W ' 37 (^^^ - (79) 



and 



H*gg.f^„U=0 , (80) 



in which r\ is the surface elevation relative to the Stillwater level, 

 U is the horizontal velocity component, h is the local depth, g is the 

 acceleration due to gravity, oj is the radian frequency, oj = 2tt/T, of 

 the incident waves and f, is a linearized friction factor defined by 

 equation (A-23) 



if |U| 

 ^ 2 w' ' , 



V = h (^^^ 



in which f^ is a wave friction factor relating bottom shear stress, 

 T, , fluid density p, and the velocity, i.e., 



T, = i pf lulu . (82) 



b 2 w' ' 



The linearized equations (eqs. 79 and 80), are solved by assuming 

 a periodic solution of radian frequency, to, and introducing complex 

 variables defined by 



n = Real{?(x)e^'^'^} (83) 



and 



U = Real{u(x)e^'^''^} , (84) 



in which the amplitude functions ? and u are functions of x only, 



i =/^, and only the real part of the complex solution constitutes the 



physical solution. 



In the constant depth region, h = h , in front of the slope, bottom 

 friction is neglected (i . e. , f^^ = f^j = for x > i^) and the general 

 solution reduces to the solution given in Section II. 2 for x < 0. With 

 the change of the orientation of the x-axis (positive away from the 

 slope as seen from Figure 14) the general solution for x > I is 



49 



