and this vanishes either if A - (which implies continuity of A . 

 and hence absence of the wake) or, for an> value of A, if X » u/U. Thus 

 the general condition on <t> . across the wake is that 



♦total «".<« " Ae^ /U (9.3) 



for some value of A. The tangential velocity is (iuA/u)e above 



the wake, (-iwA/U)e below it, and hence, the wake is an oscillatory 



vortex sheet of strength 2uiA/U, modulated by tbe phase factor 



exp i — (x - Ut) I , which shows that any element of the vortex sheet 



propagates downstream at the flow speed U. 



We wish to solve the two-part mixed boundary value problem poied 



by (9.1-9.3), subject here to the restriction to subsonic flow 



(M - U/c < 1) and tc a radiation condition, but leaving open the ismui 

 o 



of edge conditions. We first write 



♦total " ""^oj D(_q) "P^q* -5(-q)y) +4> (9.4) 



where we shall write generally 



It 



c - 







r- 



(k o + 



Ms) 



£-q) 



-«.- 



-q 





D - 



s 



1 + «a 



0) 



- l + 



Ms 



k 

 o 



D(-q) 



- 1 - 



3H 

 u 





(9.5) 



(9.6) 



The function £ replaces the y In the no-flow case, while D is a kind 



Tl S 8 



of Doppler factor for vsvenumber s. The object of writing A in 

 the form (9.4) la that the incident field associated with the velocity 



63 





