Wu 



flexible but inextensible, which is propelling itself by small transverse progres- 

 sive waves. Taylor considered a wave of displacement y = b sin (kx - cot) 

 propagating in the +x direction with phase velocity c = oj/k, and found that this 

 motion induces a velocity in the fluid at infinity of 



(kb)^ + 0(kb)' 



(74) 



also in the +x direction. A limitation of Taylor's analysis is that the Reynolds 

 number R = oj/vk^ must be small enough for the application of Stokes's equa- 

 tions. This limitation was removed by A. J. Reynolds (1965), but his result is 

 incorrect. This has been pointed out by Tuck (1968), who provided the correct 

 result as 



where 



F(R) 



1 1 + F(R) 



- (kb)2 + 0(kb)' 



2 2F(R) 



1 + (1 + R2)' 



(75) 



(76) 



a function which increases monotonically from unity at R = 0, tending to infinity 

 like Ri/2 as R— co. Thus, the effect of inertia appears to be to decrease (rather 

 than to increase, according to Reynolds) the propulsion velocity above that found 

 by Taylor at r = 0. 



The analysis has been somewhat simplified by Tuck. We use a stream func- 

 tion i// satisfying u = ip^, v = -i/;^, ^ = -vV, and the Navier- Stokes equation 



vV^l, 



11 



uC + vL 



(77) 



The boundary conditions (Taylor, 1951) are 



(78) 



u = h^kco cos (2kx- 2cot) + OCb"*) , 

 V - -cob cos (kx-ojt) + 0(b'^) , 



on the moving surface 



y = b sin (kx- cot) . 



We now make the expansion 



4j = 5J[>/'i(y) e-il^'^+i-t] + W^^y) + ^[^//^(y ) e-2ikx + 2i^t j + 0(b3) , (79) 



where the first term of (79) is o(b) and satisfies a linearized version of the 

 Navier-Stokes equation (77), while the remaining second-order terms are 



1196 



