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ON THE MOTION OF WAVES IN A VARIABLE
CANAL OF SMALL DEPTH AND WIDTH.

The equations and conditions necessary for determining the motions of fluids in every case in which it is possible to subject them to Analysis, have been long known, and will be found in the First Edition of the Mec. Anal, of Lagrange. Yet the difficulty of integrating them is such, that many of the most important questions relative to this subject seem quite beyond the present powers of Analysis, There is, however, one particular case which admits of a very simple solution. The case in question is that of an indefinitely extended canal of small breadth and depth, both of which may vary very slowly, but in other respects quite arbitrarily. This has been treated of in the following paper, and as the results obtained possess considerable simplicity, perhaps they may not be altogether unworthy the Society's notice.

The general equations of motion of a non-elastic fluid acted on by gravity {g) in the direction of the axis z, are,

«.

supposing the disturbance so small that the squares and higher powers of the velocities &c. may be neglected. In the above formulae p = pressure, p = density, and is such a function of x, y, z and t, that the velocities of the fluid particles parallel tot the three axes are

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To the equations (1) and (2) it ia requisite to add the conditions relative to the exterior surfaces of the fluid. and if A—0 be the equation of one of these surfaces, the corresponding condition is [Lagrange, Mec. Anal. Tom. ir. p. 303.

(1)1

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. dA dA dd> dA d<f> dA dd> . . . „. ... 0 = ^+^-£ + ^-| + ^^^en^ = °MA>

The equations (1) and (2) with the condition (A) applied to each of the exterior surfaces of the fluid will suffice to determine in every case the small oscillations of a non-elastic fluid, or at least in those where

udx + vdy + wdz

is an exact differential.

In what follows, however, we shall confine ourselves to the consideration of the motion of a non-elastic fluid, when two of the dimensions, viz. those parallel to y and z, are so small that $ may be expanded in a rapidly convergent series ic powers of y and 2, so that

* - +*'f + & J + + + + &c

Then if we take the surface of the fluid in equilibrium as the plane of (x, y), and suppose the sides of the rectangular canal symmetrical with respect to the plane (as, z), <f> will evidently contain none but even powers of y, and we shall have

i - & + fa +f Jl+£,.-£l+&c. (3).

Now if y ± ftx

represent the equation of the two sides of the canal, we need only satisfy one of them as

since the other will then be satisfied by the exclusion of the odd powers of y from

The equation (A) gives, since here A =*y fS,

w

Similarly, if a - 7, = 0 is the equation of the bottom of the canal.

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If moreover z » 0 be the equation of the upper surface,

<?z dxdx dt But here jl = 0; .•. also by (2) S^=-^ Substituting from (3) in (b) we get

or neglecting quantities of the order 7s,

m

Similarly (a) becomes and (c) becomes, since £ is of the order of the disturbance,

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provided as above we neglect (disturbance)'.

Again, the condition (2) gives by equating separately the coefficients of powers and. products of y and z.

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If now by means of (a), (J'), (c) we eliminate from (2'}, there results

It now only remains to integrate this equation.

For this we shall suppose /3 and y functions of x which vary very slowly, so that if written in their proper form we should have

/S = -f (a>a>), where to is a very small quantity. Then,

%=« r"' (««), ^ = »VM» &c

Hence if we allow ourselves to omit quantities of the order w*, and assume, to satisfy (4),

<f>a = Af{t + X),

where A is a function of x of the same kind as /3 and 7, we have,

omitting (^J,

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