The Huygens' Principle and Doppler Effect in the Presence of Time-Dependent Surfaces

Abstract

Kirchhoff's surface integral representation for the solution of the wave equation, also commonly referred to as the Huygens' principle, is presently considered for the case of time dependent (i.e., moving) surfaces. This is motivated by problems involving moving sources and scatterers prescribed on moving surfaces. A general formulation and discussion of simple examples are given here. The surface integral representation for the solution of the scalar and vector wave equations, for harmonic and arbitrary time dependence, are used as a starting point. The extension of these integrals to the case of time dependent surfaces and its validity are discussed. In general one can always extend the representations to arbitrarily moving surfaces, but the explicit integration with respect to the retarded time relevant to the sources on the moving surface cannot be performed in general. In the special case of motion of surface elements in the direction of the normal, it is shown that the integration is feasible. The retarded time involved in moving surface integrals is shown to be closely related to the well-known Doppler frequency shift. Using Special Relativity, operational forms are found for these integrals, relating the field measured in the laboratory frame of reference to the source fields defined in the co-moving frame of reference of the elements on the moving surface.