Thus 



Van Mater and Neat 



^UI,j,k'^i + l,j,kPi+l,j,k - ■^ijk^iJkPijk 



'lM.i.K-1iiKU„,>,„,i.J • (33) 



Similarly , 



1/2 

 ' * «-"i+l,j,k^^i + l,j,kJ 



In the computer program it is the wave system envelope, B', 

 that is carried forward to the next station and the new value at that 

 station computed from Eq, (34), The phase term is then applied to 

 obtain the wave elevation. 



The envelope function B'((jo) is symmietrical about the SWL 

 by definition. This corresponds well to the observed envelopes at 

 the starting station in deep water, but as the system moves into 

 shallow water the envelope and the phase waves develop asymmetries 

 about the SWL which we seek to describe by the application of the 

 cnoidal theory as previously discussed. 



The first problem is to calculate the elliptic modulus, k, 

 for once this parameter is known, all other cnoidal properties may 

 be computed directly. Two difficulties are immediately realized 

 in the determination of the elliptic naodulus. First, Eq, (18) does 

 not admit an explicit solution in k. Secondly, the form of cnoidal 

 waves becomes quite sensitive to k as k approaches unity. For 

 example, there is a noticeable difference between the form of the 

 wave determined by k^ = 0,99990 and that determined by k^ = 

 0.999990. Further, explosion parameters of interest require the 

 determination of modulus values as large as k =1-10' , Thus 

 the following procedure was employed to efficiently and accurately 

 determine k frona Eq, (18), 



Write Eq. (18) as 



We then seek the roots of the equation 



g(k) = (35) 



266 



