and the boundary condition on the free surface becomes 



2 



2 9 *1 2 9 *1 9( h 



-co <>. + 2 i U u -r — + U — r- + g -r-± = 



1 dx „ 2 6 9z 



dx 



(121) 



The Green's function for this boundary condition is 



G(x,y,z;x\y',z') = — + 



1 2 



00 00 



*J*f 



-! I dm || dn exp [ 



'_oo "0 



m +n - i m(x-x' ) ] 



• {cos(nzH-e) cos(nz'+e) - cos nz cos nz f }//m +n 



oo 



+ §j" dm e xpt(z+z ., <W/«-|™'|/. 2 -WH«>V 



-im(x-x')] (mU-Ho) 2 //m 2 -(mU+a)) 4 /g 2 



(122) 



where 



r x = /(x-x') 2 +(y-y') 2 +(z-z') 2 

 r„ = /(x-x') 2 +(y-y') 2 +(z+z') 2 



(123) 



tan e = - (mU-o)) /gn 



If the radical in the last integral becomes imaginary, it has an appropriate 

 sign in accordance with the radiation condition. The outer solution is 



47 



