Where 04 ■*■ 90° the wave rays become trapped very close to the shore- 

 line so that Tp -*■ R g . As a becomes smaller, Tp becomes smaller so 

 that the caustics are farther from the shoreline. As 0^ -*■ 0, Tp -*■ 0, 

 meaning that a wave ray reflected along an orthogonal to the shoreline 

 will pass through the center of curvature. 



Equations (264) and 265) are used to investigate wave rays at any 

 angle 0^ . From equation (265), c > 1 as tanh [n 2 d*/ (rg c)] £ 1. To 

 determine the limiting values of n 2 which will provide solutions, it 

 may be noted that, in equation (264), 



R* 

 -£> 1 



(266) 



From equation (265), as c + 1, n, ■> °°. From the definition that 1/M « T 2 

 (eq. 262), and that 1/M = c/n 2 , it can be seen that waves would be trapped 

 where T -> 0, which is a restatement of the fact that all waves would be 

 trapped where the coastline is concave; e.g., a large, circular bay. 

 Finding the caustic location, i.e., the radius, Tp, when n -> °° and T -*■ 

 is of interest. From equation (264), when n -*■ °°, and a. > 0, 



tanh 



which then gives 



and as c -> 1, 



which reduces to 



1 + c z tan' 



tan' 



1/2 



i 2 d* 



2 S 



1 + c z tan' 



tan' 



1 + tan' 



tan' 



-» 1 



1/2 



1/2 



(267) 



(268) 



(269) 



(r*) . 



p rmn 



R* sin a. 

 s 1 



(270) 



Where the angle oil is known, defining the angle between the reflected 

 wave ray at the shoreline and the normal to the shoreline (see Fig. 35), 

 the wave energy will always be trapped between the radius, rt, defining 

 the caustic and the radius, R*, defining the shoreline if the concave 

 shoreline extends a sufficient distance. Where the wave period becomes 

 longer, it will be trapped closer to the shoreline. 



It is of interest to note that equation (270) provides a solution 

 independent of water depth or shelf slope. Equation (270) defines the 

 distance from the center of curvature to a chord across a circular arc, 

 where a, is the angle between the chord and a radius drawn to the end 



119 



