On the generalizations of the doppler factor, local frequency, wave invariant and group velocity

Abstract

The concepts of Doppler factor, ocal Doppler-shifted frequency, Blokhintsev wave invariant and group velocity, which are well-known for sound in a uniformly moving medium, and in the ray approximation to the acoustics of nonuniform flows, are generalized to waves of arbitrary frequency in a steady, nonuniform potential flow of arbitrary Mach number. The generalized concepts coincide with the usual Doppler factor, local frequency, wave invariant and group velocity, in the ray approximation, and their definition is made unique by the requirement that, outside the ray approximation, the following relations remain valid: (i) the generalized wave invariant remains an adiabatic invariant, in the sense that it equals the total (i.e., kinetic plus compression) energy divided by the local frequency; (ii) the latter is related to the wave frequency through a generalized Doppler factor; (iii) the energy velocity is the ratio of the energy flux to the energy density, which like the preceding depends on convection and energy partition factors. These concepts are introduced on the basis of the acoustic energy equation, which can be derived from a variational principle, which also yields wave equations.