At the minimum value of n„ , 







d* 



s 



d* 



2 S 



R | S 2 



2 R * 



S 



1 



d* 

 S 





R* S^ 



S 2 



(285) 



and at the maximum value of n 9 , 



d* 

 2 s 



S 



(286) 



For a solution to exist for given values of d| and Rg, the maximum 

 value of n| d|/R| defined by equation (285) must be less than or equal 

 to 1, so the minimum value of S 2 for wave trapping to occur is given by 



d* 



s 



R I S 2 



= 1 (287) 



d* 



s 



which reduces to 



^ S 2 



d* 

 s 



S R* 

 2 S 



< 0.5 (288) 



The parameter d*/ (S R*) is a shelf parameter which determines wave 

 trapping. A continuous band of solutions exists for equations (264) and 

 (265), for values of n 2 between the minimum and maximum values defined 

 by equations (283) and (284). Solutions are not limited to integer values 

 of n„ , which define resonant periods for a circular island. 



Solutions for equations (264) and (265) are plotted in Figure 38. 

 Equation (264) is plotted on lines of constant n| d|/R|; equation (265) is 

 plotted on lines of constant d|/(R| S 2 ) . Solutions for trapped waves are 

 obtained where the two families of curves intersect. From equation (288) 

 it is seen that solutions will only exist where d*/ (R* S ) <- 0.5. From 

 equation (262) note that 



2tt M 

 gT2 2tt r* 



122 



(289) 



