approximation represents an even worse approximation than the third 

 order. Similarly, it is found that second-order cnoidal theory is 

 worse than first-order cnoidal theory for large wave steepness. This 

 is inherent to the point that both cnoidal and Stokesian power series 

 expansion in terms of the small parameters h/d and H/L respectively 

 are nonuniformly converging series since the functions of d/L attached 

 to each power term blow up when d/L tends toward small values. 



It is interesting that Yamaguchi and Tsuchiya (1976) found that 

 the shoaling coefficient given by Le Mfehaute and Webb (1964) (first 

 definition of Stoke 's phase velocity) almost coincides with the 

 shoaling coefficient obtained from cnoidal theory developed by 

 Chappelear (1962) (second definition) . 



Shuto (1974) attempted to make a synthesis of all these theories 

 in a simple and practical form by empirically matching these solutions. 

 Subsequently, he proposes the following law for practical purposes: 



= constant 







•? 



< 



30 



2tt 



The 



small -aiT 



30 



2tt 



LoH 

 d^ 



< 



50 



2w 



Use 



Hd2/7 



50 



2tt 



■f 



< 



^ 



Use 



Hd^/2 



o 



1/2 



- 2/3 



constant (5) 



These equations seem to be the most realistic to remember from all 

 the theoretical approaches. In the range where both cnoidal and third- 

 order Stokesian theory apply, the values of the shoaling coefficient 

 are very close to each other as shown in Figure 6 (Flick, 1978). 



3.0 



^ 





7=2. 48s 



2.5 



^\: 





Ho/Lo=0032 







CNOIDAL 





>- . 





STOKES third order 



2.0 



^ 







1.5 



- 



^^^^ 





1.0 



- 



"~- 



^^^-^^^ -^^=^_._ 



5 



< 1 





1 1 < I 1 



006 .008 .01 



d/Lo 



Figure 6. Comparison between Stokesian third order and 

 cnoidal shoaling coefficient with experiments 

 (from Flick, 1978). 



16 



