Theory of the Doppler effect: Fact, fiction and approximation

Abstract

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 quasistationary 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

Non-Relativistic Scattering by Time-Varying Bodies and Media

“…To the first order in v/c we thus have (13) where in (13) we also include the quasi-relativistic assumption that the formulas hold for spatiotemporally varying velocity. In the same sense as (9), (10), also (13) is invertible. It has been shown [12][13][14], that first order in v/c relativistic boundary conditions as in (13), and nonrelativistic considerations based on the Lorentz force formulas lead to the same boundary conditions.…”

“…Anyway, in general the position of a point along a trajectory should be given as an integral of the incremental distances covered by this point in time, which is not satisfied by simplistically using v = v(R) in (1), (2). This point has been mentioned before [9]. Consequently, a new modified first order quasi-Lorentz transformation suggests itself…”

“…The effect is usually associated with the celebrated Fizeau experiment (e.g., see [7]). Accordingly the phase shifts measured in a moving medium are different from those in the same medium at rest, or in free space, and are subject to the "drag effect" according to (9) and represented in the term of A (1) below.…”

“…This will be used for similar expressions below where the expressions refer to the vicinity of the boundary. The new frequency ω exT in (16) is recognized as the first order in v/c relativistic Doppler frequency effect, as given in the second line (9).…”

“…To compute the phase shift between z T = 0 and other points, specifically z T = Z, we need to consider the effect of the moving medium, i.e., we need to include the Fresnel drag effect [5,7,15] embodied by the first line (9). The effect is usually associated with the celebrated Fizeau experiment (e.g., see [7]).…”