According to Iwagaki (1968), this theory yields sufficiently accurate 

 results for Ursell parameter U > 47. However, as pointed out by 

 Svendsen (1974), the theory of Iwagaki deserves to be regarded as a 

 practical solution to second-order cnoidal waves when the deepwater 

 wave steepness is smaller than 0.02 and the relative water depth is 

 smaller than 0.05. The matching of the Iwagaki hyperbolic wave with 

 the third-order Stokesian wave is shown in Figure 3. 



The shoaling of the true cnoidal wave has been investigated by 

 Svendsen and Brink-Kjaer (1972), Svendsen (1974), and Svendsen and 

 Hansen (1977) . They also give H/H^ as function of d/Lo and Hq/Lq 

 (Fig. 4) and a computer-printed table. It can then be shown that for 

 large values of Ursell parameters the shoaling coefficient Kg -^ d-1 

 instead of d"l/4 as given by the Green law (long wave linear theory) . 

 Concurrently, Shuto (1974) arrives at very similar results. 



Yamaguchi and Tsuchiya (1976) also carry out the same calculation 

 based on the two definitions of the Stokes wave velocity for the cnoidal 

 theory of Laitone (1961) and that of Chappelear (1962) . However, an 

 arithmetic error has been found in the Laitone theory (Le Mehaute, 1968) . 



2. Comparison and Matching Between Various Theories. 



As a wave propagates from deep water to shallow water it is 

 theoretically possible to determine the variation of wave height, wave- 

 lengths, etc. This could be done by applying the principle of con- 

 servation of energy flux to either the linear wave or the nonlinear 

 Stokesian wave, or the cnoidal and solitary wave. Since a Stokesian 

 wave rather applies in deep water, the transformation of water wave 

 should be followed with that theory for the largest value of relative 

 depth d/Lp and then switched to the cnoidal theory when d/Lo becomes 

 small. However, such a scheme implies that the theories can be matched 

 continuously, but there is a priori no reason why the ratio H/Hq should 

 be the same for the value d/L^ which corresponds to the limit of validity 

 of both theories. On the other hand, if the wave heights are matched, 

 then the energy flux will present a discontinuity (Fig. 5). The signifi- 

 cant feature is that the cnoidal wave height grows faster with decreasing 

 depth, although at intermediate depth its value is up to 10 percent less 

 than predicted by a Stokesian theory. Waves with wave steepness larger 

 than 2 to 3 percent will break at a depth where the cnoidal wave height 

 is only slightly larger than that of a Stokesian wave. Waves with small 

 wave steepness, however, such as swells, reach much smaller depth before 

 they break and consequently a major part of their shoaling process is 

 governed by the cnoidal wave theory. For these waves, the two theories 

 such as the Stokesian (first order or linear theory) and cnoidal wave 

 at a second order will yield significantly different results. 



13 



