Abstract
As an extension of M. Hayes' argument (Hayes, 1977), a general relationship is derived between group velocity and fluxes quadratic in wave amplitude for single, weakly-modulated small amplitude plane waves propagating in an inhomogeneous, non-conservative, dispersive system. If a quadratic (4-) flux constructed from wave quantities with a common single harmonic phase is inserted into its linear governing equation, the resultant equation consists of a wave part with a hi-harmonic phase and a mean part, but both parts must vanish separately to satisfy the equation. The amplitude of the former gives the dispersion relation, H(kμ)=0; this in turn gives the complex (4-)group velocity vector νμg=∂H/∂κμ by varying the complex (4-)wavenumber vector κμ(μ=0, 1, 2 and 3). On the other hand, the latter gives a governing equation for the mean of the quadratic flux, and also gives, by taking the same variation, another mean quadratic flux. The latter flux is proportional to the group velocity νμg. The general relationship is illustrated by application to Rossby wave motions. The group velocity and energy flux relation obtained by Longuet-Higgins (1964) is rederived as a special case.
