A class of reduced-order models in the theory of waves and stability

Chapman, CJ; Sorokin, SV

Abstract

This paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental. The approximations have two defining characteristics: (i) they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the (frequency, wavenumber) plane and (ii) the approximate dispersion relations are polynomials that pass exactly through points on this grid. Thus, the method is interpolatory in nature, but the interpolation takes place in (frequency, wavenumber) space, rather than in physical space. Full details are presented for a non-trivial example, that of antisymmetric elastic waves in a layer. The method is related to partial fraction expansions and barycentric representations of functions. An asymptotic analysis is presented, involving Stirling's approximation to the psi function, and a logarithmic correction to the polynomial dispersion relation.

Citation

Chapman, C., & Sorokin, S. (2016). A class of reduced-order models in the theory of waves and stability. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, https://doi.org/10.1098/rspa.2015.0703

• Acceptance Date: Jan 11, 2016
• Publication Date: Feb 10, 2016
• Journal: Proceedings of the Royal Society A - Mathematical Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Publisher: The Royal Society
• DOI: https://doi.org/10.1098/rspa.2015.0703
• Keywords: barycentric representation, elastic wave, Euler truncation, Fourier series, Lamb Wave, Rayleigh wave
• Publisher URL: http://rspa.royalsocietypublishing.org/content/472/2186/20150703.article-info