and the longshore component is given by 



— PS 2 



P. = P cos a sin a = -%■ H C cos a sin a 

 X. a g 



or, since cos a sin oc = 1/2 sin 2a 



P. = tI- H^ C sin 2a 

 £ 16 g 



The approximation for P at the breaker line is written 

 For linear theory, in shallow water, C « C and 



g 



P. = -?^ h2 C^ sin 20^ 



(4-36) 



(4-37) 



(4-38) 



where Hj. 



ib 16 

 and a, are the wave height and direction and Ci is the wave 



speed from equation (2-3) evaluated in a depth equal to 1.28 Hr^ , 



Equations (4-34) and (4-37) are valid only if there is a single wave train 

 with one period and one height. However, most ocean wave conditions are 

 characterized by a variety of heights with a distribution usually described by 

 a Rayleigh distribution (see Ch. 3, Sec. II). For a Rayleigh distribution, 

 the correct height to use in equation (4-37) or in the formulas shown in Table 

 4-9 is the root-mean-square height. Hovever, most wave data are available 

 as significant heights, and coastal engineers are used to dealing with 

 significant heights, therefore the significant wave height is substituted into 

 equation (4-37) to produce 



- .ei- H^ 



sin 2c 



(4-39) 



The value of 



£s 



te 16 "sfc ^gb "" ^% 



computed using significant wave height is approximately 



twice the value of the exact energy flux for sinusoidal wave heights with a 

 Rayleigh distribution. Since this means that P„ is proportional to energy 

 flux and not equal to it, P is referred to as the longshore energy flux 



as 



factor in the following sections. 



Tables 4-9 and 4-10 present variations of P. and P 



is 



depending on the 

 type of wave data available. Table 4-11 describes some" of the assumptions 

 used for Table 4-10. Galvin and Schweppe (1980) derive these equations in 

 detail. Possible changes in wave height due to energy losses as waves travel 

 over the continental shelf are not considered in these equations. Such 



changes may reduce the value of P 



£s 



when deepwater wave height statistics 



are used as a starting point for computing P (Walton, 1972; Bretschneider 

 and Reid, 1954; Bretschneider, 1954; Grosskopf, 1980). 



The term in parentheses for equation (4-41) in Table 4-9 is identical with 

 the longshore force of Longuet-Higgins (1970a). This longshore force also 

 correlates well with the longshore transport rate (Bruno and Gable, 1976; 

 Vitale, 1981). 



4-93 



