Theory of the Doppler effect: Fact, fiction and approximation

The theory of the generalized Doppler effect, concerning the scattering of waves from arbitrary time-varying objects, is still a wide open class of problems. Over the years, two major approaches have been used for the analysis of the Doppler effect: (1) the “Einstein recipe” of coordinate transformations and (2) the direct solution of the time-dependent boundary value problem. In addition, numerous approximate methods have been devised. Some approaches, e.g., the so-called quasi-stationary method, are inherently inconsistent, as explained here. As long as uniform rectilinear motion is involved, we stand on relatively firm ground, and various methods and approximations can be compared and evaluated. For nonuniform motion in the presence of plane waves and plane interfaces (including the logical extension to geometrical optics), some results are available, but the problem already involves heuristic assumptions. Some special problems have been solved for two- and three-dimensional configurations, under very restrictive conditions. The general problem of scattering by a time-varying obstacle is still an open problem. A delineation of some methods for solving such problems numerically is given and discussed.