The generalized doppler effect and applications

Scattering Doppler effect is generalized to include certain classes of problems involving non-uniformly moving boundaries. The one-dimensional problem is considered for waves on a string and plane electromagnetic waves perpendicular to plane boundaries. The related quantum-mechanical problem is considered, for the simple case of constant velocity, in order to point out the difficulties involved in this class of problems. The solutions are derived without using space-time transformations. This facilitates the analysis of arbitrary modes of motion, e.g. harmonically moving, and uniformly accelerated boundaries. Two methods are given for solving such problems. One method relies on the D'Alembert solution for the one-dimensional wave equation, the other starts with a general spectral representation, and the boundary conditions determine the exact structure of the spectrum.