All-101 9i 



where p is p evaluated at z = r] , and p is p evaluated at 

 z = -h. 



If the free surface be considered a surface of zero 

 pressure, then p = 0. 

 Defining 



P = 1 p dz 

 J-h 



we have for equation (h) and (5) 



Py U = - II . 8| p_^ . AAV + Xy (8) 



A stream function -^ can be defined by U = - M , V = + $li 



dy 9x 



so that (6) is satisfied identically. Taking the derivative of 



(7) with respect to y and (8) with respect to x and subtracting, 



we obtain 



^ '^^x ax ay ay ex oj^ ax '^^ 



Since z = -h is the depth v/here the velocities are 

 negligible, the third equation of motion below this depth re- 

 duces to the hydrostatic pressure equation, - -^^ = gp ^ if p 



is constant along z = -h(x,y,t). Then ^dl = gp|h P::h 



?y ay ax 



= gp— . Vh'.th these results substituted into (9), we have 



AAA;]. - p.j;^ . --JS _ _Z (10) 



