d = Stillwater depth 



K(m) = complete elliptic integral of the first kind 

 The dispersion relationship may be written as 



\^ =v/F^^^~uT|7) (11) 



o o 



2 2 2 3E 



The deepwater wavelength L = ^ T and A = Afm) = 1 - —r- 



^ ^ o 2tt ^ ^ m mK 



The complete elliptic integral of the second kind is designated E. 



Invoking energy flux conservation between wave rays and using 

 linear theory in deep water, the wave height transformation is given 

 as 



s;<r0'''(e"'^H 



In equation (12) f„ = f^(U) = U 

 H H 



and 



B = B(m) ^ U- ^m^ - 5m + 2 + (4m - 2) |j 



(13) 



Equations (12) and (13) define the shoaling of cnoidal waves and are 

 used to determine the shoaling coefficient. 



To compute the wavelength and refraction the Stokes wave theory 

 is used. Linear waves are described by the dispersion relationship 



1 , 27rd r-, A~. 



— = tanh (14) 



o 1 



For the third-order Stokes approximation from Le Mehaute and Webb (1964) 



^ = I- tanh p^ (1 + C, xh (15) 



^3o a , 2 



^ = fv CI ^ ^3o3 U6) 



where 



f a' . X- = ;!!° (17) 



8 3o 3o LT 

 3o 



27 



