Conservation of Action and Modal Wave Action

For a system governed by a Lagrangian density a local action density and flux are defined for a family of solutions periodic in a parameter, and this action obeys an absolute conservation law. In the linear case the local action density and flux may be defined analogously for a pair of solutions and are essentially the components of the classical bilinear concomitant. For wave propagation the parameter is interpreted as a phase shift in the solution of interest. For wave trains the wave action density and flux are the integrals over the mode of the local action density and flux, and may be identified as Whitham's wave action density and flux. A perturbation Lagrangian for ideal fluid flow is applied to wave propagation in an acoustic duct with winds. The appropriate extensions to multiple wave systems are discussed.