An Unsteady Cavity Flow 



A straightforward expansion shows that, as |t| > R , 



-1/4 r-r -I 1/4 Tri/8 -3/4 



2x (t - Vt^ - 1 - i) ~ 2 ie r . (98) 



And it is easy to see, as some related explanation has been given in 

 the paragraph after (52) , that as | T | > R. 



X [g|(T) + B, g3(T)] ~ B,.2 le t 



(99) 

 X [g2('^) + ^\'^py^ ~ B|2 le T . 



Let us denote the part of the potential represented by the first inte- 

 gral in (87) by ff«' , 



f (e) ^ 2c (e/x)'^^ 



' (Koj)'/^ (t - Vt2 - 1 - i) 



X ] ^^^ e^ {Ai[(lK.)'''e] Ai[(iKu)'^' e'^'^] 



ooe 



- Ai[(lKu.)'^^e'^''''^e]Ai[(iKco)^^?]} (?/x )'^'^[ g,(cr) +B,g3((r)] dcr, 



(100) 



For any t with |t| > R| and :S Arg t < Tr/2 , the path of inte- 

 gration in (100) can always be chosen to be L|, which is a path 

 coming from ooe"^'''^ to T, lying completely on the right-hand side 

 of the hyperbola S and outside the circular arc |o-| = R|. A 

 typical L| is shown in Fig. 6. Since along L| |(r| > R|, we may 

 substitute the expansion given in (97), (98) and (99) into (100) and 

 obtain 



,(e) eB, i P^ iw(t-KT +Ka ) 0/4. ^ ^ 



^i ~— tt^Ht^j w4^ a^/%inh^(e,e)d.. 



Tr(2Ku)) T '^coe 



L 



(101) 



where 



*(|,|) = 4e-^*yK=(6'^^r^^. (102) 



209 



