Tides and
Waves.

value of

Tide a

where X had a finite value even when dz was indefinitely small: this would lead to suppositions w The differ- of infinite forces, &c., such as were not contemplated at all in our investigation. There must, therefore, be no Sect sudden change in the values of X at the beginning and the end of our wave-function. In like manner there Theo dX must be no sudden change in the value of dz

ential co

efficients to a certain order must vanish at

the ex

d2 X

dz2

Wav

at the beginning and end of the wave-function, as such sudden

Sub

Whet

tremities of change would make the values of infinite. In like manner there must be no sudden change in the values

the wave.

of

Forms of

function

which satisfy

these con

ditions.

Then, as the equations of (224.) hold equally for still water, and for water in wavemotion, and as there is to be no sudden change in the values of the first and second and third differential coefficients, it follows also that there must be no sudden change in the value of the fourth. If, however, we are willing to suppose the sudden introduction of a finite force at a particular point of the wave, we may dispense with the last condition.

(229.) Now for the still water which precedes and follows the wave, all the differential coefficients are 0. Hence we must have,

The higher orders of differential coefficients may have any values whatever.

(230.) There is no difficulty in finding forms for X which will satisfy these conditions. For instance, if we

dX 630 b
then, on making z=0, X will =0: and, on making z=a, X will b. The value of is
dz

. 2. (a-z);

and this quantity, and its three next differential coefficients, vanish when z=0 or z≈a.
(231.) Similarly, if we had chosen to assume that for values of z greater than a, X should =0, that condition,
with all the others, would be satisfied by the assumption for the wave-function X= c.z3. (a — z)3.

(232.) It may, however, be more convenient to assume a form depending on sines and cosines. Thus,
suppose the condition to be that all particles resume, after the passing of the wave, the same place which
they had before its approach, or that X shall =0 for all values of z<0, and for all >a. It is easily seen that
dX d2X dX_d'X
the condition that X,
shall 0 for z=0 and z=a, will be satisfied by the assumption
dz' dz2' dz3' dz♦

X-sin3 ; or by any higher power of the sine; or by sin increased by any number of such quantities as

number (whole or fractional) greater than 1, and where the argument

may be carried through as many whole multiples of as we please, provided that the first value of z be >0, and the last <a. (This amounts to the same as supposing that any number of short waves, possessing the characteristic property in regard to the differential coefficients at their beginnings and ends, are piled upon the longer wave.) But if the condition is to be, that the particles are to be removed to the distance b and left 2b (3 πZ there, it will be satisfied by the assumption X=f, sin =

A higher power of the sine might have been taken; and the function might have been increased by any number of the supplementary functions mentioned above. Thus a single wave of any degree of complexity might be produced., We shall, however, for simplicity, confine ourselves to the simple form

of

octal

(233.) But we have not yet examined whether these forms of wave are consistent in all respects with the Tides and equations of wave-motion. For this purpose we must substitute the assumed value of X in the equations

Waves.

the and thereby ascertain whether any force F is necessary to maintain the particles of water in the assumed state of movement. Now taking the last assumption, and putting t-x for z; and putting "(y) for the factor 26

Verti

(it q”

Parces. depending on y, it being understood that at the surface, or when y=k, p" (k) =

where (y) is con

3п

The sum of the two last expressions, with sign changed, is the value of F, the horizontal force which must be applied to maintain this state of undulation.

(234.) It is evident that the value of F cannot generally be =0, since the different parts of the coefficients of sin(vt-x) and sin (vt-x) are not in the same proportion, and therefore those coefficients cannot vanish

2T

a

4π a

together. Therefore, in general, this discontinuous wave cannot exist without the application of force. But if
the wave be long, so that the terms divided by a may be neglected in comparison with those divided by a2, F
'g
will =0 if —2x2g p'(k)—$'(0)+

26

a2

a2

22 v

26

"(y)=0. This implies that p" (y) is constant and =

3

Expression

for force which is

necessary to main

tain this

wave.

therefore, (y)=y, and p′(k) —$'(0)= k; and the equation becomes The same thing would have been found to be true if the wave-function had consisted of any number of sines. Thus it When the appears, that a single discontinuous wave of any degree of complexity may travel on water without any force to wave is maintain it, provided, in the first place, that it satisfies the conditions laid down with regard to the differential very long, coefficients at its terminations, and in the next place, that the wave is so long that a succession of simple waves, necessary. each of that length, would travel sensibly with the velocity due to waves of infinite length.

(235.) If the single wave is moderately long, a small force will maintain it as a discontinuous wave: but if it
be short, the force must be (in proportion to the various pressures acting on the water) considerable. In fact,
each of the different terms in the wave-function represents a wave of different length; and, when the waves are
short, each of these would tend to travel on with its own peculiar velocity, which velocities are very different
for the different waves.
But when the waves are long, the peculiar velocities are very nearly the same for the
different waves.

no force is

Tides and
Waves.

