Impulsively Generated Waves Propagating into Shallow Water 



Using the following transformations, intermediate values of 

 the parameter m are reduced or increased such that the above 

 approximations are applicable. 



To increase the parameter, let 



1/2 , , 1/2 2 



4m _ / 1 - m ^ 



TTp^- 



Then 



sn(u.m) = (1 t p^) ^"'"^;t;^^^{"'P' (47) 



dn(a,m) = (l-(l-p',^))^|^ (48) 



cn(u.m) = (l-(l-pr))^l^ (49) 



To decrease the parameter, let 



1/2 



/ 1 - m, \ u 



P " [ , 1/2 ) ' ^^ , , 1/2 



^i-m. ' 1+P 



Then 



/ \ /< X '/2\ sn(v,p) ,r-ri\ 



in(u,m) = (i+p ) [7^ — ^ (50) 



1 + p sn (v,p) 



J / \ - ti ^l/2\ sn (v,p) 



dn(u,m) = (1 - p ) ,/2 2 



1 + p''^sn'^{v,p) 



(51) 



cn(u,m) = ""<"'P)' f^^'P^ (52) 



1 + p sn (v,p) 



Note that in both the descending and ascending Landen trans- 

 formations , sn and dn are required in order to compute en. In 

 the computer program values of m greater than 0.6 are computed 

 using the ascending transformation. Values of m less than or equal to 



269 



