4 
LORD RAYLEIGH OH THE CIRCULATION 
r 
In all the applications that we shall have occasion to make, an approximate value 
of k' is admissible. On account of the smallness of v, n]v is very large in comparison 
with k 2 , that is to say, the thickness of the stratum through which the tangential 
motion can be propagated in time r is very small relatively to the wave-length X. 
We may therefore neglect l A in the equation 
and take simply 
Again 
P 4 =& 4 +-1 
v 
P *=n/v. 
(sin a — cos a) 2 = 1 — sin 2 a = I— ~ , 
n- 
so that the difference between cos a and sin a may he neglected, 
write 
k'=fi(l-\-i) . 
where 
n 
2v 
We will therefore 
■ • • • ( 12 ), 
.... (13). 
We must now distinguish the cases which we have to investigate. In the first we 
suppose that a wave motion is in progress in a vessel whose horizontal bottom 
occupies a fixed plane y= 0. We may conceive the fluid to be water vibrating in 
stationary waves under the action of gravity, the question being to examine the 
influence of the bottom upon the motion. If there are no other solids in the 
neighbourhood of the bottom, we may put D = 0, y being measured upwards, and 
f3 being taken positive. 
The conditions to be satisfied at y= 0 are that u and v should there vanish. Thus 
so that 
and 
A + B+C = 0, — &A+&B — fc'C = 0, 
\jj =Cl — cosh Icy -by sinh ky-\-e L 
= C{ —k sinh ky + k' cosh ky — k'e ^}. 
At a short distance from the bottom, u=k r C. If we denote by u 0 the maximum 
value of u near the bottom, we have 
and then 
k'C=u 0 e ini cos kx, 
. . [ cosh ky sinh ky e~ k 
xji=h 0 e’" ! cos kx\ ————-j 
k 
k' 
(in. 
