110 Shallow Water Wave Transformation through External Factors 

 The wave profile of these cnoidal waves has the form 



r, =2Aaf-(lK X ^, (V.l) 



in which en represents the Jacobi elliptical function and K the complete 

 elliptic integral of the first kind. The module k is computed from the relation 



v 



2A 

 h 



AKk 



1/3 



(V.2) 



where h is the depth (from wave trough to bottom) and 2A the wave height 

 measured from trough to crest. We can replace the module k by sin a. For 

 a = 0, k = and for small values of X we obtain in a first approximation 

 sine waves, which means a harmonic wave profile in the sense of the Stokes 

 theory. If, on the other hand, a = 90°, then k = 1 and, with K = oo, X = oo. 

 The wave profile nears a sech-function and thus corresponds to that of the 

 solitary wave (p. 116). The theory of Korteweg and de Vries is important 

 because it bridges the gap between "long" and "short" waves. 



To test the theory on observed waves, Thorade (1931, p. 183) proceeded 

 as follows: From the observed wave length I and the values h and A the 

 value at the left side of (V.2) is computed. From Table 15 or from a graphical 

 presentation the auxiliary angle a is obtained. With this value we find in 

 the Table on p. 134 of functions by Jannke and Emde (1933) the numerical 



Table 15. For determining cnoidal wave profiles 



values of the elliptic integral F(k, <p) for y = 1°, 2°, ... , 90°, where F(k, \n) = K. 

 We then obtain the values of x corresponding to each y from x = (X/2K)F(k, <p) 

 and then ?; = 2 A cos 2 95. 



Figure 47 gives for standard waves X = 1 and 2A = 1 the corresponding 

 wave profiles for several values of a between 10° and 89°. With a cons- 

 tant wave length and a constant wave height, a approaches rapidly 

 the limit of 90° with decreasing depth, and the wave crests become narrower 

 and steeper, the wave troughs wider and flatter. If a = 89°, the wave train 



