Ogilvie 



linearize the free- surface conditions. More precise^v, we could 

 assume the existence of asymptotic expansions, <|) ~ Ij^n ^^id 

 ^ ~ /jt,n> ^rid let the first term in each be o(l) as 6 -* 0. The 

 linearized free-surface conditions take their usual form: 



[A] = g; + <^j, on z = 0; 



[B] = - <|)2 + Ct» on z = 0. 

 These can be combined into the following: 



^^-v^ = Q, on z = 0, (3-22) 



where v = to /g = 0(€"'). In the far field, it is very difficult to 

 guess how differentiation alters orders of magnitude. If the oscilla- 

 tion frequency is very high, then the resulting waves are very short; 

 it would be reasonable, perhaps, to try stretching the coordinates, 

 and there would be no obvious basis for doing this anisotrophically. 

 The approach which I take here is somewhat different: Solve the 

 above- stated linear problem exactly, then observe the behavior of 

 the solution for high frequency of oscillation. In other words, the 

 problem is not stated in a consistant manner, but when we have the 

 solution we rearrange it and make it consistent. 



The desired potential function can be written in the following 

 form: 



itut ■ 



(t>(x,y,z,t) = Re {^(x,y , z)e'*^'} , (3-23) 



where: 



^(x,y,z) = - 2^^ di^ii)^ ^ e'^^jQ(kV[(x-e)' +y'']) (3-23a) 



poo r^ At '•^y+z^A'*^*-'^) 



= _ J_ \ dk e'l^V (k) \ ^ ,^ . (3- 23b) 



4tt J-oo J-cx) ^(k^ -^ i'^) - V 



The form in (3- 23a) can be obtained readily by superposing a distri- 

 bution of free-surface sources: Jq is the ordinary Bessel function 

 of order zero, and the wiggly arrow shows that the integral is to be 

 interpreted as a contour integral, indented at the pole in the obvious 

 sense indicated. Form (3-^ jb) is obtained by a transform method; 

 (j-*(k) is the Fourier transiorm of cr(x); details may be found in 

 Ogilvie and Tuck [1969] . Again, the inner integral is to be inter- 

 preted as a contour integral; there are two poles in this case. In 

 both formulas, the path of the contour has been chosen so that the 



750 



