significance, it is used to express the relationships between the various 

 wave parameters. Tabular presentations of the elliptic integrals and 

 other important functions can be obtained from the above references. The 

 ordinate of the water surface Xg measured above the bottom is given by 



Jt 



+ Hcn^ 



2K(k)^-| U 



(2-59a) 



where y^ is the distance from the bottom to the wave trough, en is the 

 elliptic cosine function, K(k) is the complete elliptic integral of the 

 first kind, and k is the modulus of the elliptic integrals. The argument 

 of cn^ is frequently denoted simply by ( ), thus. Equation 2-59a above 

 can be written as 



y^ + H cnM ) 



(2 -59b) 



The elliptic cosine is a periodic function where cn2[2K(k) ((x/L) - (t/T))] 

 has a maximum amplitude equal to unity. The modulus k is defined over 

 the range between and 1. When k = 0, the wave profile becomes a 

 sinusoid as in the linear theory, and when k = 1, the wave profile 

 becomes that of a solitary wave. 



The distance from the bottom to the wave trough. 

 Equations 2-59a and b, is given by 



yt' 



as used in 



yj 

 d 



d 



H 

 d 



16d' , , H 



— — K(k) [K(k) - E(k)] + 1-7 

 3V d 



(2-60) 



where 



is the distance from the bottom to the crest and E(k) is the 



complete elliptic integral of the second kind. Wavelength is given by 



L = 



'I6d^ 

 3H 



kK(k) , 



(2-61) 



and wave period by 



kK(k) 



1 + 



H 



y? 



E(k)\ 

 K(k) 



(2-62) 



Cnoidal waves are periodic and of permanent form thus L = CT. 



Pressure under a cnoidal wave at any elevation y, above the bottom 

 depends on the local fluid velocity, and is therefore complex. However, 

 it may be approximated in a hydrostatic form as 



p = Pg (y^ - y) 

 2-48 



(2-63) 



