in Whitham's Lagrangian theory and has proved to be a valuable concept. 

 In moving media, such as the currents being considered here, it is a 

 wave-related property that is conserved in the absence of dissipation. 



The first derivation of the concept of wave action was via a 

 Lagrangian. No convenient Lagrangian is available for rotational free- 

 surface flows. However, Stiassnie and Peregrine (1979), using 

 conservation of mass, momentum, and energy, show that wave action is 

 also conserved if the large-scale flow is rotational. (The "local" wave 

 motion is irrotational.) See also Christof fersen and Jonsson (1980) who 

 include dissipation. This result, derived for fully nonlinear waves by 

 Stiassnie and Peregrine, means that wave-action conservation can be 

 applied to any large-scale current system. 



For linear waves, wave action is 



A = E/a = Sspga^/a (22) 



where E Is the usual energy density of linear wave motion, and a Is wave 

 amplitude. As before, a Is the wave frequency relative to the current. 

 (The result (eq. 22) does not extend to nonlinear waves.) The standard 

 refraction analysis for water waves in shoaling water normally has a 

 constant frequency u), which equals a for the case of no currents. Thus 

 in the better established case of still water, the conservation of 

 wave action is equivalent to the conservation of wave energy. (Note 

 that this is not so for nonlinear waves, partly due to wave-induced 

 currents.) 



In mathematical terms the conservation of wave action is expressed 

 as 



9A/8t + V -[(u + Cg)A] = (23) 



For applications equation (23) can be rewritten as 



[9/9t + (u + Cg) • V]A = -[V • (u + Cg)]A (24) 



The left side of equation (24) is the rate of change of wave action 

 along a ray, and the right side shows that the rate of change varies 

 with the divergence of the rays. The operator on the left side of 

 equation (24) is the same as that in equations (17) and (18). Thus 

 these three equations have the same characteristics and may be 

 integrated along the rays described by equation (20). 



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