this reaches its maximum, so that it is high
or low water respectively, when w = 8, 8 -f P0, 8 + 180.
m , - ,,, H cos 28 , .
Iheretore V = ~ - ; and since at these points the
AT
net force = 0, we have sin 28 = - = -f G - , and tan 28
JJ. Zr
remains as before = - 7000/. The rise of the tide in a sea
three miles in depth, will be 2 '4 cos 28 inches.
CASE OF GLOBE COVERED WITH WATER. 15
The effect of the vertical component may be estimated
as follows :
As shown above the whole attraction of the moon
= SlFx-60 5 = 29M60 nearly " 3 is re P resented in the
figure by ME. The vertical component of the disturbing
force is represented on the same scale by xh which is greatest
when x coincides with (7, and is then
ME g g
"60" = 293650 T~30 = 8800000' nearly>
This is the proportion in which the weight of a particle
directly under the moon is diminished. It is less than two
grains in a ton, or equivalent to less than T^th of an inch
in a depth of three miles with the water at a temperature
of 50 F. ; an increase in temperature of one-tenth of a
degree would produce more than a hundred times this
effect.
In the preceding demonstrations we have supposed the
water to be limited to an equatorial canal, the moon also
being in the equator. It is desirable to consider what
modifications will be introduced, first, by supposing the
earth to be uniformly covered with water ; and secondly,
by taking into account the moon's declination.
It will save repetition if we state once for all certain
general principles which we shall have to employ :
1. First, suppose an accelerating force acts alternately
in opposite directions; the effect (measured by velocity)
increases as long as the force acts in either Direction ; and
therefore the velocity in that direction is greatest at the
16 THE TIDES.
moment that the force changes its direction. Thus a
pendulum is moving fastest at the lowest point of its
oscillation.
2. Secondly, the velocity (diminishing under the coun-
teraction of the force) continues to be in the same direction
until this counter force has undone all the work accom-
plished in that direction by the previous force. If the
circumstances are alike in both directions, this will be
when the force has done half its work. This again is
precisely the case of the common pendulum. For the
present case this will be at efgh, fig. 3.
3. Thirdly, in the case before us, the water rises when
the particles behind are moving faster than those before.
The rate of rise is greatest when this difference is greatest ;
but as the effect is cumulative, the whole amount of the
rise is greatest at the moment when the difference = 0, and
is about to change to the opposite.
4. Fourthly, as in 2, this difference ceases to increase
(i.e. is greatest) when the force (or difference of forces)
producing it ceases to act ; but it is not reduced to until
the opposite force has done half its work. At this moment
the accumulation is greatest.
5. Fifthly, in the case which we are now considering,
the effective force depends on the form of the surface, and
vice versa. If, then, when this form is spherical the dif-
ference mentioned in 3 were always in the same direction,
it would continue to act until a certain permanent alte-
ration was produced. If the difference were constant,
a state of equilibrium would be attained ; but if it alter-
nately increases and diminishes, then the mean form of
GLOBE COVERED WITH WATER. 17
the surface will be the same as would be produced by a
constant force equal to the mean amount of the actual
force. The alternate excess and defect of the latter will
cause a periodical motion, just as if it were an independent
force.*
First, then, the moon being still supposed to be in the
equator, let the earth be uniformly covered with water.
The tangential force may be resolved into two compo-
nents one touching the parallel of latitude (i.e. east and
west), the other meridional. These may be regarded as
giving rise to distinct waves one east and west, the other
north and south.
The actual amount of these forces may be found as
follows :
By the previous construction (fig. 4) (ME being moon's
force at E) 9 the disturbing force at A is represented by
- | r sin 2 A M = - H sin 2 A M.
Resolved along the parallel of latitude, this is
3r sin AM cos AM sin 0.
But by the right-angled spherical triangle (fig. 6)
sin AM sin = sin MB (hour angle from moon),
and cos AM = cos MB cos AB (latitude).
