Figure 31. Shelf resonance, 



for the shelf is defined by the Mathieu function Cej (£, 0j) where 

 0, = 7.51361. The period is given by 



T l = 



2a 



3.2417 



t 



l 6 s 



/gd^ 



(189) 



The second mode of oscillation is proportional to the Mathieu function 

 Ce 2 (£, 2 ) where 2 = 21.29863, and the period T 2 is 



T 2 = 1.9254 



(190) 



The first and second modes are shown in Figure 31 . The values obtained 

 by Hidaka for resonant period's are for a shelf extending a long distance 

 offshore; i.e., the shelf width £ g >> L, where L is the wavelength of 

 the incident wave. These results have not been verified by other inves- 

 tigators, but Hidaka' s results of the variation of wave amplitude agree 

 very well with those obtained by Wilson (1972) for a constant (linear) 

 shelf slope, as shown in example problem 14. 



92 



