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NOTE ON THE MOTION OF WAVES IN CANALS.
In a former communication* 1 have endeavoured to apply the ordinary Theory of Fluid Motion to determine the law of the propagation of waves in a rectangular canal, supposing f the depression of the actual surface of the fluid below that of equilibrium very small compared with its depth; the depth 7 as well as the breadth /9 of the canal being small compared with the length of a wave. For greater generality, # and 7 are supposed to vary very slowly as the horizontal co-ordinate x increases, compared with the rate of the variation of f, 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 Eussel 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 ;i3
With regard to the effect of the breadth /8, this is expressly stated by Mr Eussel (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 formula 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 resume the formula},
where we have neglected the function /, which relates to the wave propagated in the direction of x negative.
Suppose, for greater simplicity, that fi and 7 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 formulae, take the values,
+ = F(x-t\'gd (1),
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; (x, y, z) the co-ordinates of P at the end of the time t, and <b*sj'<f>dt when (x, y, z) are changed into (a, i, c). we have (vide Micanique. Analytique, Tome 11. p. 313),
* = a + ^, 2/ = b + w, i-«t^ (2).
Applying these general expressions to the formula? (1) we get
Suppose « = length of the wave when I = 0; then £(a) = 0, except when o is between the limits 0 and a. If therefore we consider a point P before the wave has reached it,
/"<*»?(« - t v'gy) -J" A*?(a) - F;
the whole volume of the fluid which would be required to fill the hollow caused by the depression f below the surface of equilibrium when t = 0. Hence we get
x being the horizontal co-ordinate of P, before the wave reaches P.
Also, let a" be the value of this co-ordinate after the wave has passed completely over P, then
J dai(a-t •fgy) -0, and x'r = a.
If f were wholly negative, or the wave were elevated above the surface of equilibrium, we should only have to write — V for V, and thus
x' — a , and x" = a.
We see therefore, in this case, that the particles of the fluid by the transit of the wave are transferred forwards in the direction of the wave's motion, and permanently deposited at rest in a new place at some distance from their original position, and that the extent of the transference is sensibly equal throughout the whole depth. These waves are called by Mr Russel, positive ones, and this result agrees with his experiments, vide p, 423. If however f were positive, or the wave wholly depressed, it follows from our formula, that the transit of the fluid particles would be in the opposite direction. The experimental investigation of those waves, called by Mr Russel, negative ones, has not yet been completed, p. 445, and the last result cannot therefore be compared with experiment.
V The value — which we have obtained analytically for the
extent over which the fluid particles are transferred, suggests a simple physical reason for the fact For previous to the transit of a positive wave over any particle P, a volume of iluid behind P. and equal to V, is elevated above the surface of equilibrium. During the transit, this descends within the surface oi equilibrium, and must therefore force the fluid about P forward through the space
admitting as an experimental fact, that after the transit of the wave the fluid particles always remain absolutely at rest.
Mr Russel, p. 425, is inclined to infer from his experiments, that the velocity of the Great Primary Wave is that due to gravity acting through a height equal to the depth of the centre of gravity of the transverse section of the channel below the surface of the fluid. When this section is a triangle of which one side is vertical, as in channel (H), p. 443, the ordinary Theory of Fluid Motion may be applied with extreme facility. For if we take the lowest edge of the horizontal channel as the axis of x, and the axis of z vertical and directed upwards, the general equations for small oscillations in this case become
we have, also, the conditions
V==dy ° (When y = 0) W'
^ = ^ = J(when^"cOta^ (6)'
a being the angle which the inclined side of the channel makes with the vertical.