Equation (H-l) can be rewritten in terms of the dimensionless quantities as: 



2Tr sin ai 2it sin a? ^ . /tt o\ 



However since the period is conserved, i.e., T^ = T^ 



sin ai _ sin gg „ . ,„ ., 



""LJ; 14 " Const 3 (n-4) 



The energy flux relationship. Equation (II-2) can be expressed as: 



or recognizing that the period is conserved 



2 

 -i^l F'^ L' cos a = Const^ (H-S) 



Lq J TE 



Equations (11-4) and (11-5) describe the shoaling-refraction process in terms of available 

 dimensionless parameters, and were solved as described in the following paragraphs. 

 Solution 



It was found convenient to characterize a particular incoming deepwater wave by the 

 direction, a^, and deepwater steepness, H^ /L^. The problem is to determine wave 

 steepnesses at other relative depths h/L^ such that Equations (114) and (II-5) are satisfied 

 recalling that L' and F'j.^ both depend on h/L^ and H/L^. For each relative 

 depth, h/L^, four values of L' and Fy^ are available (for H/H^ =0.25, 0.5, 0.75, and 

 1.0, c.f. Figure 23) whereas a continuous distribution is required for the purpose here. For 

 each relative depth, h/L^ , continuous distributions were obtained by fitting straight Unes 

 between the four available points; for H/H^ = 0, it was assumed that the simple linear wave 

 theory applied, see Figure II -2 for an example for h/L^ = 0.02. 



For given H^ /L^ and G^ , the constants in Equations (II-4) and (II-5) are defined. The 

 wave steepness H/L^^ and direction a at any relative depth are determined by iteration of 

 the two following equations. 



k+1 . -1 f/^.f^k sinoo"! ,^ f. 



Ln' 



jjk-Hl |'(Ho/Lo)^ (''tE^° -^0 "^^^ "°" 



^° ~ [ (F^g)^ (L')^ cos a^ 



h 



(n-7) 



105 



