4.0 



3,0 - 



2.0 



.0 



Ho/Lo=.OOI 



0.001 

 Figure 5 , 



Linear 



Cnoidal with Fqv 



continuous 

 Cnoidal with H 



continuous 



0.01 

 d/L. 



0.1 



Matching Stokesian (first order) and cnoidal 

 wave theory (Svendsen and Hansen, 1977) . 



These results (Fig. 5) show that no continuous transition is pos- 

 sible between the two theories. This means that it is not possible to find 

 a value of the water depth, d, where the curves for the two theories fit 

 smoothly together. If the Stokesian theory is used in deeper water and 

 changed to a cnoidal theory when the wave enters shallow water, there 

 will be a discontinuity in the variation of either wave height or wave- 

 length, or both, depending on which water depth is chosen for the switch. 

 Of course, the same will appear for all other quantities such as particle 

 velocities, pressure, etc., and the rate of change of these. Svendsen 

 (1974) shows that the limit of applicability of the cnoidal theory is 

 d/Lo < 0.1193 when H is small. Koh and Le Mehaute (1966) also showed 

 that the limit of applicability of the fifth-order Stokesian wave theory 

 is d/L^ > 0.10 when H/L^ =0.05 and d/Lo > 0.13 when H/Lq =0.10 (see 

 Fig. 2). 



There is a large difference between Stokesian and cnoidal wave 

 between d/Lg equal 0.1 and 0.3. In this region no known wave theory 

 fits very well. It could have been expected that a higher order 

 Stokesian theory would be the answer, but the investigation by Koh ^ 

 and Le Mehaute (1966) shows that when d/Lg decreases the fifth-order 



15 