In a subsequent article (410.), we shall give the theory of a single wave, acted on by any force, and travelling Tides with a velocity different from that mentioned above.

Ware

(236.) We may proceed in the same manner for the discussion of the motion of a single wave of considerable d X

Sect. 1 Theory Wares.

depth and of great length, observing that the equation for that case is -=F+gk.

d X
dt

Depth of

canal va

riable.

Assumption on

which a plausible solution may be found.

action

cal Fo

(237.) Hitherto we have supposed the depth in every part of the canal to be the same. We shall now Horiz suppose that the depth is different at different points of the canal; the variation, however, being supposed to be and gradual. We have already seen (157.), that the equations cannot be satisfied in this case: and our investigation, therefore, cannot be quite so satisfactory in its character as the investigations undertaken where the equations can be satisfied. Still we conceive that the following will be found sufficiently certain and accurate to enable us to judge with confidence of the effect of the variation of depth upon the general circumstances of the Problem. The depth being supposed variable: to find what alteration takes place in the magnitude, length, and velocity of the waves, in passing from one part of the canal to another.

waves.

my

(238.) In our former investigations, in which the horizontal bottom of the canal was taken as the axis of r, we found that the horizontal disturbance X might be represented by a collection of terms, each of which is of Emk -mak E the form A. (y +ε ̃my) cos (nt — mx), where m and n are connected by the equation n2=mg. If, Emk +επικ instead of taking the bottom of the canal, we had taken some other horizontal line for the axis of r, and if the ordinate of the bottom of the canal had then been ŋ, the expression for X would have been A (ɛTM(y—") +ɛTM("-Y)) COS (nt − mx); ɛ (k−n) - ɛTM (7—k) and the equation connecting m and n would have been n2=mg. ¿m(k−n) +ɛm (9−k)* In this case, n is constant.

η

Suppose now that in our canal of slowly varying depth there are, at different parts, portions of sensible length, whose depth is uniform through those lengths; then, through each of these lengths, the expression for X and the equation between n and m will have the same form as those above. It seems, then, not unreasonable to conjecture that the same form may apply to the parts of variable depth, or the parts where n is a function of a, and where, consequently, m will be a function of x, (for n, upon which the period of the waves depends, must be invariable through the whole extent of the disturbed water.) A must also be a function of x, whose form is yet to be determined. In regard to the term mr under the cosine, a necessity for change will be obvious. In a canal of uniform depth, ma represents the decrease of phase due to the space x, and, therefore, mh would represent the decrease of phase due to the small space h: if, then, (going upon the principle already announced,) we make the phase decrease for each small part of the canal of variable depth in the same manner as if that depth were continued uniform, we must not use mr for the decrease of the phase, but s, m. Let this integral M: then our supposition will be

dM

dx

where n is a given function of r, m in consequence is implicitly a given function of x, =m, and A is an unknown function of r, whose form is to be determined so as to satisfy as nearly as possible the equations of

Now it is well known that, as x, y, and t, are independent of each other, the differentiation expressed by

may be performed upon the quantity under the sign and the integration with regard to y may also be

d

dx

{

dr m

=m (y—n) — ¿TM (1−3)) cos nt — M

d (n3A dx m2

nt—

(generally) = {(+g"(+-) cos nt=M }

+

d In A dr m2

d (gA dx

{

m

m (y-n) m

m

_m (k-n)

'+ɛTM (7−y)) cos nt — M

+ɛTM (7−k)) cos nt —M

M}.

- ɛTM (n−k)) cos nt — M

M}.

be observed, that the two last lines destroy each other, by virtue of the equa- Expression
d2X
dt2

(241.) To facilitate the differentiation of the last term, we will remark, that the alteration of depth is supposed
dn
to be gradual, so that is small, and therefore and which depend on it are small, and therefore their

powers and products and the second differential coefficients will be extremely small, and may be neglected. If
noA
m2

then for the moment we make (εTM (y−") +ɛm(n−y))=P, in differentiating P.cos nt - M we may omit

for force necessary to maintain this

motion.

¡Expanding the differential coefficient, we should have one term multiplied by A and another by

; and by

Horiz

dx

and V

cal F

we should be able to make the expression for F=0 for any one value of y; but

Whole

force, from

to the surface,

it is not possible to make it =0 for all values of y; and thus it appears that some force, though perhaps extremely small, is necessary to maintain the sort of undulation which we have supposed.

(242) Among the different conditions on which we may fix for the determination of dA

the following appears > dx

the bottom the most reasonable; that upon the whole, from the bottom to the surface of the water, the horizontal force necessary to maintain the assumed wave-motion shall be 0, or that f, F shall =0. We cannot here perform the integration under the differential sign, because the limits of the integration will be ŋ and k, the former of which is a function of x. Expanding the differential therefore, we find F=

assumed

to be 0.

(244.) The expression for the vertical displacement of particles at the surface, or K, consists of a large term