

dz 



8x 



-h 



-h 



-h 



The boundary conditions to be satisfied by these equations are 

 that the pressure be zero at the free surface, z = ri(x,t), 



p = p =0 atz = n, CA-4) 



and the kinematic boundary conditions 



1^ . U^ 1^ -W^ = at z = n (A-5) 



and „, 



W, + a ^ = at z = -h , (A-6) 



b b dx 



where subscripts n and b refer to the conditions at the free surface 

 and at the bottom, respectively. 



To derive the continuity equation in the form normally encountered 

 in work involving long waves, equation (A-1) is integrated over the 

 depth of water 



dz + W -W, = , (A-?) 



and the remaining integral is evaluated using Leibnitz' rule 

 (Hildebrand, 1965) 



B(a) 



|i dz = / f dz -f (B) |i + f (A) |A , (A-8) 



A(a) ^° ^'^ -'a ^ 3a ^ ^ 3a ' 



in order to obtain 



3^ [ U dz + W -U |a _w R |ll = . CA-9) 



3x J n n 3x b b 3x l" ^J . 



-h 

 Introducing the concept of a mean velocity, U, defined by 



- 1 r 



^ -h 



106 



