where 



x and y = two orthogonal horizontal coordinate directions 



c(x,y) = the wave celerity (= a/k) 



a = angular wave frequency (defined to be 2tt/T) 



k(x,y) = wave number given by the dispersion relation, 

 2 

 o = gk tanh (kh) 



T = wave period 



c (x,y) = group velocity (= 3a/ak) 



4>(x,y) = complex velocity potential 



g = acceleration due to gravity 



h(x,y) = still-water depth 



8. Numerical solution of this equation for the velocity potential field 

 is an effective means for solving the complete wave propagation problem. The 

 equation can be solved using either finite element (Berkhoff 1972 and Houston 

 1981) or finite difference methods (William, Darbyshire, and Holmes 1980). 

 Since transmission and reflection boundary conditions are easily implemented 

 into these solution schemes, this approach is a popular one for modeling tsu- 

 nami propagation and for solving problems involving the response of harbors to 

 short and long waves. This method becomes computationally infeasible for 

 large scale, open coast, short-wave problems because of its great expense. 

 Numerical solutions of Equation 1 are only practical, as a rough rule of 

 thumb, when the dimensions of the spatial area of interest are no more than 



10 times the length scale of the wave lengths being considered (Berkhoff, 

 Booy, and Radder 1982). 



9. An alternative method based upon a simplification of this equation 

 has recently been developed. This method alleviates the computational bur- 

 den imposed by a direct solution of Equation 1. The velocity potential can 

 be separated into a forward scattered and a reflected component. By neglect- 

 ing the reflected part and assuming that diffractive effects in the direc- 

 tion of propagation are much less than those perpendicular to the direction 

 of wave advance, the following equation for the forward scattered wave can be 

 derived: 



ax 



ik - 



it. — (kcc ) 



2kcc 3x g 



2kcc 



d - fee &) 



ay V g ay/ 



(2) 



