Impulsively Generated Waves Propagating into Shallow Water 



Fig. 1. Bottom topography and wave refraction 



The approach will be to consider the energy patch between 

 two adjacent wave rays, S| and Sg, and between two adjacent 

 frequencies, w and w + dco, and to establish the dimensions of this 

 patch as a function of path position. While the total energy of the 

 patch remains constant the energy density changes with patch size 

 and this, in turn, determines the local wave amplitude. Establish 

 a rectilinear coordinate system, (x,y), with x- axis normal to the 

 parallel bottom contours and the beach. Orient the curvilinear 

 coordinate system (s,n) shown in Fig. 2 to the median between 

 rays. Since k = /c(ci),h) only the magnitude of AC,, K^t and K will 

 be the same but due to the curvature of the orthogonals the directions 

 will be different. Let K\ have components (i, ,mj) parallel to 

 (s,n). Since the patch 6t, 6^, is small ly ^ l^ = I m, = mg- iri* 



It may be shown by applicatinn of Snell's Law that the head of 

 the vector /c, will shift to the right along line AB in Fig. 3 main- 

 taining a constant projection on the beach as the patch moves inshore. 

 Then, from Fig. 3, 



cos 9, = m cos 9 = m^ cos 9(, 



(5) 



and also» 



m^= K^b%, 



(6) 



245 



