so the vertical dimensions in equations (252) to (259) are assumed to 



where the term r 



P 



represent dimensional values divided by r p e . 



the vertical length scale. Shen (1972) takes eM = 1 . It is assumed 



e -> 6, where < 6 << 1, so that M must be large. Shen defines M 



M = 



(cj*) 2 (horizontal length scale) 



which gives 



=(r) 2 (?) 



(262) 



For long-period waves (e.g., tsunamis) where the period T is large, 

 the caustic radius r£ must be large in order for M to be large. In 

 general, the solution is for cases where the shoreline radius Rg (and 

 therefore the caustic radius r„) is much greater than the wavelength. 

 Using Shen's work, equation (2b6) now reduces to 



1 + c z tan z a, 



(R*) 2 tan 2 a, 

 s L 



(r*) 2 



P 



1/2 



tanh 



1 + c z tan z a. 



'(R*) 2 tan 2 a, 

 s_ 1 



(r*) 2 



P 



1/2 



i d* 

 2 S 



= 1 



(263) 



Shen has defined M = n /c so that e = 1/M = c/n, 

 reduces to 



Equation (263) further 



1 + c z tan z a. 



tan 2 a. 



1/2 



tanh 



1 + c z tan z a 



1/2 



l' 2 d* 

 2 s 



tan z ot. 



(264) 



in its dimensional form, where d| is the dimensional depth at the toe 

 of the shoreline slope, R* the dimensional radius of curvature of the 

 shoreline, and r£ the dimensional radius to the point where the wave 

 ray turns parallel to the bottom contours. Also the dimensional form of 

 equation (259) becomes 



c tanh 



i 2 d' 



y c 



= i 



(265) 



where d* is the water depth at radius r^ 



Consider first a concave coastline, such as a large bay, where the 

 radii rt and R* are measured from the center of curvature offshore, 

 and Tp < R g . In the limiting case, a concave coastline would form a 

 closed circular basin with radius R g . Therefore, all wave rays could 

 obviously be trapped by this type of coastline. 



118 



