On mechanical waves and Doppler shifts from moving boundaries

We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, $X_\mathrm{s}(t) = vt$ is the location of the source with constant velocity $v$. In the present work, however, we consider a source co-located with a moving boundary $x=X_\mathrm{s}(t)$, where $X_\mathrm{s}(t)$ can have an arbitrary functional form. For "slowly moving" boundaries (\textit{i.e.}, ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving-boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extending the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (a) $\dot X_\mathrm{s}$ for non-uniform boundary motion can be inserted in place of the constant velocity $v$ in the classical Doppler formula and (b) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated.