Quasi Doppler Effects Associated With Spatio-Temporal Translatory, Moving, and Active Boundaries

Abstract

Moving sources and spatiotemporally dependent boundaries have been introduced in the past, in order to facilitate analyses of the so called "Doppler effect" phenomena. Here a model is introduced for generalized situations involving translatory and moving surfaces on which certain boundary or source conditions are prescribed. The ambiguity arising from analyses of Doppler-like effects in electromagnetics as well as acoustics, in which the (mathematical) translatory surface is not explicitly distinguished from the (physical) moving object is carefully discussed here, and the role of physics, e.g., in the form of Einstein's Special Relativity theory, is considered. The present approach facilitates the general reformulation of the Doppler effect class of problems and suggests meaningful first order v/c (relative velocity) approximations which can then be employed for more complicated problems. Quasi Doppler effects are introduced in order to replace the inherent "inverse problem" nature of the scattering Doppler effect with a "forward problem" formulation which allows for a broader scope of problems and approximations. This facilitates the representation of relativistically exact but complicated solutions in terms of simpler expressions involving first order velocity effects. In turn, this facilitates new approximate solutions for problems not considered previously. By further distinguishing amplitude and phase effects, even simpler expressions, inconsistent in v/c , can be used. This is also helpful in assessing the validity of some heuristic approximations suggested in the past. We start with a general analysis of the Doppler effects initiated by complicated surfaces, providing some general guidelines and insight for our ability to analyze increasingly complicated problems.