Wang 



in (32) , we can obtain, from (i) , that on the free surface 9<j>|/8n = 

 o(z"') as |z I — ► oo. These results indicate that along the free sur- 

 face near the point at infinity both f and df/dz should vanish. 

 Since the unsteady disturbance is mainly a surface phenonnenon, it 

 is not unreasonable for us to assume that f and df/dz also vanish 

 in the interior of the flow field near the point at infinity. Therefore, 

 we assume that 



H^ as |z I -* oo. (33) 



This rules out the possibility that there is any induced circulation 

 around the point at infinity. Based on the assumption that h is 

 bounded at infinity and the result that on the free surface 9<J)|/8n = 

 o(z" ) as |z I -♦ oo, an integration of (1) will show that if 



■5F 



= 0(r ) 6 > (34) 



in the neighborhood of the points A, B with r the distance from 

 these points , h will be bounded at A, B. Condition (34) is also 

 necessary in order that the pressure be integrable over the plate 

 AB. 



To facilitate the determination of H, let us introduce a 

 transformation 



t4(;4) 



; = T-(T^.i)'/2 



(35) 



where the cut in the T-plane is taken along the straight line between 

 -1 and Land {r^ - i^^ -* r as |t|-*oo, - it < Arg t ^ tt. The 

 mapping (35) maps the entire basic flow onto the upper-half T-plane 

 as shown in Fig. 3. In terms of the variable t, the function y 

 becomes 



V = 2kMT^-l) ! J^^^^'^'^^-^'^tz'^^i-^')'^' +4T 



+ ir + 2 cos'' t] + (Kw)^[T(l-Ty^(4-5T^ 



- 12t^ - ^ T^ + IOt + tt + (2-3t2)cos'' t] +2jKwt { , (36) 



where 



196 



