24 
MR. D ARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
„ 7r , mx 
6= r: 
a+h—y. 
Also in the small terms we may put r—a. Thus the potential becomes 
V =const. -\-gy + F cos [to (a?— vt) + e]-f G sin [m(x — vt) + e]. 
Again, to find the equation to the bottom of the canal, we have to transform the 
equation 
r = s in 2 6 cos [2((f)—cot) 
If y' be the ordinate of the bottom of the canal, corresponding to the abscissa x, 
this equation becomes after development 
y' — li — E cos [to (x-vt)-\- e]. 
We now have to find the forced waves in a horizontal shallow canal, under the 
action of a potential Y, whilst the bottom executes a simple harmonic motion. As 
the canal is shallow, the motion may be treated in the same way as Professor Stokes 
has treated the long waves in a shallow canal, of which the bottom is stationary. In 
this method it appears that the particles of water, which are at any time in a vertical 
column, remain so throughout the whole motion. 
Suppose, then, that x-\-£=x is the abscissa of a vertical line of particles PQ, 
which, when undisturbed, had an abscissa x. 
Let rj be the ordinate of the surface corresponding to the abscissa x. 
Let pq be a neighbouring line of particles, which when undisturbed were distant 
from PQ a small length k. 
Conceive a slice of water cut off by planes through PQ, pq perpendicular to the 
length of the canal, of which the breadth is b. Then the volume of this slice 
is 6 xPQxNto 
Now PQ = /t —E cos [rn(x— vt) + e]—y, 
and Nn=i^l+|^. 
Hence treating E and rj as small compared with h, the volume of the slice is—- 
