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