Taking the radius at the shoreline as R s , and the radius where the 

 wave ray turns parallel to the bottom contours as r„, at the shoreline 



tan a 1 = ±(n 2 R 2 - c 2 )" 1/2 



'l + c 2 tan 2 a,\ 1/2 



R 2 tan 2 a. 



1 + c A tan z a. 

 R 2 tan 2 a 1 



1/2 



tanh 



1 + c z tan z a. 



R 2 tan 2 ct, 

 s 1 



1/2 



2 

 — d 

 c s 



Where the wave ray turns parallel to the bottom contours 



(n 2 r 2 - c 2 ) = 



V 



(254) 

 (255) 



1 (256) 



(257) 

 (258) 



Substituting into equation (253) , 



tanh 



P 



= 1 



(259) 



where d is the water depth at radius r p . Equation (256) can be solved 

 to determine c for any integer value n 2 satisfying the equation, and 

 equation (259) can then be solved to obtain a value for r p corresponding 

 to each value of c which will provide a solution. Shen and Meyer (1967) 

 indicate that a number of caustics may exist. For a circular island, reso- 

 nance will occur between the shoreline and the caustics for wave periods 

 defined by integer values of n 2 for which solutions exist. 



Equations (252) and (253) were derived for dimensionless variables 

 where the dimensional values of length had been divided by some horizontal 

 length scale. Camfield (1979) gives the following development to express 

 the solution in terms of several dimensionless parameters. Shen and Meyer 

 (1967) infer that the radius r„ of the caustic (i.e., where the wave ray 

 turns parallel to the bottom contours) is an appropriate length scale. 

 Using an asterisk (*) to define dimensional values, the radius of curva- 

 ture r„ is normalized so that its dimensionless value is 



V 



The derivation of the equations also assumes that 



vertical length scale 

 horizontal length scale 



(260) 



(261) 



117 



