The coefficient A n can be determined by the method well known in the two- 

 dimensional theory of a heaving cylinder. Then we can determine a n by 

 solving the above equation. However, we need not solve the equation 

 exactly. The reason is as follows. If the frequency is very low or to the 

 contrary very high, the integral on the right-hand side can be neglected. 

 So we can put 



a (x) = V(x) A Q (x) (168) 



It may be assumed that the deviation from the above relation at inter- 

 mediate frequencies is not large. Therefore, we approximate 



a Q (x) = 



J6 

 V(x)- \ K {V'Cx'HqCx^+VCx'^Cx')} N(K|x-x* | ) sgn(x-x' )dx' 



-I 

 x A Q (x) (169) 



In order to determine other coefficients, we substitute V(x)A(x) for a Q in 

 the boundary condition and put 



W(x) = -| K {V , (x')A (x , )+V(x , )A^(x')} N(K|x-x' |)sgn(x-x')dx' (170) 



-I 



Then the boundary condition becomes 



™(2D) . 



fr = [V(x)-W(x)] £r (171) 



The solution of the two-dimensional problem with this boundary condition 



determines other coefficients a„ . We can rewrite 



2m 



W(x) in the following form for numerical purposes. 



determines other coefficients a„ . We can rewrite the function a„(x) and 



2m u 



62 



