is used; i.e., the average horizontal water particle velocity over a 

 wavelength is zero. The results of such investigation are presented 

 in Figures 1 and 2. 



The correction AH due to nonlinear effects never exceeds 5 

 percent and is more commonly of the order of 1 percent. These in- 

 vestigations show that: 



(a) The nonlinear shoaling coefficient is initially less than 

 the linear coefficient when d/Lg > 0.4, then becomes larger toward 

 shallow water until the wave breaks. 



(b) The Stokesian power series is not uniformly convergent, i.e., 

 the function of d/L of higher order tends toward infinity when d/L tends 

 to small values. Therefore, the "best" order of approximation is not 

 necessarily the highest order. For relatively deep water d/L > 0.25, the 

 fifth order of approximation would be the best insofar as wave height 

 transformation is concerned; for very shallow water d/L < 0.01, the 

 linear theory would be best. In the intermediate range the third-order 

 theory would be best, and therefore should be preferred overall because 

 of its range of applicability. 



The second definition of Stokes for the phase velocity can also be 

 used; the average momentum over a wavelength is zero by addition of a 

 uniform motion. Yamaguchi and Tsuchiya [1976) indicate that the results 

 yield slightly larger values, at most a 7-percent increase for the 

 shoaling coefficient, than the results obtained by Le Mehaute and Webb 

 (1964) . 



The principle of conservation of energy flux has also been applied 

 to a cnoidal wave, and like the Stokesian wave the results depend on the 

 order of approximation and the definition of phase velocity. All these 

 investigations on cnoidal waves are based on an energy flux such as ex- 

 pressed by equation (1). Masch (1964) was the first to deal with this 

 subject; however, his wave theory is not consistent, even erroneous, 

 (in the table of functions used by Masch in the shoaling of cnoidal 

 wave, the water depth below NIWL should be substituted by ht , the water 

 depth under trough) , and the results are presented in a form which is 

 difficult to use. The relation to deepwater wave and sinusoidal theory 

 is not discussed and no attempt is made to follow the shoaling of a 

 specific wave. 



A significant contribution to the shoaling of cnoidal waves is 

 given by Iwagaki (1968) . Iwagaki treats the case of an approximate 

 solution of cnoidal wave in which he used the second definition of 

 phase velocity as given by Laitone (1961) . The approximation is on the 

 value of the elliptic integral which is replaced hy a simple function 

 of empirical coefficients. Iwagaki shows that this simplification 

 actually covers a wide range of cases and allows him to simply in- 

 vestigate the shoaling of what he calls "hyperbolic waves." When the 

 energy flux in deep water (as computed using small -amplitude theory) 

 is equated to the energy flux in shallow water, described by first- 

 order hyperbolic waves, Iwagaki obtains 



