Van Mater and Neat 



have come from Keller [ 1948] , Benjamin and Lighthill [ 1954] , 

 Laitone [ I960] , and Laitone [ 1962] . Masch [ 1964] has computed 

 the shoaling characteristics of cnoidal waves assuming constant power. 

 Iwagaki [1968] has sinnplified cnoidal wave equations to a form which 

 he calls "hyperbolic waves." Masch and Wiegel [ 1961] have pro- 

 vided extremely useful tables of cnoidal wave functions. 



A paper by Le M^haut^ , Divoky, and Lin [ 1968] motivated, 

 in part, the choice of the Keulegan and Patterson theory for incor- 

 poration in this prediction scheme. These authors reported shallow- 

 water wave experiments and compared the results with twelve differ- 

 ent wave theories. Their finding was that none of the theories was 

 uniformly satisfactory but that the cnoidal theory of Keulegan and 

 Patterson was the most generally satisfactory. For the shortest 

 waves linear theory was the best but failed rapidly as the wave length 

 was Increased. Stokes' second order and Laltone's second order 

 were consistently worst. Stokes' third and fifth order were better 

 but not as good as linear theory. In terms of the wave profile the 

 Keulegan and Patterson cnoidal profile gave the overall best agree- 

 ment and was accurately placed with respect to the still water line. 

 This latter finding has also been confirmed by Adeymo [ 1968] . 



The central equations adapted from Keulegan and Patterson 

 for use here are: 



Tl'(r',t') = - Ti2' + H cn^ [^^ (/cr' - o;t'), k] (14) 



n,' _ K,(k) - E(k) . tiz'., ni' (iw 



i^H. 16fl^K{k)]' (17) 



h^ ^ 



f -¥-[™]^-[(^)-(f^(l?)(-3f|)](18, 



(primes Indicate that the variable Is In the dimensional coordinate 

 system.) The expression, L H/h , may be recognized as the 

 Ursell parameter, although. In fact, Stokes was the first to Identify 

 It. The parameter gives a measure of the linear or non-linear nature 

 of the system. Linear theory Is generally applicable for values less 

 than i while cnoidal theory Is most appropriate for values greater 

 than 10. 



The en function displays a character that is particularly useful 



250 



