90 Lord Rayleigh on the Propagation of 



At the wall of the tube r~y, a known function of x ; and 

 the boundary condition, that the motion shall there be 

 tangential, is expressed by 



dx dx dr ' ^ 



in which (r=y) 



<fa ~ <fa 2*U** )dx + 2 2 A 2 \dx 2 + tC / <te '"' " w 



dr 2W f "j l+ 2UU 2 / W 



Using these in (3), we obtain an equation which may be 

 put into the form 



As a first approximation we may neglect all the terms on 

 the right of (6), so that the solution is 



Ae~ ik * + Be ikx ... 



F 0) = » ( 7 ) 



where A and B are constants. To the same approximation, 



** + W.--*p™ (8) 



dx* y ax ax 



For a second approximation we retain on the right of (6) 

 all terms of the order d 2 yldx 2 , or (dyjdx) 2 . By means of 

 (8) we find sufficiently for our purpose 



\dx 2 J dx y dx dx 2 ' 



\dx 2 + ) ""■ <te 2 \j/ 2 i dx) ydtfy 



