Impulsively Generated Waves Propagating into Shallow Water 



so that, 



_ cos 9o ^- - . 



m = -^ oQ- . (7) 



cos 9 w ^ ' 



Since dco is small the frequency dependent change in the trajectory 

 of the phase wave orthogonals may be neglected. The change in the 

 patch width due to the geometric spreading of the rays as the patch 

 moves through a distance, ds , will be 



ds 



pds . (8) 



Next, consider the patch length, 6^. If 6t is the interval required 

 for the patch to pass a given point, i, on s, then 



6i = Vj 6t 



where v. is the group velocity at point i. The time interval, 6t, 

 may also be represented as follows 



6t = ^— dco 

 oco 



Since v = doi/dK, 



5t=-dto)^ _1_|X ds 



247 



