and equations (A-8) and (A-17) to get the second of the equivalent forms for 

 P^ given in Table 4-7 of the SPM, i.e., equation (4-32) in the SPM: 



P^ = C (i E, sin 2a,) (A-19) 



The third equivalent form for Pjj, is equation (4-33) in Table 4-7 of the SPM. 

 It is obtained from equation (A-15) by using equations (A-12) and (A-8) , 



P = k2 C f^E sin 2a') (A-20) 



I R o \4 o I 



The fourth and last of the equivalent forms of P^ is equation (4-34) in Table 

 4-7 of the SPM. It is obtained by substituting equations (A-8), (A-11), and 

 (A-18) into equation (A-9) , 



'.=(fr)(§;)^.(i^=^"-») '-^» 



2. Equations for P 



Is 



Up to this point, all results are for small-amplitude linear theory. How- 

 ever, the assumed relation between longshore transport and energy flux in the 

 surf zone requires that P^ be evaluated at the breaker line where small - 

 amplitude theory is less valid. To indicate approximations for waves entering 

 the surf zone, the symbol P^g will be used in place of P^ . This approxima- 

 tion is called the energy flux factor, ^iq, in the SPM, and like P^ , it is 

 measured in units of energy per second per unit length of shoreline; e.g., foot- 

 pounds per second per foot. One expression for Pj^g will be derived from each 

 of the four equivalent expressions for Pj^ (eqs . A-9, A-19, A-20, and A-21) 

 to obtain the four equations in Table 4-8 of the SPM. 



The energy density appears in all four equations for P£. In foot-pound- 

 second units, and for saltwater (w = 64 pounds per cubic foot), the energy 

 density is, from equation (A-2), 



TT WH2 



- 8 H^ (foot-pounds per square foot) (A-22) 



In shallow water, group velocity equals wave speed, and near breaking, wave 

 speed depends on depth measured from the crest elevation, as in solitary wave 

 theory (Section 2.27 in the SPM). 



The equation for wave speed near breaking, Ci, , is an approximation. 

 Several approximations are possible, but the solitary wave approximation 

 used in obtaining the SPM equations for P^g is as follows (symbols defined 

 in Fig. A-3) . The first approximation for the speed of the solitary wave is 



20 



