14 
LORD RAYLEIGH ON THE CIRCULATION 
Jc/3~ We, p,J sin 2kx f .. . , , _ 1 
- -j^X 1- %)( sm fty—cos fiy)+\e • • • 
in which we may write 
(53), 
A/3~II 2 /,’/3 l H 2 nkW 
n "" ?i/3 2 — _ 4P/3 2 ' 
Combining (53), (46), and (51), we get finally 
it 
, ^ sin 2kx \ . „ 1 3i> 0 2 o7 \ _oz.„ • o7 
=—^3 -j ~ 2 sm Py~i e j+ 2 ^) e sm 
; C|„_2faq _ e - ft(2s ; n ^ + | e -»» )+ | (1 _ %)( ,-^j. , 
( 54 )> 
which expresses the mean particle velocity. 
When fiy is very small, (54) gives 
Vr? sin 2 kx 
u — 
2Y 
(-^+- • .)• 
(55). 
from which it appears that quite close to the plate the mean velocity is in the 
opposite direction to that which is found outside the frictional layer. 
§ 3. In the third problem, relating to Kundt’s tubes, the fluid must be treated as 
compressible, as the motion is supposed to be approximately in one dimension, parallel 
(say) to x. The solution to a first approximation is merely an adaptation to two 
dimensions of the corresponding solution for a tube of revolution by Kirchhoff,* 
simplified by the neglect of the terms relating to the development and conduction 
of heat. It is probable that the solution to the second order would be practicable 
also for a tube of revolution, but for the sake of simplicity I have adhered to the 
case of two dimensions. The most important point in which the two problems are 
likely to differ can be investigated very simply, without a complete solution. 
If we suppose p=a?p, and write cr for log p — log p 0 , the fundamental equations 
are 
die du 
dt U dx 
— v 
du 
dy 
-\-v^ 2 u-\-v 
(56), 
with a corresponding equation for v, and the equation of continuity, 
du dv da da da 
Tx + dy + Yt +U dx +V ly = ° 
(57). 
* Pogg. Ann., t. cxxxiv., 1868. 
