shown that the concepts of wave action and wave rays are also funda- 

 mental for real flows. An inconsistency in PHILLIPS' (1977) book is 

 pointed out in this connection. 



Looking at a ray tube, a new formula for the calculation of wave 

 amplitudes on large-scale currents over gently sloping seabeds is 

 described. It gives the amplitude, divided by the initial amplitude, 

 along a wave ray as a product of four factors: a Doppler coefficient, a 

 shoaling coefficient, a refraction coefficient, and a friction coef- 

 ficient . 



This is a natural extension of the pure depth refraction case, and 

 the Doppler coefficient is the new concept. This simple formula is 

 probably not available in other publications. 



It is explained how, on the above basis, the total wave field on a 

 current can be calculated, when the following quantities are given: the 

 bathymetry, the absolute frequency, and appropriate boundary conditions 

 in the horizontal plane for the waves and the current. 



All expressions are correct to second order in wave amplitude. The 

 current velocity is assumed to be vertically uniform and of zero order. 



Coastal Engineering Significance . The theoretical developments which 

 have shown the importance and convenience of the concept of "wave 

 action" were all developed for conservative systems without dissipation 

 (BRETHERTON and GARRETT, 1968; STIASSNIE and PEREGRINE, 1979). This 

 paper extends the value and applicability of the concept to flows with 

 dissipation. The flows and wave fields are assumed to be steady but the 

 extension to unsteady flows is not difficult. There still remains the 

 difficulty of finding reliable and confirmed estimates of dissipation 

 and bed friction. 



8. CHRISTOFFERSEN, J.B., and JONSSON, I.G., "An Energy Reference Line 

 for Dissipative Water Waves on a Current," Journal of Hydraulic 

 Research, Delft, The Netherlands, Vol. 19, No. 1, 1981, pp. 1-27. 

 (See also Report No. 177, Danish Center of Applied Mathematics and 

 Mechanics (DCAMM) , Apr. 1980). 



Keywords . Bottom Friction; Conservation Equations; Current Depth 

 Refraction; Currents, Large-Scale; Depth Refraction; Streamlines; Total 

 Head Line; Wave Dissipation. 



Discussion. A well-known principle for the calculation of wave heights 

 in current depth refraction of water waves is the adoption of wave 

 action conservation along wave rays, e.g., see CHRISTOFFERSEN and 

 JONSSON, 1980. This is a natural extension of pure depth refraction, 

 where wave heights are calculated along wave orthogonals. 



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