* If the reader wishes to apply these considerations to the case of an
equatorial canal assumed above, it must be observed that there the elevating
force is the excess of easterly force acting on any particles of water above
that which affects those in advance, i.e. to the east of them. This excess is
positive from 45 west of the moon to 45 east (i.e. while the moon passes
from 45 east zenith distance to 45 west), then negative for 90, and so on.
C
18 THE TIDES.
Hence the component parallel to equator
= - H cos lat. sin 2 (hour angle)
(H being the greatest disturbing force). This is less than
the force in the equatorial canal in the proportion of cos
lat. : 1. But the velocity of rotation is less in the same
proportion ; hence the rise of the tide will be the same. If
this force were alone (that is, if the water moved in canals
N
M
Fig. 6.
parallel to the equator), the ocean in every circle of latitude
would take the form of an ellipse with its short axis towards
the moon. But these ellipses would not be similar unless
the depth of the sea varied as cos lat.
The effect of the meridional component is of a different
kind. Its value is
- H sin 2AM cos = - 2H cos AM sin AM cos 0.
But sin AM cos = sin AB cos MB,
and cos AM = cos MB sin AB (as above).
Therefore this component
= -2H sin AB cos AB cos 2 MB
= - H sin 2 lat. cos 2 (hour angle)
77" 77"
- sin 2 lat. + -^ sin 2 lat. cos 2 (hour angle).
The mean value of this is the first term, the effect of which
is to cause a permanent accumulation at the equator.
MOON'S DECLINATION CONSIDERED. 19
The second term, which is the variable part, represents
the tide-producing part of the force. This is positive as
long as the hour angle from the moon is less than 45 on
either side ; and in that case from the equator to lat. 45
this is an elevating force, being greater as the particles are
further from the equator : from 45 to the poles it is
depressing. In the remaining quadrants this term is
negative. Hence, by 5 and 4, the elevation at the
equator (and up to lat. 45) will be greatest (i.e. it will be
high water), 90 from the moon. Beyond lat. 45 the
depression will be greatest under the same circumstances.
In these latitudes, therefore, the effect of the former com-
ponent would be partially counteracted. It is easy,
however, to see that the variation in the meridional force
(and it is only the variation that affects the tide) is in any
latitude less than that in the force parallel to the equator
in the proportion of sin lat. : 1 ; for the latter varies from
H cos lat. to - H cos lat. and the former from H sin lat.
cos lat. to - H sin lat. cos lat. Hence while the height of
the tide would be lessened, the place of high water would
be as before. The actual magnitude of the tide may be
ascertained as follows :
TT
The force being - - sin 2\ cos 2tn (\ being lat., and m
6
hour angle).
In order to apply the same method of summing as in
fig. 5, we write this
- ^sin 2 A sin 2(45 - w).
02
20 THE TIDES.
Then, as in fig. 5,
77"
velocity = j- sin 2X cos 2(45 - m).
(T being as in p. 12 the angular velocity relatively to the
moon).
Now, this increase in the height of the water depends on
the difference in velocity at two points, of which the lati-
tude is X and X + a where a is very small. In fact the
difference in the amount of the water entering the section
and leaving it is equal to the area of the section multiplied
by the difference of velocity, and the decrease or increase
of height is equal to this difference of amount divided by
the area of the surface, i.e.
. . . , depth x increase of velocity
Decrease of height =
ra
(the height increasing when the velocity is diminishing,
and vice versa).
But a being small,
sin 2(X + a) - sin 2X = 2a cos 2X ;
7) TT
.'. decrease of height = - cos 2X sin 2m (D being depth),
DHcoB2\
and total rise or fall = - - cos 2m.
r 4r 2
This is = cos 2X x half the total rise or fall in the equa-
torial canal with the moon in the equator.
After passing 45 latitude, the decrease in the circles of
latitude becomes important. If we assume our meridional
canal to be of uniform width, then the canals will gradually
overlap, the tide thus diminishing until at the pole, as is
obvious, there will be no tide.
MOON'S DECLINATION CONSIDERED.
