More studies have been made of the simplified case in which the 

 currents and waves are steady in time. For this case, the right-hand 

 side of equation (18) is zero and OJ is constant along rays. Conditions 

 are usually chosen so that to is equal to the same constant on all rays. 



For some applications, information on wave number and frequency is 

 sufficient, but in most cases, wave amplitude is also required. There 

 are several ways of deriving equations for wave amplitude. Early 

 attempts to do so were incorrect (see the pioneering paper by Johnson, 

 1947) since the fact that energy can be exchanged with the current was 

 not appreciated. The matter was resolved for water waves by Longuet- 

 Higgins and Stewart's (1960, 1961, 1962) papers which are still of 

 Interest because of the examples given of various applications. The 

 final paper in the series (Longuet-Higgins and Stewart, 1964) summarizes 

 the important aspects of their work. (For acoustics, the corresponding 

 equations had been correctly formulated by Blokhintzev, 1956.) 



The mathematical approach which shows the effect most clearly is to 

 average the equations of motion over the period of the wave motion and 

 examine the resulting terms. This is set out in Phillips (Sec. 3.5, 

 1977). Averaging the effect of the oscillatory velocity field due to 

 the surface waves leads to an effective stress field in the fluid. 

 Mathematically, this is the same as the Reynolds stresses obtained by 

 averaging a turbulent flow. These effective stresses act on the mean 

 flow. Longuet-Higgins and Stewart call 



S - = [ (pu u + p6 .)dz - Jspgd^s (21) 



aB J a B ctB aB 



--d 



the radiation stress tensor, in which ri(x,t) is the free-surface 

 elevation, u the oscillatory horizontal particle velocity due to the 

 wave motion, p the pressure, and 5 g is either 1 or (the Kronecker 

 delta). The final term in equation (21) is simply the hydrostatic force 

 corresponding to mean water level. (The sign of S „ is opposite to that 

 which is usual for Reynolds stresses.) 



Energy is transferred because, as water particles move, they move 

 across a velocity gradient and interact with it. The physical 

 interpretation of radiation stress is best described in Longuet-Higgins 

 and Stewart (1964), and its role in interaction with currents is 

 described in the book by Lighthill (Fig. 78, p. 329, 1978), where it is 

 called a mean momentum flux tensor. 



From a different approach, building on Whitham's (1965, 1967) work 

 on averaging nonlinear waves, and using an averaged Lagrangian, 

 Bretherton, and Garrett (1968) drew attention to and showed the 

 importance of a quantity they called wave action. It arises naturally 



31 



