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Title: 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 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 s(t), where X s(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (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. Here, 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 extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) $$.\atop{x}_s$$ for non-uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) 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.
ORCiD logo [1] ;  [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Purdue Univ., West Lafayette, IN (United States). School of Mechanical Engineering
  2. Univ. of Louisiana, Lafayette, LA (United States). Dept. of Mathematics
Publication Date:
Report Number(s):
Journal ID: ISSN 0170-4214; TRN: US1703127
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Mathematical Methods in the Applied Sciences
Additional Journal Information:
Journal Volume: 40; Journal Issue: 12; Journal ID: ISSN 0170-4214
Research Org:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE National Nuclear Security Administration (NNSA)
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Doppler effect; accelerating source; multiple-scales expansion; wave equation; moving boundary
OSTI Identifier: