vx(vs) = (6) 



The phase function gradient can be written in vector notation as 



Vs = |vs| cos e i + |vs| sin 9 j (7) 



->■ -* 



where i and j are unit vectors in the x- and y-directions, respectively, 



and e(x,y) is the local wave direction. Equations 6 and 7 can be combined 

 to yield the following expression: 



fc(H Sin 9 / "ly (H C ° S 9 ) = ° (8) 



If the magnitude of the wave phase gradient is known, local wave angles can be 

 calculated from Equation 8. Similarly, Equation 7 can be substituted into 

 Equation 5 to yield 



( a cc I vs I cos 9 ) + — [ a cc Ivsl sin 9) = 



\ g ' ' / ay V 8 J ' / 



This form of the energy equation can be solved for the wave amplitude function 

 a once the wave phase characteristics Vs and 9 are known. The wave 

 height can be determined and is proportional to the amplitude function, since 

 wave frequency is constant. 



15. Equations 4, 8, and 9, along with the dispersion relation, describe 

 the combined refraction and diffraction process for linear plane waves subject 

 to the restrictions that the bottom slopes are small, wave reflections are 

 negligible, and any energy losses are very small and can be neglected. The 

 numerical solution scheme used to solve these equations is presented in the 

 next section. These equations are assumed to be valid outside the surf zone. 

 The method used to determine wave characteristics inside the surf zone is de- 

 scribed later. 



Numerical solution 



16. The three governing equations (Equations 4, 8, and 9) are solved 

 using numerical methods. Partial derivatives within the equations are approx- 

 imated using finite difference operators. Finite difference solution methods 

 require the construction of a computational grid system or mesh. Solution ac- 

 curacy is directly related to resolution within the grid system. Discussion 



10 



