1. Conservation Law Balances . 



a. Momentum . Consider the idealized setting defined in Table 2. A 

 schematic and definition sketch is shown as Figure 23, where the plan view 

 shows two refracted wave rays a unit distance apart (exaggerated) , The wave 

 height and MWL variations shown are taken along any wave ray which are all 

 identical for a plane, infinite beach. Clearly, there can be no gradient in 

 S in the y-direction for this case since time-average normal stresses are 

 identical in the y-direction. The radiation shear stress S (i.e., flux of 

 y-direction momentum across plane perpendicular to the x-direction) is not 

 the same on two sides of a differential element as shown in Figure 23. This 

 is because all three factors (E, n, and a) in equation (25) for S vary in 

 the x-direction. The gradient dS /dx thus becomes the driving stress in the 

 y-direction momentum balance and is resisted only by the bed shear stress, t . 

 The overbar is for a time-averaged value. An additional y-direction stress is 

 due to the gradient of the lateral shear force over the total depth, T due 



to turbulent mixing, i.e. wave orbital velocity interactions in the x- and y- 

 directions. For the direction of wave incidence and coordinate system shown 

 in Figure 23, the y-direction momentum balance becomes 



dS _ dT 



The longshore^ current velocity, v appears in the time- averaged bed shear- 

 stress term x , and in the lateral shear force term T . With appropriate 

 expressions for these quantities and for S^^y, it is possible to integrate 

 equat^^on (42) to derive an expression for the distribution of longshore cur- 

 rent v(x) across the nearshore zone as schematized in Figure 23. This pro- 

 cedure again means a decoupling of the x-direction equation (28) from the y- 

 direction equation (42) . 



b. Energy Balance . It is informative to also consider the energy 

 balance equations in the nearshore zone. A good summary is found in Longuet- 

 Higgins (1972a, b) for the idealized case in Figure 23. The general flux of 

 energy per unit length of shoreline toward shore is given by 



F = EC cosa = EC cosa (43) 



X g n 



where E is the local energy density of the waves (eq. 14) and Cg, the group 

 celerity. Assuming negligible wave reflection and no wave-current interac- 

 tions, then the energy balance in the x-direction gives 



— + D = , (44) 



where D is the local rate of energy dissipation per unit area. 



But earlier, using linear wave theory, the shear component of the radia- 

 tion stress was given by equation (25) , 



S = E n sina cosa 

 xy 



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