6 
LORD RAYLEIGH ON THE CIRCULATION 
u 0 sin kx f k cosh ky 
ke^ y 
v= 
da 7 
— = U () COS KX 
ay 
^ /9 sin (nt — \tt) + sinli ky sin nt + ~ - sin {nt — £ 7 t— fry) j- , 
c° S I^ cos {nt — ^7r)-|-&sinh ky cosn£ + ^2.(3 e~ $y cos {nt-\-^-ir—/3y) k 
Ps/< 
of which the two first terms may be neglected relatively to the third (containing the 
cLil 
large factor (3). The product of y and ^ will consist of two parts, the first indepen¬ 
dent of t, and the second harmonic functions of 2nt. It is with the first only that we 
are here concerned. The mean value of the velocity parallel to x is thus 
0 2 sin 2 kx e py 
4:71 
k cosh ky cos (3y-\- \/2.(3 sinh ky sin {/3y—^Tr) — k e Py 
On account of the factor e ?y , this quantity is insensible except when ky is extremely 
small. We may therefore write it 
U ° Sm ^ X - e — jeos fiy+fiy (sin /3 y— cos (3y) — e & j .... (20), 
V (equal to k/n) being the velocity of propagation of waves corresponding to k and n. 
The only approximation employed in the derivation of (15) and (16) Is the neglect 
of the right hand member of (4), and the corresponding real values of u and v could if 
necessary be readily exhibited without the use of a merely approximate value of k'. 
To proceed further we must calculate the value of 
u d\/ 2 \jr 
v dx 
+ 
V d\/~\fr 
dy 
( 21 ) 
in (4), for which it will be sufficient to take the values given by the first approxima¬ 
tion. Thus 
and by (17) 
VV = Vu/q 
1 d^y* 
v ~di’ 
d~k„ 
dt 
nu n cos kx e 
-Py 
/V2 
sin {nt—lTT—py), 
from which we find as the value of (21), 
nku 0 2 sin 2 kx e~^ y \ ( k /3\/2\ . . 
“47^/2— j \^y 2 — ir ) sm 1 ky sin v 2 cosh h J cos py+ v 2 
-ft/ 
+ terms in 2 nt. 
