APPENDIX 

 Hyperbolic Wave Shoaling 



It is well known that near breaking surface gravity water waves are 

 highly nonlinear and therefore not well characterized by linear theory. 

 For prediction of breaking wave characteristics it is necessary to use 

 higher order wave theories and, in particular, a higher order cnoidal wave 

 solution could appear to be attractive because of the improved fit to data 

 found when using first-order cnoidal theory. Iwagaki (1968) derived a 

 practical asymptotic solution to second-order cnoidal waves called 

 "hyperbolic waves." This approximation takes m = 1 and E(m) = 1 for 

 K(m) > 3 where K(m) and E(m) are the complete elliptical integrals of the 

 first and second kind. 



If the wave height transformation is written in the usual manner 

 as a product between a shoaling and refraction coefficient 



then Iwagaki derived 



H K . K (A-1) 



OS r 



L/3 



-1/3 /h\" 2m/3 



'1 



11 M_A 11 1^ /lil\^ /T -11 ^11 1114 



2 K "" d^ 1^5 " 2 K ■" j^2J ^ ^d^J \^ 112 " 160 K " 4 j^2J 



-2/3 



(A-2) 



where 



d = water depth 



2 

 T = gl_ fi + \^) = deepwater wavelength (third-order Stokes 

 o 2tt „„„-, 

 wave J 



H' = unrefracted deepwater wave height 

 o 



K = complete elliptic integral 



H = wave height 



2 



d = d{l - -^5- + T^^ C5-) > = water depth under trough 

 t K d 12 K z 



41 



