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 colocated 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 multiplescale asymptotic analysis of the moving boundary problem for the linear wave equation. Here, we obtain a closedform leadingorder (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 nonuniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the nonuniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated.
 Authors:

^{[1]}
;
^{[2]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Purdue Univ., West Lafayette, IN (United States). School of Mechanical Engineering
 Univ. of Louisiana, Lafayette, LA (United States). Dept. of Mathematics
 Publication Date:
 Report Number(s):
 LAUR1525121
Journal ID: ISSN 01704214; TRN: US1703127
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Mathematical Methods in the Applied Sciences
 Additional Journal Information:
 Journal Volume: 40; Journal Issue: 12; Journal ID: ISSN 01704214
 Publisher:
 Wiley
 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
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Doppler effect; accelerating source; multiplescales expansion; wave equation; moving boundary
 OSTI Identifier:
 1406197
Christov, Ivan C., and Christov, Christo I.. On mechanical waves and Doppler shifts from moving boundaries. United States: N. p.,
Web. doi:10.1002/mma.4318.
Christov, Ivan C., & Christov, Christo I.. On mechanical waves and Doppler shifts from moving boundaries. United States. doi:10.1002/mma.4318.
Christov, Ivan C., and Christov, Christo I.. 2017.
"On mechanical waves and Doppler shifts from moving boundaries". United States.
doi:10.1002/mma.4318. https://www.osti.gov/servlets/purl/1406197.
@article{osti_1406197,
title = {On mechanical waves and Doppler shifts from moving boundaries},
author = {Christov, Ivan C. and Christov, Christo I.},
abstractNote = {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, Xs(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source colocated with a moving boundary x=Xs(t), where Xs(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 multiplescale asymptotic analysis of the moving boundary problem for the linear wave equation. Here, we obtain a closedform leadingorder (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 nonuniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the nonuniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated.},
doi = {10.1002/mma.4318},
journal = {Mathematical Methods in the Applied Sciences},
number = 12,
volume = 40,
place = {United States},
year = {2017},
month = {2}
}