Wave Transformation Outside the Surf Zone 



Theoretical Basis 



12. The velocity potential function for linear, monochromatic, plane 

 waves can be represented by the expression 



(3) 



where 



a(x,y) = wave amplitude function equal to gH(x,y)/2o 



H(x,y) = wave height 



s(x,y) = wave phase function 

 Here again, the velocity potential function only describes the forward scat- 

 tered wave field. No considerations are given to wave reflections. By sub- 

 stituting this expression for the velocity potential into Equation 1 and solv- 

 ing the real and imaginary parts separately, two equations can be derived 

 (Berkhoff 1976), namely, 



. 2 2 

 13 a a a 1 



a , 2 ,2 cc 

 lax ay g 



Va-v(cc ) 



+ k 2 - Ivsl 2 = (4) 



p 

 v-(a cc Vs) = (5) 



where the symbol v denotes the denotes the horizontal gradient operation. 



13. Together, these equations describe the combined refraction and dif- 

 fraction process. Diffraction is often erroneously described as the propaga- 

 tion of energy along wave crests which are defined to be perpendicular to the 

 wave phase function gradient Vs . Equation 5 shows energy is still propa- 

 gated in a direction perpendicular to the wave crest. Diffractive effects do 

 change the phase function as a result of significant wave height gradients and 

 curvatures. These changes cause the local wave direction to vary. If dif- 

 fractive effects are neglected, Equations 4 and 5 reduce to those describing 

 pure refraction in which the wave number represents the magnitude of the phase 

 function gradient. 



14. Linear wave theory assumes irrotationality of the wave phase func- 

 tion gradient. This property can be expressed mathematically as 



