In the small -amplitude theory assumed, energy is not dissipated and does 

 not cross wave orthogonal s. Therefore, the energy flux in the direction of 

 wave travel must remain constant between orthogonals, i.e., between deep water 

 and point i . 



P*b = P'Jb. = total wave power between orthogonals (A-4) 



From the geometry of Figure A-2, it is obvious that b^ changes with 

 position on the wave path. However, the distance between adjacent orthogonals, 

 s, measured in the longshore direction, does not change because orthogonal 2 

 must be identical to orthogonal 1 in all respects except being displaced along 

 the coast by a distance s. Therefore, at any point, i, in the wave path 



b^ = s cos o^ (A-5) 



For the straight parallel contours assumed, the distance s is arbitrary. 

 Therefore, divide both sides of equation (A-4) by s, and set s equal to the 

 unit of distance. In the SPM, this unit is 1 foot; in metric units, it would 

 be 1 meter. This procedure defines a total wave power, per unit length of 

 shoreline, indicated by P 



P = P* cos a-i (A-6) 



In the Appendix, i indicates any point on the path where equations (A-1) 

 to (A-5) are valid. With this understanding, the subscript i is dropped in 

 the derivation below. Since P has both a magnitude and a direction at any 

 point, a longshore component, P£, can be defined. Using equations (A-1) and 

 (A-6), this longshore component of energy flux can be written 



Pj^ = (ECg cos a) sin oi (A-7) 



By use of the identity 



sin 2 a = 2 cos a sin a (A-8) 



equation (A-7) can be written as the first of the four alternate forms for 

 ?^ given in Table 4-7 of the SPM, i.e., equation (4-31) in the SPM: 



P, = 2 C^ (^E sin 2a) (A-9) 



The equation is given in this form so that the term in parentheses corresponds 

 to the longshore force term used in Longuet-Higgins (1970). The energy density, 

 E, is proportional to the local height squared, H^ , (eq. A-2). The local 

 height can be related to the deepwater height, H , by equations (7.8) and 

 (7.9) in Wiegel (1964) 



" " Vs% (A- 10) 