21
Let us now consider the case of the moon having a
declination, which for simplicity I shall suppose less than
22 30'. This limitation will not affect our results. We
shall, as before, take the two components separately.
With respect, then, to the component which acts parallel
to the equator. Near the equator itself the considera-
tions previously applied still hold good. Next consider
a place a, whose polar distance is less than the moon's
declination, to which therefore the moon is circumpolar,
and (with the assumed declination) alternately north and
Fig. 7.
south of the zenith. If abed be the circle of rotation of
such a place, the distance of the moon from any point in
this circle is less than that from the earth's centre. If>
then, the direction of the rotation be abed it is obvious
that the water will be accelerated through the whole semi-
circle, abcj and retarded through cda. The same reasoning
as already employed will show that it will be low water
at c and high water at a. Now take an intermediate place
whose circle of rotation is Imno. Here the water is retarded
and rising from I to m and from n to o ; and accelerated
22 THE TIDES.
and falling from m to n and from o to /, and the interval
olm is less than mno. Hence the tide is lowest at n and
not so low at /, and it is high water at m and o. Hence
we have a diurnal tide in addition to the semi-diurnal,
this diurnal tide becoming of more importance as we recede
from the equator until the co-latitude = moon's declination,
when the semi-diurnal tide disappears.
At the equator the meridional component acts during half
a rotation towards the equator, and during the other half
from it, and in each case is an elevating force, which, as
before, has its greatest effect 90 from the moon. At all
places whose latitude is not greater than the moon's declina-
tion there is a permanent accumulation. In the circle abed
this component is directed towards the north at a and
towards the south at c, the points of change being where
the great circles from M touch abed. This gives rise to
a north and south oscillation. The southerly force being
the greater, there will be a residual depression of the water
in this region. The depressing force, however, varies,
being greatest at a and at c* while the elevating force
is greatest where the great circle tangents from M
meet the circle. Hence, by 4 and 5 the tide will b&
lowest at the latter points and high at the former, and
there will be a diurnal tide, as in the former case. Com-
bining this with the former result, the effect of both
components together will be to give high water at a.
* If the moon's declination were greater than 22 30', c might be less than
45 from M, in which case the force there would be an elevating one.
Again, at a place whose latitude was greater than 22 30', and less than the
moon's declination, the moon's least nadir distance (= IN] would be greater
than 45, and the force depressing.
MOON'S DECLINATION CONSIDERED. 23
It is not necessary to enter into a detailed examination
of the state of things at intermediate places. It is not
difficult to see that, as long as the moon's declination is
small, there will be an accumulation effected by the me-
ridional component extending from the equator to about
lat. 45, and that, as the moon's declination increases, the
accumulation becomes less at the equator and greater
towards 45, until the declination reaches 45. With a
declination greater than 45 there would be an accumu-
lation at the poles; and obviously, if the moon were at
the pole, the ocean would take the form of a prolate
spheroid.
The place of high water at any latitude, as far as
this is due to the meridional component, would be easily
found ; but the proportionate effect of the meridional
and equatorial components depends partly on the latitude
and partly on the moon's declination ; and it does not
come within the scope of the present essay to solve this
problem. It is sufficient to observe that the importance
of the meridional component increases with the declination
as well as with the latitude. If the moon were at the
pole this force would be alone ; and, whatever the declina-
tion, it alone produces an effect at the pole.
The same reasoning applies, mutatis mutandis, to the
solar tide.
It was remarked, on p. 10, that the disturbing force is
slightly greater on the side nearer the moon than on the
remoter side. The effect of this inequality is to produce
a small diurnal tide.
24 THE TIDES.
ON THE EFFECT OF THE TIDES ON THE LENGTH OF
THE DAY.
1. Historical.
In the year 1754 the Berlin Academy proposed, as the
subject for a prize essay, the question, " Does any Cause
exist tending to Eetard the Rotation of the Earth? " What
the result of the competition was I do not know ; but the
question led to the publication by Kant of a short essay,
in which he suggested that such a retarding cause existed
in the tides. He worked out this suggestion in a rough
way, there being, as he truly said, no ascertained data on
which any trustworthy calculation could be built.
