corresponding value of B<J>/8y on y » 0, x > is 0(x 2 ) as x+ OK 

 The second term grows ac s ■*• », and must aris<: as the generalized Fourier 

 transform of a function which has a nor.-integrable singularity >": x - 0. 

 According to Lighthill [11, p. 43] 



f" 



\ -K- (1-A) sgns 



exp isx dx - e* (-X)! |s| A ~ l 



for real s, and the correct interpretation of this in R is 



¥ a -*> 



(-X)l » A " 1 (8.26) 



.X-J 



where the branch cut for the function s is to go from to » 

 in the lower half-plane. 



Hence *j(s,o) - 0(s N+2 ) 



<=> P- (x,0) = 0(x" N ~ 2 ) as x •*■ 0+. 



3 



Therefore the velocity has a singularity at least as bad as x near 



x = 0, and the kinetic energy in a small region around x » will 



diverge to infinity. 



We argue that this singularity is unacceptable, and choose the 



solution corresponding to 



E(s) = , 



l 



thus giving the least singular behavior — like x — in the velocity at 

 x ■ 0. Note that it is impossible to impose a Kutta condition, that 

 the velocity be finite (except by abandoning the radiation condition, 

 or allowing <j> to be discontinuous, and that cannot be permitted in 

 static fluid). To see what kind of a pressure field exists near x ■» 



58 



j *®® * ® 1 ^^ 



