Van Mater and Neat 



the expression may be written for numericcd Integration as follows: 



^■j"^ =1 '^'-Ui." ' cos a ','■., ^^j^lik (38) 



i=0 i-l.J.x 



where 



• =74-- - 



ijk l_j Vjj coj 



As' 



^ijk "Z ^- cos0i.,,j,, ^ (39> 



i=0 



In Eq, (39) t\^^^ is the time of arrival of the jth frequency com- 

 ponent at location (i,k) and is printed out for each frequency at each 

 location. 



Introducing the cnoidal phase term of Eq, (14) the wave ele- 

 vation becomes 



lijk = - HZijk + HijKCn' [^^^ (4^ij,) . kj jj . (40) 



The Jacobian elliptic functions can all be expressed in terms 

 of theta functions, and can be computed from the resulting infinite 

 series. However, in this program the elliptic function en in 

 Eq, (40) is evaluated to any specified degree of accuracy using 

 Landen's transformations. 



Let m = k, m=|-m. Then for m sufficiently small 

 such that m and higher powers are negligible, we have the follow- 

 ing approximations for the Jacobian elliptic functions 



sn(u,m) = sin u - 0.25 m(u - sin u cos u) cos u (41) 



cn(u,m) = cos u +0.25 m(u - sin u cos u) sin u (42) 



dn(u,m) = i - 0.50 m sin u (43) 



2 



For m sufficiently close to unity such that m| and higher powers 



are negligible, we have the approximations 



2 



sn(u,m) = tanh u +0.25 m,(sinh u cosh u - u) sech u (44) 



cn(u,m) = sech u - 0.25 m (sinh u cosh u - u) tanh u sech u (45) 

 dn(u,m) i sech u + 0. 25 m,(sinh u cosh u + u) tanh u sech u (46) 



268 



