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NOTE ON THE MOTION OF WAVES IN CANALS.
In a former communication* I bave endeavoured to apply the ordinary Theory of Fluid Motion to determine the law of the propagation of waves in a rectangular canal, supposing & the depression of the actual surface of the fluid below that of equilibrium very small compared with its depth; the depth y as well as the breadth B of the canal being small compared with the length of a wave. For greater generality, B and ny are supposed to vary very slowly as the horizontal co-ordinate . increases, compared with the rate of the variation of %, due to the same cause. These suppositions are not always satisfied in the propagation of the tidal wave, but in many other cases of propagation of what Mr Russel denominates the “Great Primary Wave," they are so, and his results will be found to agree very closely with our theoretical deductions. In fact, in my paper on the Motion of Waves, it has been shown that the height of a wave varies as
With regard to the effect of the breadth B, this is expressly stated by Mr Russel (vide Seventh Report of the British Association, p. 425), and the results given in the tables, p. 494, of the same work, seem to agree with our forniula as well as could be expected, considering the object of the experiments there detailed.
In order to examine more particularly the way in which the Primary Wave is propagated, let us resunic the formule,
where we have neglected the function f, which relates to the wave propagated in the direction of a negative.
Suppose, for greater simplicity, that B and my are constant, the origin of x being taken at the point where the wave commences when t = 0. Then we may, without altering in the slightest degree the nature of our formulæ, take the values,
do le pe
But for all small oscillations of a fluid, if (a, b, c) are the co-ordinates of any particle P in its primitive state, that of equilibrium suppose; (ic, y, z) the co-ordinates of P at the end of the time t, and Ø = pdt when (x, y, z) are changed into (a, b, c): we have (vide Mécanique Anolytique, Tome II. p. 313),
Applying these general expressions to the formulæ (1) we get
$=-WoF (a - eN9Y), und -=-*F (a-ing»). Neglecting (disturbance)', we have
$=-VF" (a- *V97), and consequently,
"5(a –eN99).da=-VF(a-1797), supposing for greater simplicity that the origin of the integral is at a = 0. Hence the value of x becomes
x=a+"dağ(a – Vg2). m otor