APPENDIX n 



DEVELOPMENT OF COMBINED SHOALING-REFRACTION COEFFICIENTS 

 Introduction 



This appendix describes briefly the method employed to calculate the combined 

 shoaling— refraction coefficients. 

 Background 



The shoaling— refraction coefficients developed are vaUd for a bathymetry characterized 

 by straight and parallel bottom contours and for a wave system which suffers no energy 

 losses. The two principles employed are SneU's Law and the concept that there is no energy 

 flux across a wave ray, see Figure II-l. 



SneU's Law governs refraction and relates the wave propagation speed, C, to the wave 

 direction, a , 



sin ai _, . s in az ,„,, 



Ci ^ = Const 1 = c. <°-l) 



in which the subscripts pertain to any arbitrary depths. 



The requirement that no energy is propagated across wave rays may be written as: 



F^g cos a = F^jg cos a = Constz (n-2) 



in which Fj^ represents the energy flux per unit width in the direction of wave 

 propagation and the cos a term represents the width between adjacent wave rays. 

 The Ff^ term could be expressed as the product of the wave energy density, TE, and the 

 group velocity, Cq , although this will not be helpful in the effort here. For linear wave 

 theory, it is possible to separate the refraction and shoaling effects because neither the 

 celerity, C, (governing refraction) nor the group velocity, Cq (governing shoaling) depend 

 on wave height. For our case, inspection of Equations (II-l) and (II-2) will show that the 

 two phenomena are coupled through the dependency of C and Cq on the wave height. 

 Method 



The method employed here utilizes the dimensionless energy flux, F^^ (Table XI, 

 Item 5) and the dimensionless wavelength, L' (Table XI, Item 1), where 



TE 



TE H^ L 

 "^8 T 



, _ L 



^ (gTV27T) 



103 



