Bessho 



--Z(x,y) = _0(x,y,O) , (3) 



ox oz 



and 



32 ^ 



— — 0(x,y,O) = -g — -0(x,y,O) outside of S, (4) 



ox dz 



where p is the water density, g is the gravity constant, and Z(x,y) is the water 

 surface elevation. 



Then, it has the well-known (6) representation 



''^(''•y-z) = 4;^ JJ P(x',y') — S(x,y,z; x',y',0) dx'dy' , (5) 



s 



where (7) 



" '- r r 



♦0 " J., J„ 



exp[kz + ik(x - x' ) cos 6 + ik(y-y') sin 6\ 



S = - 1 im ^::- I I x dkd5 



k cos 6 - g + i/Li cos 6 



= 4go!2^ [g(x-x'), g(y-y'), -z] , (6) 



and where 



^- g^ S= 2g'-^^, r^= (x-x')^. (y-y')^. z^ (7) 



ox oz dz 



Then the condition on the ship surface s becomes 



3x Bz ^e Bx ^ i Bx2 '^' 



(8) 



or 



Z(x,y) = - - p(x,y) + 4;^||p(x',y') ^^ S dx'dy' . (g) 



s 



The solution of this integral equation has been examined by Maruo for small 

 values (7,8). 



For large g values, it is well known that the second term of the right-hand 

 side of Eq. (9) is small and 



Z(x,y) % -p(x,y)/pg (10) 



except near the periphery of s. 



776 



