parameters. Tabular presentations of the elliptic integrals and other impor- 

 tant functions can be obtained from the above references. The ordinate of the 

 water surface y measured above the bottom is given by 



y = y + H cn'^ 

 's -^ t 



2K(k) f-| . k 



(2-59a) 



where 



yt 



en 



K(k) 



k 



distance from the bottom to the wave trough 

 elliptic cosine function 



complete elliptic integral of the first kind 

 modulus of the elliptic integrals 



The argument of cn^ is frequently denoted simply by ( ); thus, equation 

 (2-59a) above can be written as 



yg = y^ + H cn2 ( ) 



(2-59b) 



The elliptic cosine is a periodic function where cn^ [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; when k = 1, the wave profile becomes that of a solitary 

 wave. 



The distance from the bottom to the wave trough y^., as used in equations 

 (2-59a) and (2-59b), is given by 



c 

 d 



H 

 d 



16d2 H 



— r- K(k) [K(k) - E(k)] + 1 -- 

 3l2 d 



(2-60) 



where y is the distance from the bottom to the crest, and E(k) the com- 

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



'4 



16dj 

 3H 



kK(k) 



(2-61) 



and wave period by 



kK(k) 



1 + 



H /: 



a 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 



2-45 



