Analysis of Ship-Side Wave Profiles 



where U.m(£,r) denotes the source density at the point Q(£,i?,0 

 on the distribution surface S . The Green function G(x, y, z;£,»7,f)is 



G(x, y, 7,-X V,n = -^y-^-+H (3) 



where 2 , t ,2 . ,2 , n Z 



7l = (x-£) + (y-v) + (a-f) 



^2 



2 = (x-£) 2 + (y-r,) 2 + (z+f) 2 (4) 



1_ lim f J k exp[k(z+r) + ik(x-£cosfl + y-gsinfl)] 



7r m->0 II n 2 



7 7 k - K /sec - ijusectf 



-7T O O 



By making use of the well-known free surface condition 



(5) 



f( x v ) = L <=L£_ 



{ ' Y) KjSx 



z = 



the integral equation (2) can be converted to 



dS (6) 



r(x ' y) ~~ -^^/f m( ^ r) ^T G(x ' y ' z; ^ 7? ' r) 



z 



= 



where f (x, y) denotes the surface elevation in general. 



For simplicity, let us confine ourselves to the specific limi- 

 tations, 



(a) the distribution surface is the rectangular, vertical 

 central plane (-1<£<1, -t<?<0) 



(b) the distribution function is draughtwise uniform. 



Further, the ship-side wave profiles K •. (x, y) are selected as 

 the given information of the wave elevation f (x, y) . 



Thus we have the fundamental integral equation 



o^J-to J-l 



f h< X '^ = "437 / df / ^fG( X ,y,, ; S,o,f) d£(7) 



n*' /_tn J_ 1 OX 



For numerical solution of Equation (7), the modified Fourier expans- 

 ions are introduced as follows 



689 



