Wave propagation in elastic medium with heterogeneous quadratic nonlinearity

Guangxin Tang, Laurence J. Jacobs, Jianmin Qu

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

This paper studies the one-dimensional wave propagation in an elastic medium with spatially non-uniform quadratic nonlinearity. Two problems are solved analytically. One is for a time-harmonic wave propagating in a half-space where the displacement is prescribed on the surface of the half-space. It is found that spatial non-uniformity of the material nonlinearity causes backscattering of the second order harmonic, which when combined with the forward propagating waves generates a standing wave in steady-state wave motion. The second problem solved is the reflection from and transmission through a layer of finite thickness embedded in an otherwise linearly elastic medium of infinite extent, where it is assumed that the layer has a spatially non-uniform quadratic nonlinearity. The results show that the transmission coefficient for the second order harmonic is proportional to the spatial average of the nonlinearity across the thickness of the layer, independent of the spatial distribution of the nonlinearity. On the other hand, the coefficient of reflection is proportional to a weighted average of the nonlinearity across the layer thickness. The weight function in this weighted average is related to the propagating phase, thus making the coefficient of reflection dependent on the spatial distribution of the nonlinearity. Finally, the paper concludes with some discussions on how to use the reflected and transmitted second harmonic waves to evaluate the variance and autocorrelation length of nonlinear parameter β when the nonlinearity distribution in the layer is a stochastic process.

Original languageEnglish
Title of host publicationReview of Progress in Quantitative Nondestructive Evaluation
Pages43-50
Number of pages8
DOIs
StatePublished - 2011
Event37th Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE - San Diego, CA, United States
Duration: 18 Jul 201023 Jul 2010

Publication series

NameAIP Conference Proceedings
Volume1335
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Conference

Conference37th Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE
Country/TerritoryUnited States
CitySan Diego, CA
Period18/07/1023/07/10

Keywords

  • Elastic Media
  • Finite Amplitude
  • Heterogeneous Nonlinearity
  • Longitudinal Wave
  • Nonlinear theory

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