Generalized doppler effect in time-varying media

Doppler effects for time-varying inhomogeneous media are considered. The ray path is represented in terms of a two-parameter function, this takes into account its time evolution. Time dilation along the path is discussed. This provides, in a systematic way, an alternative derivation to a heuristic expression previously given by Gill. The phase function is integrated along the ray path and yields the various Doppler effects in a straightforward way. It is shown that the generalized Doppler effect is produced by the relative motion of source and receiver with respect to the medium, and the cumulative rate of change of frequency and propagation vector along the path.