On wave-action and its relatives

Conservable quantities measuring ‘wave activity ’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary, in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves. The sense in which the general wave-action equation follows from the classical ‘ energy-momentum-tensor ’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy , which in turn are related to special cases such as Blokhintsev’s conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when waves ’ are defined respectively as departures from ensemble-, spaceand time-averaged flows. The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.