On the Doppler Effect in a Medium

Abstract

The problem of calculating the frequency of the wave scattered by a body moving in a medium is formulated from field-theoretic considerations. The Doppler equation for a homogeneous dispersive medium is obtained on the basis of the fact that the frequency and the wave vector of a plane wave form a 4-vector. It is found that the solutions of the Doppler equation can be classified into two kinds. In one kind, the solutions are close to the frequency of the incident wave. In the other kind they appear near the poles of the refractive index of the medium on the ω-axis. In the case of an isotropic plasma, the monochromaticity of the incident wave is shown to be preserved after the wave is scattered by a moving body. However, in the case of a magneto-active plasma, the scattered wave contains more than one frequency for a monochromatic incident wave. The physical interpretations of these frequencies are given. In an inhomogeneous medium the Doppler equation has to be derived from a different starting point. The crucial point of the derivation is to perform spectral decompositions of the transformed fields and then to apply, under the assumption of gradual inhomogeneity, the method of stationary phase to determine the critical points. It is shown how the phase functions of the fields can be obtained by transforming Maxwell's equations into equations of Riccati-type. Approximate solutions of the Doppler equation are obtained for isotropic as well as for gyroelectric stratified media.