Lee 



APPENDIX B 

 Evaluation of the First-Order Velocity Potential 



There are two first-order potentials, <p. and <p^, involved 

 in this work. Since their solutions are essentially identical (they 

 differ only in the frequencies), ^, will be chosen as the representa- 

 tive first-order potential. As shown in Appendix A, the boundary- 

 value problem for <p^ is 



VV, = 0, (B-1) 



^,y(x,0) - K,^, =0, (B-2) 



^, (x ,y )f'(x ) - 9), = - bo-, , (B-3) 



^,y{x,-oo) = 0, (B-4) 



^|(x,y) = ^|(-x,y), (B-5) 



2ind the radiation condition can be explicitly written as 



lim Rej(^,^T jK,9»,) = 0. (B-6) 



X— *±oo 



There are two methods for the solution of the above problem. One 

 of them is the method of multipole expansions (see Ursell [ 1949]) 

 which is essenticilly an eigenf unction expansion of the unknown function. 

 The other is the method of source distribution (see Frank [ 1967]) 

 i.e. the method of Green's function. A brief description of the 

 method of multipole expansions will be given. First we consider 

 the problem^ without the boundary condition on the body given In Eq. 

 (B-3). If we transform the problem Into the t,-plane we find that 



vS^(X.,a) = 0, (B-7) 



K,a JX - J ^^" ^.^Jr ^ nt ' j M(X.O) - ^ = 0, (B-8) 



'Here, It should be recalled that the transformation given by Eq, (2) 

 maps the %- and ii-axes Into the x- and y-axes and maps the contour 

 of the seml-clrcle in the t,-plane onto* the contour of the cylinder In 

 the z -plane. 



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