Wang 



f,(z,t) = f(z)e^*'^ (20) 



If we substitute (20) into (10) we may write the free surface 

 boundary condition as 



Re H = (21) 



where 



„ d^f ./_. d , dwo\ df r^,. d , /I dwoNl, /oo\ 

 H = 5+ \2jw In 2-) CO + ju> ln( — 2 ) f , (22) 



df| ^ dfo dfo/dfo I dfo Vwo dfo^J 



which is an analytic function defined in the flow field. We may also 

 write 



„ 1 d^f , 1 r,. d , / dwQ\ldf 

 H = — n — 5- T — 2jw - In I Wn a ] 1 — 



Wo dz^ Wo L dfo ^ dfo^Jdz 



- r,,2^j^_d_^ /_l_dwo)1f^ (23) 



L dfo ^wo dfo^-^ 



Since the differential operator L is purely real on the plate AB, 

 the boundary condition (17) may be expressed in terms of the analytic 

 function H. By a straightforward application of L on (17) and by 

 the use of (19) and (20), the boundary condition on AB may be written 

 as 



Im H = Y. (24) 



where 



V = c j [- .^x . j.(w, . i-)] |- m 1^ + px 



< 1- ^ Wo^-* dfo dfo ^ 



- J"(3w„ - -^)1 -^ In Wo + jA + co2(wo + ~) [ . (25) 

 ^ Wo^J dfo ^ Wq^ \ 



The boundary conditions expressed in the forms of (21) and (24) may 

 be used to determine H for points in the interior of the flow field. 

 However, to obtain a physically acceptable H, other boundary con- 

 ditions have to be imposed on H. 



We shall assume that the free surface displacennent due to the 

 unsteady disturbance of the plate AB has to be bounded everywhere. 

 This condition can be satisfied if the free surface displacement is 

 bounded at the separation points A, B and at the point at infinity. 



194 



