Lee 



should represent outgoing waves as |x| — oo. 



These boundary- value problems are almost identical to those 

 for the first-order velocity potentials. Thus we find them by the 

 same method used to find the first-order potentials which Is described 

 in Appendix B. It is often called the "multipole- expansion method" 

 since the potential Is expanded in an infinite series of poles, located 

 at the origin, of increasing order with unknown strengths. Each 

 pole satisfies Laplacian equation everywhere except at the origin, 

 the linear free-surface condition of the type 4>y(x,0) - K$ = and 

 the infinite- depth condition, and is even in x. However, since each 

 pole vanishes as |x| -^ oo the radiation condition of outgoing plane 

 waves is not satisfied. To circumvent this a source singularity 

 which has all these properties plus the property of outgoing waves 

 at |x| = oo is added to the multipole -expansion series. The unknown 

 source and multipole strengths are found by satisfying the remaining 

 condition which is the boundary condition on the body. Specifically 

 we assume the solution for G|^ to be 



00 



G^(x,y)= ^ (bK„, + JsJM,^(x(X.a),y(X,a))e-'*'K. (43) 



m = 

 Here b^ = Q,^= unknown strength of a source at the origin, c,^^ = 0, 



M,jQ = - \ ^ p - °K ^^ ^P " ^^^ *^°® ^^ ^^^^ 



= a source of unit strength at the origin, 



where j- indicates that a Cauchy principal value is to be used, 

 - . cos Zma sin (2 m - i)Q! 



I 



(2ntl)a,„.,. sin(2m+2nM)Qr ^> ^ ^^5^ 



2m + 2n + i . 2m*2n-H 



= multlpoles of unit strength at the origin, 



h^^ and c^^ for m ^ 1 are unknown multipole strengths, and q^ 

 represents unknown phase relations between the forced motion of the 

 body and the pulsating singularities at the origin. The expression 

 for the harmonic conjugate of G^^ is 



00 

 Gt(x.y) = ^ (bk„+jc,JM*„(x(X,a),y(X.c.)) 



(46) 



m=0 



920 



