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Title: 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 » of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated.« less

Authors:
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:
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}
}

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Cited by: 8 works
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Works referenced in this record:

Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation
journal, February 2005


Moving boundary value problems for the wave equation
journal, July 2010


Method for Solving Moving Boundary Value Problems for Linear Evolution Equations
journal, May 2000


On invariance properties of the wave equation
journal, February 1987


Effect of Low-Frequency Modulation on the Acoustic Radiation Force in Newtonian Fluids
journal, January 2011


An Asymptotic Solution for the Wave Equation in a Time-Dependent Domain
journal, January 1981


Exact Representations for Acoustical Waves When the Sound Speed Varies in Space and Time
journal, February 1987


On Mapping Linear Partial Differential Equations to Constant Coefficient Equations
journal, December 1983


An Alternative Example of the Method of Multiple Scales
journal, January 2000


Experimental demonstration of Doppler spectral broadening using the PC sound card
journal, February 2007


Linear and nonlinear generalized Fourier transforms
journal, October 2006


Self-adjoint acoustic equations with progressing wave solutions
journal, September 1990


Asymptotic Solution of the Wave Equation with Variable Velocity and Boundary Conditions
journal, July 1972


Mikhel’son effect in a plasma
journal, January 2000


Elastic Wave Propagation in Homogeneous and Inhomogeneous Media
journal, June 1959


Transformation method for electromagnetic wave problems with moving boundary conditions
journal, May 1989


A unified transform method for solving linear and certain nonlinear PDEs
journal, July 1997


Boundary-value problems for linear PDEs with variable coefficients
journal, April 2004


Some remarks on nonlinear poroacoustic phenomena
journal, September 2009


Multiple-timescale asymptotic analysis of transient coating flows
journal, September 2009


The Spaghetti Problem
journal, December 1949


A new twist on the Doppler shift
journal, February 2014


Three‐dimensional acoustic waves in the ear canal and their interaction with the tympanic membrane
journal, March 1988


Exact solutions for wave equations of two‐layered media with smooth transition
journal, January 1988


Asymptotic Behavior for the Vibrating String with a Moving Boundary
journal, March 1993


Numerical solution of the wave equation describing acoustic scattering and propagation through complex dispersive moving media
journal, December 2009


Multiple Scale and Singular Perturbation Methods
book, January 1996


Analytical Method for Solving the One-Dimensional Wave Equation with Moving Boundary
journal, January 1998


A Method for Obtaining Exact Solutions to Partial Differential Equations with Variable Coefficients
journal, June 1988


The Spaghetti Problem
journal, December 1949


Linear and Nonlinear Waves
journal, November 1975


Exact Representations for Acoustical Waves When the Sound Speed Varies in Space and Time
journal, February 1987


A Method for Obtaining Exact Solutions to Partial Differential Equations with Variable Coefficients
journal, June 1988


Multiple Scale and Singular Perturbation Methods
book, January 1996


On the solution of the wave equation with moving boundaries
journal, December 1961


A string problem
journal, June 1963


Moving boundary value problems for the wave equation
journal, July 2010


On invariance properties of the wave equation
journal, February 1987


Exact solutions for wave equations of two‐layered media with smooth transition
journal, January 1988


Self-adjoint acoustic equations with progressing wave solutions
journal, September 1990


A unified transform method for solving linear and certain nonlinear PDEs
journal, July 1997


Linear and nonlinear generalized Fourier transforms
journal, October 2006


Head-on collision of two concentric cylindrical ion acoustic solitary waves
journal, March 1996


Elastic Wave Propagation in Homogeneous and Inhomogeneous Media
journal, June 1959


Three‐dimensional acoustic waves in the ear canal and their interaction with the tympanic membrane
journal, March 1988


Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. I. The Cauchy problem
journal, March 2007


Asymptotic Solution of the Wave Equation with Variable Velocity and Boundary Conditions
journal, July 1972


Effect of Low-Frequency Modulation on the Acoustic Radiation Force in Newtonian Fluids
journal, January 2011


The Spaghetti Problem
journal, December 1949


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