of the chord. This defines the path of an unrefracted wave ray (the 

 expected result when T -> 0) , and therefore verifies equations [264) and 

 (265) for a concave coastline such as a large bay. 



For a convex coastline (e.g., a circular island), the wave rays 

 which would probably be trapped are those where a± is large, i.e., 

 the wave rays most nearly parallel to the shoreline. Letting o^ -*■ it/2, 



tan ai 



so that the term 



1 + c z tan z a, 



tan' 



1/2 



(271) 



From equation (264) 



c tanh I — 



n 2 d a* 



R" 



K* 



St V 

 Substituting equation (265) in equation (272) above, 



4 d s 



tanh I 



c R* 



s 



R* 



2 d* 

 tanh I 



(272) 



(273) 



as r* > R* for a convex coastline, then R*/r* < 1, 



P S S tr 



d* 

 d* s 



Therefore, 



(274) 



as a condition of wave trapping on a convex coastline. This means that 

 the slope of the shelf must be greater than some minimum value defined 

 by d|/R| in order to have a caustic, i.e., to have waves trapped on the 

 shelf. This is necessary in order to have the rate of curvature of the 

 wave ray exceed the rate of curvature of the bottom contours, a necessary 

 condition of wave trapping. Where a circular island has a small radius, 

 R* in relation to the water depth at the shoreline, d|, there is a 

 greater probability of the wave rays spiraling off into deep water than 

 there would be for an island with a large radius. 



The minimum and maximum values of n 2 which will produce solutions 

 can be found as follows: 



From equation (264) , where | c 



,,.2 



c tanh 



n- d* 



2 S 



c R* 



R* 



— , tan a, > + 6 



(275) 



120 