Laplace examined the question from the historical side,
with the help of the records of ancient eclipses, and came to
the conclusion that the period of rotation had not altered.
Eecently, in consequence of the improvement of the
lunar tables, astronomers have seen reason to re-open the
question. It has been inferred from the records of ancient
eclipses that the day is lengthening at the rate of one
second in two hundred thousand years. At first sight
this may seem to be an amount too small to leave any
trace in history. It must be remembered, however, that
in calculating what part of the earth's surface came into
the shadow of a given total eclipse, say 2500 years ago, we
have to "unwind" 2500 times 365i (= 913,125) rotations,
and a difference amounting to an eightieth of a second
between the first and last of these would in the whole
period have a very considerable effect.* M. Delaunay
* About 100 minutes: see Ball, "Elements of Astronomy," p. 377.
UNIVERSITY
EFFECT ON LENGTH OF DAY: 25
attributes the retardation to the moment of the moon's
disturbing force on the tidal prominences. He started
from the assumptions that without friction it would be
high water under the moon and anti-moon, and that friction
retards the time of high water. Both these assumptions
were erroneous ; but they so far counteracted one another
as to leave the place of high water in the same quadrants
as the true theory, viz. in the quadrants east of the
moon and anti-moon, in which the moon's force is
retarding.
Sir George Airy corrected these errors, and working out
the equations, found two terms which indicate a constant
current westward one term (the smallest) depending on
the vertical, and the other on the horizontal, displacement
of the water.*
In my own Essay on the Theory of the Tides (Quarterly
Journal of Mathematics, 1872, and Philosophical Magazine),
the effect of friction was indicated, but there was no at-
tempt to estimate it quantitatively. I am not aware that
any attempt has been made to solve this problem ;f and
indeed it would be absurd to pretend to do so with any
degree of accuracy. "What I propose to do is to estimate
the effect so far as to enable us to form a judgment as to
the actual importance of the tides as a cause retarding the
earth's rotation.
It will be convenient first to prove the following propo-
sition respecting the effect of obstacles :
* Monthly Notices of the Royal Astronomical Society, 1866, p. 221.
t See, however, Lord Kelvin, Philosophical Journal, 1866, p. 533.
He mentions also a Paper by Wm. Fernel, Astronomical Journal of
Cambridge, U.S.A., December 8, 1853, which I have not seen.
26 THE TIDES.
2. Obstacles which check the motion of the water towards
a certain point retard the time of high water, and
increase the height.
If the obstacle is a complete barrier, the tide will rise as
long as the motion of the water is towards it, and will fall
as long as the motion is from it. Hence, at 45 east of
quadratures it will be high water on the east of such an
obstacle, and low water on the west of it. The influence
of this on the time of high water at other places will
extend as far as the pressure is felt.
An obstacle not sufficient to stop motion altogether
will produce a similar effect, but of course much smaller,
in consequence of the continuity of the surface. If the
obstacle be such as to destroy half the velocity of the water,
then high water on its east side would be 30 after quadra-
tures. In both cases the height would obviously be
increased.
It appears from this that the effect of such obstacles is
in both respects the reverse of that of friction.
3. Effect of the moment of the moon's attraction on the
tidal prominences in an equatorial canal with the
moon in the equator.
This is the way in which the retardation was supposed
by Delaunay to be produced, and Thomson and Taithave
adopted the same view.*
* The statement that the earth rotates in a "friction collar," which
seems to put the matter in a nutshell, obviously assumes that the passage of
the tidal wave is the passage of a mass of water. But this is true only so
far as there is a residual westward current, which is certainly not self-
evident.
ATTRACTION ON TIDAL PROMINENCES. 27
Now, in order to estimate the greatest effect possible,
let us suppose that the greatest elevation is in the middle
of the quadrant, i.e. 45 before quadratures ; and further
that the elevation is not diminished by friction.
Let H = the moon's greatest horizontal force.
w = angle from the moon.
e = greatest elevation.
