Lee 



first-order potentials and can be best described by the expressions 

 of asymptotic behavior of the first-order potenticils such as 



K.y j(K Ixl-q ) 

 ^^~ - j"rrQ,^e " e "^ " for k=i,2 as |x|-*oo. 



It is apparent from Eq. (30) that hg is not absolutely integrable. 

 This implies the necessity of further consideration in deriving the 

 solution of W which is associated with h . 



D 



4. 1 Solution for a Case Where Free-Surface Pressure Distribution 

 is Specified 



In this section we consider a potentlal-flov/ problem with a 

 given pressure distribution on the free surface. We restrict our 

 attention to the pressure distributions which have harmonic tlnne 

 dependence and are even In x. Furthermore we consider the two 

 special cases: the one where the pressure distribution decays In 

 the manner of l/x^ as |x| — ^ oo and the one where the pressure 

 distribution behaves like that for outgoing plane waves as | x | — ^ co. 

 Let w(x,y,t) be a harmonic function defined In y < and with Its 

 time dependence of the form 



w(x,y,t) = W(x,y)e''*^ 



where W = Wj. + jWg and co is an angular frequency. The free- 

 surface boundary condition Is 



Wy(x,0) - KW = h(x) (31) 



where K = w /g and h Is a known pressure distribution and is even 

 In X. We expect that the solution of W should represent outgoing 

 plane waves as |x| — * oo and furthermore that Wy(x,-oo) = 0. We 

 seek solutions to this problem In two cases. 



Case 1: h(x) = 0(i/x^) as |x| -* co. (32) 



The solution for this case Is given In Wehausen and Laltone [ I960] 

 In the form of a complex potential 



F(z) = W(x,y) + lW*(x,y) 



= lA J_ h(e)e-"^^'-^^E,(-iK(z-e)) de + 21 £ h(e)e"''^^'"^^ de 



00 



00 



(j-i)y_ 



h(e)e""^^'"^^de. (33) 



916 



