Wave propagation in slowly varying one-dimensional random waveguides using a finite element approach

Abstract

This work investigates structural wave propagation in one-dimensional waveguides with randomly varying material and geometric properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering due to any changes in the properties of the medium. This variability plays a significant role in the so-called mid-frequency region, but wave-based methods are typically only applied to homogeneous and uniform waveguides. The WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude of a wave propagating through a non-uniform waveguide, but it is typically restricted to analytical solutions of the equation of motion. In this paper a Wave and Finite Element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution is available. The wavenumber is expressed as a function of the position along the waveguide and a Gauss-Legendre quadrature scheme is used to the numerically integrate the phase. The WFE method is used to evaluate the wavenumbers at each integration point, and these are kept to a minimum to minimise computation cost while being able to capture the non-homogeneity to a given accuracy. The wave amplitude is calculated using conservation of power flow. The numerical example of a straight rod with a single propagating wave mode is considered. Random field properties are expressed in terms of a Karhunen-Loeve expansion. The forced response to a point excitation is calculated and results are compared to a standard Finite Element (FE) approach and to the WKB analytical solution. Results show good agreement and require only a few WFE evaluations, providing a suitable framework to account for spatially correlated randomness in waveguides.

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ORCID for Neil Ferguson: orcid.org/0000-0001-5955-7477

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