Then the tangential force at any point = ZTsin 2o>, and the
elevation = e sin 2<o. For it is proportional to the velocity,
which as we have seen (p. 14) = Fcos 2 (w - ), and if
8 = 45 this = Fsin 2w. Multiplying by the element of
the equator, and the unit of width, we get the moment
= He sin 2 2o> x rdw x r. Now sin 2 2w is always positive,
varying from to 1 and from 1 to 0. Substituting its
equivalent ^(1 - cos 4w) we divide the moment into a
periodical part ^ Her* cos 4wtfw which varies from - %Het*du>
to + -^ Her*(fo, and therefore produces no permanent effect,
and a constant part which is = -J- Hei*d&.
Summing this round the circle, and multiplying by the
coefficient of friction, we have for the whole moment Heirr z f.
Taking the density of the earth as 5, the moment of
inertia of the equatorial section of the earth is - irr*.
Dividing the former by the latter, we have for the angular
tion ~T-^-
1 depth of sea
(negative) acceleration ~T-^-
XT rr
NOW ' H 365000 6440
If we assume the depth of the sea to be 3 miles, the angu
s
lar (negative) acceleration becomes nearly = .. . .... - .
93 billions x r
28 THE TIDES.
Multiplying by the number of seconds in 100,000 years
(about 3 billions), we obtain ^j- nearly.
Now, the velocity of the earth's surface at the equator
relatively to the moon is about 1470 ; the angular velocity
1470
, .
therefore is
r
I earth's velocity
If the earth's velocity is diminished in this propor-
tion, the length of the lunar day will be increased by
89432
/ seconds = nearly T96 / seconds. To find the
effect on the actual (solar) day we must take the angular
velocity = - , when the last fraction will become .,.,, ,., /
r 47151 '
or, approximately, 1.836 seconds. The solar tide need
not here be taken into account since its effect on the pro-
minences due to the moon is alternately positive and
negative. If the displacement instead of being 45 is
= then the elevation will be (as a first approximation)
e sin 28 and the retardation l*7/sin 2$ seconds.
Now, in the case supposed, /is exceedingly small, the
friction being ultimately that of water on water. Hence we
conclude that in an unobstructed equatorial canal the
effect of the moon's attraction on the tidal prominences in
retarding the rotation would be quite insignificant, even
on the supposition above adopted, that the place of high
water is 45 before quadratures. If this place were affected
only by friction, the displacement could never reach 45,
for tan 28 = 7000 /and /must be less than unity. The
EFFECT OF KESIDUAL CURRENT WESTWARD. 29
elevation also is diminished by friction in the proportion
of cos 28 to 1. Eemembering this, it is easy to see that
the figure just given should he multiplied by ^ sin 48, so
that with the greatest possible displacement, the result would
be practically 0. In order that 8 should be = 22 J, since
tan 28 = 7000 /, / should be = -- (which is far beyond
its value) . We should then have 1 '83 f Jsin 48 = 7 nearly.
The retardation in this case would therefore be less than
one second in 700 million years. It will be seen here-
after that /is millions of times less than this.
There is another way of viewing the matter, which
does not introduce /. The following consideration explains
this :
4. Of the effect of the residual current westward due
to the change in the time of high water.
The constant force found above, = \He sin 28, produces
an accumulating westward tendency in the water. This
once impressed will continue to increase until the increase
is checked by friction, that is, until friction becomes equal
to this constant force. This occurs when
sin 28 x t x/= He sin 28,
that is, when t = -. After this the velocity of this west-
t/
ward current is constant, the constant force being expended
on counteracting friction. Therefore when we take a
sufficiently long time we may assume that the total moment
30 THE TIDES.
(of the water) is ultimately not affected by the coefficient
/. In fact the retarding force fo is then = \He sin 28.
This being premised, I shall now examine the question
from the point of view suggested by Airy.
5. Effect of the changes in the disturbing force due
to the displacement of the water.
