On mechanical waves and Doppler shifts from moving boundaries

Abstract

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.Comment: 14 pages, 4 figures, Wiley journal style; accepted for publication in Mathematical Methods in the Applied Sciences; v2 corrects typographical error

Acoustic shock and acceleration waves in selected inhomogeneous fluids

“…Notice also that both Eqs. (10) and (11) are second-order wave equations with variable coefficients. A survey of such PDEs is given in [10], wherein it is shown that they also arise in propagation problems involving homogeneous media with moving boundaries.…”

“…(10) and (11) are second-order wave equations with variable coefficients. A survey of such PDEs is given in [10], wherein it is shown that they also arise in propagation problems involving homogeneous media with moving boundaries. In this section we shall, for the two most common cases of ̺ a (z) (relating to the atmosphere), investigate the following hybrid 4 initial-boundary value problem (hIBVP):…”

A frequency domain method for scattering problems with moving boundaries

“…More mathematical works related essentially to one-dimensional moving boundary problems have also been proposed e.g. by Fokas and his co-authors [28] to recast the problem as a Volterra integral equation in a fixed domain, or by Christov and Christov [20] for an asymptotic multiscale analysis of the Doppler effect in a half-space. To the best of the authors' knowledge, however, the numerical solution of the micro-Doppler PDE modeling problem has not been addressed yet.…”

Delamination propagation under high loading rate

“…The first term is due to quasi-static motion and, therefore, does not need to be adjusted for a propagating delamination. The second term, however, is related to vibration and so does need to be adjusted to account for dispersion and the Doppler effect [24].…”

“…These frequencies need to be modified due to the Doppler effect [24]. The frequencies of flexural waves observed at the crack tip decrease with increasing delamination propagation speeds.…”