V • ( X c c V(J> ) + !?-!*. 4, = (2) 



where 



V = horizontal gradient operator 



A = complex bottom friction factor 



c - wave phase speed 



c g = wave group speed 



<p = velocity potential 



0) = angular frequency 



This equation is identical to Berkhoffs (1972) equation except for addition of the 

 bottom friction factor A. The factor A, which is a complex number with 

 magnitude greater than zero and less than or equal to one, is specified as 



1 + -Jti- e « O) 



d sinh Kd 



where 



i =(-D ,/2 



p = dimensionless bottom friction coefficient that can vary in space 



a, = incident wave amplitude 



d = water depth 



k = wave number 



Y = phase shift between stress and flow velocity 



The bottom friction factor is a factor tending to reduce local velocities propor- 

 tionately through the relationships 



44 



Chapter 4 Numerical Model 



