whose velocity varies linearly with elevation where this has been shown 

 possible with reasonable accuracy. The nonlinear finite - difference 

 technique is potentially a useful tool for dealing with currents under 

 combined wave and current flows in real cases. However, the author 

 hints that the rate of convergence needs to be improved. 



10. DALRYMPLE, R.A., "Longshore Currents with Wave Current 

 Interaction," Journal of the Waterway, Port, and Coastal and Ocean 

 Division, Proceedings of the American Society of Civil Engineers, 

 Vol. 106, No. WW3, Aug. 1980, pp. 414-420. 



Keywords. Current Depth Refraction; Currents, Nearshore; Currents, 

 Wave-Induced; Equations of Motion; Theory; Wave Breaking. 



Discussion. This technical paper examines analytically the effects of 

 larger angles of incidence and the refraction of waves by the longshore 

 current on planar beaches for the case of no lateral mixing. The long- 

 shore current velocity is included in a modified Snell's law, and the 

 momentum equation is derived, based in part on previous work by Liu and 

 Dalrymple (1978) and Iwata (1976). A perturbation analysis is done 

 which depends on the deepwater waves being incident at a small angle to 

 the normal to the beach. 



A zero -order result of the analysis is that the longshore current 

 velocity equation of Longuet-Higgins (1970) is obtained. This equation 

 is written as a product of a term Aq and terms in depth, deepwater 

 direction, and deepwater speed. The analysis then shows that there is a 

 critical value of Ap (equal to 0.78), above which the current refraction 

 in the surf zone produces a somewhat higher velocity (above the Longuet- 

 Higgins value). Below the critical value, the "large angle effect" 

 causes a decreased velocity, which in turn, causes the surf zone to 

 widen in order to balance bottom shear. However, the deviations from 

 the Longuet-Higgins value are not significant for the example shown. 



Closer examination suggests that the expected values of Aq are much 

 larger than critical. Accepting a breaker height-to-depth ratio of 0.8 

 used by the author, it appears that Ap must equal 202 m/f, where m is 

 the slope of the planar beach and f is the Darcy-Weisbach friction 

 factor. It is rare for a surf zone to have a slope flatter than 0.02. 

 Thus, Aq probably exceeds 4/f. If this is the case, f would have a 

 value of at least 5 to get A^ below the critical value, but typically f 

 is on the order of 0.01. 



Coastal Engineering Significance. The perturbation analysis produces 

 the Longuet-Higgins equation as the zero-order longshore current 

 velocity, which lends further support to the widespread adoption of that 

 equation. Under the interpretation given here, current refraction 

 effects dominate over large-angle effects in practical cases, the effect 

 being to increase the actual current above the predicted Longuet-Higgins 

 value. The amount of increase is not clear, but may be small. 



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