2. Shelf Resonance . 



Hidaka (1935a, 1935b) carried out a theoretical investigation of a 

 vertical wall at the shoreline, where the water depth at the wall was 

 d and the sea bottom sloped seaward. The depth d at any arbitrary 

 distance x from the shoreline is given by 



(■ • if 



d = d g ( (183) 



where the horizontal distance, x, is positive measured seaward from 

 the shoreline, x = at the shoreline, and a is the distance from the 

 shoreline to the depth d = /2~ d . The depth variation defined by equation 

 (183) can be compared to a linear (constant) bottom slope, S 2 , between 

 the toe of the nearshore slope (taken to be a vertical wall) and a point 

 at the distance x = a from the. shoreline. For the linear bottom slope, 

 S 2 , 



d - d 

 S 2 = —^ (184) 



or at a distance, a, from the shoreline 



2 

 from which 



fZ d - d 

 S = 2 £ (185) 



a 

 (/2 - 1) d 



(186) 



S 2 



The variables are shown in Figure 31. 

 Defining the wave by the equation 



at 2 



[<•£] 



(18.7) 



3x 



Hidaka defined the surface elevation n above the undisturbed water as 



n = U cos (— ) (188) 



and U a dimensionless amplitude obtained by dividing the amplitude at 

 any point by the amplitude at the shoreline (U = 1 at the shoreline) , 

 T the wave period, and t time. Hidaka obtained a theoretical solution 

 for wave resonance on the sloping shelf defined by equation (183) using 

 Mathieu functions (see Blanch, 1964). The primary mode of oscillation 



91 



