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A proof that multiple waves propagate in ensemble-averaged particulate materials.
Artur L Gower1, I David Abrahams2, William J Parnell3
1Department of Mechanical Engineering, University of Sheffield, Sheffield, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|October 16, 2019
Summary
This study proves that a unique effective wavenumber does not exist for inhomogeneous materials. Instead, an infinite number of effective wavenumbers are required for accurate wave propagation analysis.
Area of Science:
- Physics
- Materials Science
- Acoustics
Background:
- Effective medium theory simplifies complex materials using macroscopic parameters.
- The effective wavenumber is crucial for characterizing wave propagation in inhomogeneous media.
- Existing research focuses on calculating a single effective wavenumber.
Purpose of the Study:
- To prove the non-existence of a unique effective wavenumber.
- To demonstrate the necessity of multiple effective wavenumbers for accurate wave propagation.
- To analyze wave reflection and transmission coefficients in inhomogeneous materials.
Main Methods:
- Application of the Wiener-Hopf technique.
- Ensemble averaging over random inhomogeneities.
- Analysis of scalar (acoustic) waves in a 2D material with cylindrical inclusions.
Main Results:
- An infinite number of complex effective wavenumbers exist.
- A small subset of effective wavenumbers significantly contributes to the wave field.
- Accurate reflection and transmission coefficients require numerous highly attenuating effective waves.
Conclusions:
- The concept of a single effective wavenumber is insufficient.
- Wave propagation in inhomogeneous materials is described by a spectrum of effective wavenumbers.
- The Wiener-Hopf technique offers a robust method for calculating wave phenomena.
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