Van Mater and Neat 



0.6 are computed using the descending transformation. The transfor- 

 mations are reapplied until higher powers of m or m| are deemed 

 negligible. The currently used cutoff value is m (m| ) = 10 . Both 

 of the Landen transformations converge quite rapidly. Thus the 

 cutoff parameter value is attained in three of fewer applications of 

 the pertinent transformations. 



Some computational difficulty may be experienced in evalua- 

 ting the hyperbolic functions used in the ascending Landen's transfor- 

 mation, for large values of the argument u. This problem can be 

 alleviated somewhat by reducing the en argument, u, to its 

 principal value - 4K(k) ^ u < 4K(k), Further difficulty may be 

 resolved by using the descending transformation throughout the 

 modulus range where applicable. 



When k = u, the cn^ term in Eq. (40) reduces to cos (4;jj|j)/2 

 and Hljjn = HZjjk = (HjjkV^ = B'. Thus, for k = 0, Eq. (40) reduces 

 to 



Tiijn = B' cos 4jijk, 



which is the usual wave elevation equation. 



The matter of the frequency dependence of the trajectories of 

 the wave orthogonals has been discussed briefly. In principle it 

 would be possible to compute an initial angle Oqjk for each frequency 

 and at each station which give a path length and a path offset from 

 the axis that would fall within established error limits. Such an 

 iterative procedure would increase the computation time enormously 

 and was rejected on this basis. Several schemes were tried in 

 attempting to find a simple rule for the choice of Gojk which would 

 give reasonable conformity in a given family of trajectories. The 

 simplest rule turned out to be the best. A linear distribution of 

 j^ was chosen according to the following relation: 



QojK = QojkLl - 0.04(0) - 0.2)] (54) 



where Gojk is a control angle for the family of trajectories. The 

 choice of the above relation Is quite an arbitrary one and a different 

 and more complicated bottom topography could necessitate a different 

 function or the Iterative procedure discussed above. 



The refraction angle at each station along a given path Is com- 

 puted from Snell's Law: 



e =arcsln ^° i ^^^ ^^i*^ (55) 



270 



