Discussion. Regular waves propagating on a steady current over a gently 

 sloping bed are analyzed. The current varies linearly with depth, and 

 so has constant vorticity. The analysis is two-dimensional, and 

 dissipation is neglected. 



Definitions and expressions correct to second order in wave 

 amplitude are given for the radiation stress, wave energy density, and 

 total energy flux. 



Applying average Lagrangian techniques, the authors generalize 

 BRETHERTON and GARRETT's (1968) results for waves on a current with 

 uniform velocity profile, namely that wave action density is equal to 

 the wave energy density divided by the intrinsic (i.e., relative) 

 angular frequency. In the extended theory the relative frequency is 

 that relative to a frame of reference moving with the average-over-depth 

 current velocity. This determines the wave amplitude variation 

 explicitly. 



Also an analytical expression for the current wave setdown is 

 found. Graphs illustrate the effect of the vorticity on wavelength, 

 wave amplitude, and setdown. 



Coastal Engineering Significance . This paper demonstrates that a linear 

 current profile over depth is a good first approximation to real flows 

 with waves superimposed. It further shows that in the wave action 

 conservation equation, the relevant frequency is that relative to the 

 average-over-depth current velocity for linear current profiles. 



29. JONSSON, I.G., SKOUGAARD, C., and WANG, J.D., "Interaction Between 

 Waves and Currents," Proceedings of the 12th Coastal Engineering 

 Conference, American Society of Civil Engineers, Vol. 1, 1970, pp. 

 489-507. 



Keywords. Conservation Equations; Current Wave Shoaling; Currents, 

 Large-scale; Currents, Unidirectional; Flow, Irrotational; Mean Energy 

 Level; Setdown; Theory. 



Discussion. This paper considers interaction between regular gravity 

 waves and a steady current over a gently sloping bed. Flow is 

 irrotational and a second-order Stokes wave expansion is used. The 

 general concept of a mean energy level is introduced, and is shown to be 

 a constant horizontal level for periodic, irrotational free-surface 

 flow. 



The complete set of conservation equations for a two-dimensional 

 current wave motion is presented, leading to practical equations for 

 wavelength and height, and current wave setdown. Dissipation is 

 disregarded, and there is no comparison with measurement. 



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