By substituting, in the expression for the disturbing
force, the altered value of the ordinate of the water for the
original value (% + X, for #), Airy finds that the expres-
sion contains a constant term dependent on the distance
of high water from quadrature. The source of this con-
stant term may be understood from the following observa-
tion :
The particles are in their mean place at the moment of
high water and at that of low water ; at the former they
are travelling W. with their greatest velocity; at the
latter they are travelling E., also with their greatest
velocity. Now, the place of high water being W. of
quadrature, and the water moving W., it follows that
when the water reaches quadrature, approaching the moon,
it is behind, or west of the place which, without friction, it
would have occupied. On the other hand, at syzygy it is
in advance, or E. of its place. In both cases the disturb-
ing force is diminished by this displacement, the force being
greater the nearer the particles are to the middle point of
the quadrant. In other words, H sin 2w is diminished
throughout, o> being increased when over 45, and dimi-
nished when less than 45. In the following quadrant,
i.e. after passing the moon, the opposite change takes
EFFECT OF DISPLACEMENT OF WATER. 3L
place, since the particles enter it E. of the place they
would otherwise occupy, and leave it W. of their place.
Now, the former quadrant is that in which the moon's
force is accelerating, the latter that in which it is retarding.
The same observation applies to the other two quadrants.
Thus the accelarating and retarding forces are no longer
in equipoise, the latter predominating.
To calculate the effect : The maximum excursion of
the water without friction in the case of a canal three miles
deep would he about 136 feet. For the greatest velocity
= - (feet per second). Now if this continued for one-
o2
fourth of a day the space passed over would be ^ = 430
nearly. With the varying velocity ^ cos 2o> the space
traversed is less than this in the proportion of 1 to -^TT (as
in the calculation of the velocity in p. 11). It is therefore
= 272 feet nearly. This is the double excursion. There-
fore the maximum excursion on either side is 136 feet.
Assume that this is undiminished ; and assume, as before,
that it is high water 45 W. of quadratures. Then we
may assume the displacement at each point to be 136 cos 2w ;
and the moon's force being H sin 2o>, the change in the
disturbing force due to this displacement
2H cos 2w x cos 2(o per second.
r
I O
The constant part of this = H
Putting for H its value OCKAAJ an( i calculating the
32 THE TIDES.
effect continued for one lunar day (89280 seconds), we
have
11 136 100
n x - > or 7^ ^rr - > nearly.
46 r 65 millions
This acts on the whole mass of the canal. Introducing
the moments, as in p. 27, we have as the acceleration for
one day
200 mass of canal
_ V -- - ' - - .
65 millions mass of equatorial section of earth x r
With the assumed depth of sea, the latter factor =
Hence the daily angular acceleration
1 1
~ 325000 X 3300 x r'
Multiplying by the lunar days in 100,000 years (about
33 millions) we have nearly as before . This gives a
retardation of about 1'83 seconds in 100,000 years. Adding
the solar tide we have as the total 2*6 seconds. This is
on the hypothesis that the elevation is not diminished.
Introducing the necessary correction we have, as in p. 28,
2-6 sin 48.
For the reason before stated, it is unnecessary to multiply
this by the coefficient of friction.
There is a third way of viewing this cause. Owing to
the displacement of the place of high water, since that is
the point where the water is moving fastest westward, the
water is a longer time in the retarding quadrants than in
the others, e.g. on the previous hypotheses it is 136 feet
behind its place on entering the accelerating quadrant,
ACTUAL STATE OF EARTH. 33
and 136 feet in advance on leaving it. It is, therefore, in
272
that quadrant about y seconds = about '18" short of a
quarter of a lunar day. This would give a result similar
to that already found.
The preceding calculations are obviously applicable to the
case of a globe uniformly covered with water, since each
section parallel to the equator would give the same results.
The meridional wave would have no effect on the rotation.
It is not worth while to extend our calculation to the
case of the moon not being in the equator. The nett
result would be to diminish the retardation.
6. Application to the actual state of the earth's
surface.
