An Unsteady Cavity Flow 



To investigate the behavior of f| for |t | > R , let us divide the 

 region in the first quadrant of the cr-plane outside the circular arc 

 I a I = R| into two parts, D| and D2. D| , as shown by the shaded 

 area in Fig. 6, is bounded by the circular arc |(r| = R|, the imagii- 

 nary cr-axis and the hyperbola S' representing the equation 

 Re [ i((r^- Rfe^'^)] = 0. Dg is bounded by the real (r-axis , 1 0- 1 = R, 

 and S'. For t in D2, we may choose the path of integration in 

 (106) to be L2> which is similar to L| except now it lies on the 

 left-hand side of S- A typical L2 is shown in Fig. 6. Using a 

 process similar to that used to obtain (105) from (100), we may 

 obtain, from (106), 



f| ~ !— ge T . (107) 



4TT(Kaj) 



When T is in D|, the hyperbola S will be extremely close to both 

 positive axes of the tr-plane, S degenerates into the axes when t 

 lies on the imaginary (r-axis. For this case we have to deform the 

 simple path L2 into L3+ L4, as shown in Fig. 6, in order that the 

 factor exp [-iKa)((r -t )] will not become e^onentially large along 

 the path. L, is a path coming from ooe'^ to on the lower-half 

 a-plane, from there along the real (r-axis towards (r = 1 , turning a 

 small circle to the upper side of the real (r-axis and along it to 

 on the upper-half (r-plane, to circumscribe the cut in the (r-plane, 

 and then leaving to co&^'"^'^ . Path L4 comes from coe'^'/^ to 

 T, lying completely on the left-hand side of the hyperbola S and 

 outside the circular arc |(r| = R|. The integration along L3 is 

 convergent; near both ends of the path the integral is exponentially 

 small, near (r = M|(cr) and M.^^) , which appear in goio") . are 

 of order (r^ and cr respectively and %^^) is of order cr^, near 

 a = 1 only M2((r) contributes to g2(o') a square-root type of^singu- 

 larity. The latter property of M2((r) is the reason that (Kw) M2(o-) 

 is being kept in the integrand together with the term (Kw) M3(cr) . 

 We naay renaark here that if we did not neglect the termi ZttjKcot 

 appearing in g2(o") from g(T) in (83), we would have a term of the 

 form ZTriKcjcr. Since the presence of such a term would not affect the 

 property of the integrand along L^ near erg = and 1 , and since It 

 is one order in w smaller than the (Kw) M2(or) term. Its neglect is 

 justified. Let us now denote the contribution to fj*' from the inte- 

 gration along L_ by Z(t), 



^^^' = \7% Fi== ^ ) Ai[(-iKoo) ^] 



(Kco)"^^ (T-/;^rT_i) ^ 



3Tri/4 



i 



(X)e 



(De 

 4 



,i/4 e-''^'^ Ai[(-iKco)'^e-^-'/^(T/x)'/"[g,(cr)+^,g3((r)] d(r - 



211 



