APPENDIX A 



DERIVATION OF LONGSHORE ENERGY FLUX FACTOR 



This appendix derives four formulas for longshore energy flux, Pj^, and 

 four formulas for corresponding approximations, the longshore energy flux 

 factors, P^s • These eight formulas are given in the SPM in Tables 4-7 and 

 4-8, based on assumptions summarized in Table 4-9. For convenience. Tables 

 4-7 and 4-8 are reproduced here as Figure A-1 (p. 4-97 of the SPM). 



The basic longshore energy flux derivation is given in Galvin and Vitale 

 (1977), who made use of earlier work by Walton (1972). The derivation also 

 benefits from Longuet-Higgins ' (1970) work on conservation of momentum flux 

 in the surf zone. Equations well known from linear wave theory are presented 

 without derivation but are keyed to other chapters in the SPM, or to the devel- 

 opment in Wiegel (1964) where more detailed derivation is provided. 



1. Equations for Pjj,. 



The derivation for P^ proceeds as follows: Assume a coast with contours 

 that are parallel to a straight shoreline (Fig. A-2) . Waves approaching this 

 coast are assumed to be described by linear small -amplitude theory. In general, 

 a wave crest that makes an angle Uq with the shoreline when in deep water 

 will refract to make an angle a^ at some shallower depth (Fig. A-2), where a^ 

 is related to a^ by Snell's law. In what follows, the subscript o refers 

 to deepwater conditions, and the symbols are those used in the SPM. 



The path of a wave passing through point i is shown on Figure A-2 as the 

 dashed orthogonal labeled "wave path." The flux of energy in the direction of 

 wave travel per unit length of wave crest at point i is given by 



P* = C^ E 



= n C E (A-1) 



where C„ is the wave group velocity, C is the wave phase velocity, 



n = C„/C, and E is the energy density, the total average energy per unit 



area of sea surface, and is defined as (eq. 2-39 in the SPM) 



F = — tA-2) 



where w is the weight density of water, 64.0 pounds per cubic foot for 

 seawater, and H is the wave height. (This wave height is the height of a 

 uniform periodic wave. How it relates to the significant wave height and to 

 wave heights characterizing wave height distributions is discussed in App . B.) 



From equation (A-1), the energy flux in the direction of wave travel, at 

 point i, for crest length, b^ , is (see eq. 7 in Galvin and Vitale, 1976) 



P* b^ = E Cg hi ^A-3) 



15 



