SOLUTION OF THE PROBLEM 



The potential satisfying the conditions outlined in the previous section is given 1 by 



p r Bn Y /2 f n 



(f> (x, y, z) = / / dx dy J d< 



f 



sec i 

 n* P c "'-B/2 -L /2 



oo , 



kh cosh k (z + h) sech kh 



dk sin [k (x - x) cos 6] 



1 , 



kh - — sec d tanh kh 



F 2 



cos [k (y -y) sin 6] [1] 



1/2 L/2 77/2 



/ 



-B/2 -L/2 

 k Q cosh k Q (z + h) sech k Q h 



p B/2 L/2 77/2 



- / / d~x dy I ddsec 9 



n P C J n J r J* 



1 - — sec 2 #sech 2 k Q h 

 F 2 



cos [k Q (x - x) cos d] 



cos [k Q (y - y ) sin 8] 



where F - is the Froude number based on depth 



\/gh 



1 , 



k Q = k Q (6) is the positive, real root of kh - — sec tanh M = 



F 2 



{cos - x ( — ) for F > 1 

 for F < 1 



and f means that the integral is to be interpreted as a Cauchy principal value. 



If z is set equal to -h, and the integration with respect to (x, y) is carried out, then 

 since 



~ L/2 { sin [k(x - x) cos d] \ 2 / L \ I sin {k x cos 6)\ 



\ d *\ \ = 1 2 sin (* T cos d )\ \ 



t L/2 \ cos [k(x-W) cos 0]J *cos0 V 2 7 ( cos (A ^ cos 0) ) 



