  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.
