where the s5rmbol V denotes the horizontal gradient operator. 



29. Together, these equations describe the combined refraction and 

 diffraction process. Diffraction is often erroneously described as the 

 propagation of energy along wave crests which are defined to be perpendicular 

 to the wave phase function gradient Vs . Equation 28 shows energy is still 

 propagated in a direction perpendicular to the wave crest. Diffractive 

 effects do change the phase function as a result of significant gradients and 

 curvatures of the wave height. These changes cause the local wave direction 

 to vary. If diffractive effects are neglected. Equations 27 and 28 reduce to 

 those describing pure refraction in which the wave number represents the mag- 

 nitude of the phase function gradient. 



30. Linear wave theory assumes irrotationality of the wave phase 

 function gradient. This property can be expressed mathematically as 



V X (Vs) = (29) 



The phase function gradient can be written in vector notation as 



->■ ^ 



Vs = |vs| cos e i + |Vs| sin e j (30) 



-»■ ->• 

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



and e(x,y) is the local wave direction. Equations 29 and 30 can be combined 



to yield the following expression: 



1^ (|Vs| sin e) - 1^ (|Vs| cos e) = (31) 



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

 calculated from Equation 31. Similarly, Equation 30 can be substituted into 

 Equation 28 to yield 



— (a cc |vs| cos e) + — (a cc |vs| sin 6) = (32) 



ox g 9y g 



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

 a once the wave phase characteristics Vs and 6 are known. The wave 



23 



