An Unsteady Cavity Flow 



It is shown in W that the curvature of the free surface of the basic 

 flow is 



R 



q^ dwp 

 Wq dfo 



(26) 



where qg = 1 in our problem. From the local mapping behavior near 

 |z| = oo between the Wq- , i^- and z-planes shown in Fig. 2, we see 

 that 



Wq + i ~ a,z 



-1/2 



(27) 



and 



- iz 



as 



CO, 



(28) 



where a. is a constant. In the following analysis we always use an 

 to indicate some constant. The substitution of (27) and (28) into (26) 

 gives us 



i 



-3/2 

 = 0(|z| ) 



(29) 



on the free surface as | z | — ► oo. Since we assunne that the free sur- 

 face displacement near the point at infinity has to be bounded, then, 

 from (2) and (29), we have, near the point at infinity, 



4ii + 4^ ~ 0. 



TT as 



(30) 



For harmonic oscillations, (30) suggests that we may write 

 <t>. = A(fJeJ'^<*-*o> 



as 



oo. 



(31) 



since along the free surface of the basic flow 8/8s = 9/8fo. The 

 substitution of (31) into (2) gives 



h = ± R e as 



dfo 



oo. 



(32) 



In view of (29) and (28), (32) implies that along the free surface near 

 the point at infinity A = 0(z'' ), at most, in order that h be bounded 

 there. With h being bounded at infinity and having a forna shown 



195 



