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, Xs(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=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 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 amplitudemore »
- Authors:
-
- 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:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1406197
- Report Number(s):
- LA-UR-15-25121
Journal ID: ISSN 0170-4214; TRN: US1703127
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Journal Article: Accepted Manuscript
- Journal Name:
- Mathematical Methods in the Applied Sciences
- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 12; Journal ID: ISSN 0170-4214
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Doppler effect; accelerating source; multiple-scales expansion; wave equation; moving boundary
Citation Formats
Christov, Ivan C., and Christov, Christo I. On mechanical waves and Doppler shifts from moving boundaries. United States: N. p., 2017.
Web. doi:10.1002/mma.4318.
Christov, Ivan C., & Christov, Christo I. On mechanical waves and Doppler shifts from moving boundaries. United States. https://doi.org/10.1002/mma.4318
Christov, Ivan C., and Christov, Christo I. 2017.
"On mechanical waves and Doppler shifts from moving boundaries". United States. https://doi.org/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 co-located 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 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.},
doi = {10.1002/mma.4318},
url = {https://www.osti.gov/biblio/1406197},
journal = {Mathematical Methods in the Applied Sciences},
issn = {0170-4214},
number = 12,
volume = 40,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}
}
Web of Science
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