36& Lord Rayleigh on the 



the influence of the terms involving the squares of the 

 velocities, but would persist in certain cases even though the 

 motion were made infinitely slow. 



We will now investigate the motion in two dimensions of 



Fi°r. 6. 



a viscous incompressible fluid past a corrugated wall AB 

 (fig. 6), whose equation may be taken to be 



y = /3 cos kx (43) 



In this k/3 will be supposed to be a small quantity ; in other 

 words, the depth of the corrugations small in comparison 

 with their wave-length (27r/k). Further we shall suppose, 

 in the first instance, that the motion is slow enough to allow 

 the terms involving squares of the velocities to be neglected ; 

 in which case the equation for the stream-function may be 

 written 



V 4 ^ = (44) 



At a distance from the wall we suppose the motion to take 

 place in plane strata, as defined by 



y]r = Lf (45) 



In the absence of corrugations this value of ty might hold 

 good throughout, up to the wall aty = 0. The effect of the 

 corrugations will be to introduce terms periodic with respect 

 to x ; but the influence of these will be confined to the 

 neighbourhood of the wall. For any term in -sjr, proportional 

 to cosmx, (44) gives 



f d 2 \ 2 



b? _m 7 ^ = ' ( 46 ) 



or 



ir = A e~ m y + By e~ m y + C e m y + Dye m y; 



but the condition last named requires that of the four arbitrary 

 constants C and D vanish. Also for our present purpose 

 m is limited to be a multiple of k. 



The^ form of -^ applicable to our present purpose is 

 accordingly 



n/r = Ao + B^-f Lt/ 2 + cos^(A 1 ^-^+B 1 «/^-^) 



+ cos 2kx (A 2 e~ 21( y + B 2 y <r 2 ^) + . . . , (47) 



