boundary condition was derived from the solution for zero forward speed. 

 The solution obtained involves integrals over the plane z = 0. It seems 

 of some interest that the outer solution for the radiation potential takes 

 a degenerated form like 



l> R ~-— m(x)+2ifi(z-i|y |) ^T~J ^'i' 1 < ' I >'! ■ 1 ^ : 



The first term in parentheses means a simple harmonic plane wave propagat- 

 ing outwards in the y direction, while the second term is related to the 

 variation of the source density along the x-axis which means the inter- 

 ference between different sections. An extensive discussion of this case 

 is given by Ogilvie and Tuck and will not be reproduced here. 



Next we consider the case of high frequency with low forward speed, 



-1/2 1/2 



that is co = 0(e ), U = 0(e ). This is the case of short waves such 



that the wavelength is of the same order as that of the breadth of the ship. 



3/2 

 In this case, the order of magnitude of 1 is £ . The free surface 



condition becomes 



9 2 <J> 



1 + R -z-± = (139) 



at 2 6 3z 



1/2 3/2 



up to the order of 6 e . The next term is of the order of 6 £ , which 



includes the effect of the forward speed, but has quadratic forms. There- 

 fore, the ordinary linearized treatment can be applied to the first order 

 only, by which the effect of the forward speed cannot be taken into 

 account. The asymptotic form for the outer solution for the radiation 

 potential takes the form 



4iri icot + v(z-ilyl) , s /--i/rw 



d> ss - e ' ■" ' m(x) (140) ■ 



R g 



that means outward-going plane waves. There are no three-dimensional terms 

 involved and the solution is purely two-dimensional. The strip theory is 



53 