In attempting to apply the preceding results to the
actual condition of things on the earth's surface, the fol-
lowing points must be noted :
First. On the earth as it actually is the effect of friction
proper on the tides is trifling compared with that of ob-
stacles. Against these the tidal current impinges, and in
addition the increased elevation gives the moon an in-
creased pull, which, if acting towards the obstacle, exerts
its full moment on the earth, but only for a fraction of a
day.
Secondly. The existence of a retarding influence de-
pends, as we have seen, on the place of high water being
in what I have called the retarding quadrants, i.e. less
than six hours in time later than the moon's meridian
passage. If this condition is violated, the influence might
D
34 THE TIDES.
be accelerating. Suppose a continent whose coasts run
N. and S. (as those of America may be roughly said to do) ;
then if it is high water on the east coast less than six
hours after the moon, the effect of the pull just mentioned
is retarding ; if it is high water on the west coast more
than six hours after the moon, the effect is to accelerate.
In other cases the direct effect is nil.
Now, owing to the great irregularity of distribution of
land and water, theory will not help us in determining the
times of high water ; but on consulting the Tables founded
on observation we find, for example, the following re-
sults :
In the open part of the Pacific Ocean high water is
about 30 before quadratures ; farther from the equator at
both sides it is at quadratures ; farther still it is 30 after
quadratures.
(Confining ourselves to the direct effect of the moon's
action on the tidal prominences)
The effect on-
East coast of China, ..... None.
,, India, ,,
,, Australia, . . . . ,,
,, Africa, Retarding.
,, S. America, .... ,,
,, N. America, .... None.
"West coast of S. America (Peru), . . . Accelerating.
,, N. America (California, &c.), . ,,
,, Australia, .... ,,
(It should be observed, that for the present purpose we
ought to take the time of high water where the depth of
ACTUAL STATE OF EAETH. 35
the sea begins sensibly to diminish in approaching the
coast, but this we are unable to do for want of data.)
These instances are sufficient to show the difficulty (per-
haps amounting to impossibility) of determining whether
there is any preponderance of retardation at all. At all
events, however, it is clear that the retardation, if any,
must fall very far short of the maximum.
It is to be remembered, further, that in the case con-
sidered above of a globe uniformly covered with water,
each section of the globe parallel to the equator has its
own tidal current to encounter its own inertia, and hence
the result in the case of the equatorial canal was applicable
to the entire globe. But in the case of the earth it is
not so, and this would still further dimmish the retar-
dation.
On the whole, it would appear that no certain conclusion
as to the amount of retardation of the earth's rotation by
the tides can be drawn from theoretic considerations.
I am not aware that any estimate of the value of the
coefficient / has been published. The following has
been kindly communicated to me by the late Professor
Fitz Gerald:
Let fj. = the coefficient of viscosity of water.
Then x being the ordinate of the water, let
d z V dV
(h being the depth in centimetres)
(since tf> =fv)
36 THE TIDES.
With a depth of 3 miles, h = 483000 nearly .'. W = 233000
millions nearly. The value of /u at a temperature of
10C. = 50 F. is = -013257,
15 C. = 59F ____ -011503,
20 C. = 68F ____ -010164.
If we take the third of these values, we find
3-82 billions
Then sin 23 = tan 2g - 7000 / =
546000 millions
If we introduce this value into the formulae for the
retardation given above, we arrive at the result that if
the displacement is due to friction alone, the consequent
retardation would be in fact nil. Supposing, however,
that from whatever cause the displacement amounted to
15 (and we have seen on p. 34 that over a certain limited
region it reaches 30), the formula on p. 32 would give
1-15 seconds in 100,000 years. When we consider how
far the actual state of things is from the hypothesis of
a globe uniformly covered with water 3 miles deep, with
the moon always in the equator, the fact that the retar-
dation deduced from eclipses is about two-fifths of this
is certainly curious. But we cannot attribute much im-
portance to this coincidence.
(With the value of / here given it would take about
12,000 years to produce the effect described in 4 so as to
make this formula applicable) .
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.
NOV
JUN 10 1943
JHIO'65
REC LD
pr- --
'80
09283
