wee? 
* 
a A 
pac 
y 
ON RECENT RESEARCHES IN HYDRODYNAMICS. 7 
the ridges of the waves do not bend forwards, but are situated in a vertical 
plane, and they rise higher towards the slanting sides of the canal than in 
the middle. I shall write down the value of ¢, the integral of (A.), and then 
any one who is familiar with the subject can easily verify the preceding re- 
sults. In the following expression x is measured along the bottom line of 
the canal, y is measured horizontally, and z vertically upwards :— 
g=A(e*¥ +27 *Y) (e%* +2 %*)sin VW 9a(x—ct). . +» +» (CG) 
I have mentioned these results under the head of oscillatory waves, be- 
cause it is to that class only that the investigation strictly applies. The 
length of the waves is however perfectly arbitrary, and when it bears a large 
ratio to the depth of the fluid, it seems evident that the circumstances of the 
motion of any one wave cannot be materially affected by the waves which 
precede and follow it, especially as regards the form of the middle portion, or 
ridge, of the wave. Now the solitary waves of Mr. Russell are long com- 
pared with the depth of the fluid; thus in the case of a rectangular canal he 
states that the length of the wave is about six times the depth. Accordingly 
Mr. Russell finds that the form of the ridge agrees well with the results of 
Prof. Kelland. 
It appears from Mr. Russell’s experiments that there is a certain limit to 
the slope of the sides of a triangular canal, beyond which it is impossible to 
propagate a wave in the manner just considered. Prof. Kelland has arrived 
at the same result from theory, but his mathematical calculation does not 
appear to be quite satisfactory. Nevertheless there can be little doubt that 
such a limit does exist, and that if it be passed, the wave will be either con- 
tinually breaking at the sides of the canal, or its ridge will become bow- 
shaped, in consequence of the portion of the wave in the middle of the canal 
being propagated more rapidly than the portions which lie towards the sides. 
When once a wave has become sufficiently curved it may be propagated 
without further change, as Mr. Airy has shown*. Thus the case of motion 
above considered is in nowise opposed to the circumstance that the tide 
Wave assumes a curved form when it is propagated in a broad channel in 
which the water is deepest towards the centre. 
It is worthy of remark, that if in equation (C.) we transfer the origin to 
either of the upper edges of the canal (supposed complete), and then suppose 
hk to become infinite, having previously written A e—“” for A, the result 
will express a series of oscillatory waves propagated in deep water along the 
edge of a bank having a slope of 45°, the ridges of the waves being perpen- 
dicular to the edge of the fluid. It is remarkable that the disturbance of the 
fluid decreases with extreme rapidity as the perpendicular distance from the 
edge increases, and not merely as the distance from the surface increases. 
Thus the disturbance is sensible only in the immediate neighbourhood of the 
edge, that is at a distance from it, which is a small multiple of A. The for- 
mula may be accommodated to the case of a bank having any inclination by 
merely altering the coefficients of y and 2, without altering the sum of the 
squares of the coefficients. If i be the inclination of the bank to the verti- 
cal, it will be easily found that the velocity of propagation is equal to 
eae era ta ‘ 
F cosi - When 7 vanishes these waves pass into those already men- 
tioned as the standard case of oscillatory waves; and when 7 becomes nega- 
tive, or the bank overhangs the fluid, a motion of this sort becomes im- 
possible. 
* Tides and Waves, art. 359. 
