Analytic mode-matching for accurate handling of exceptional points in a lined acoustic waveguide

Abstract

Exceptional points (which occur when two or more modes coalesce) have long been associated with optimal attenuation in lined acoustic waveguides. In recent years, with a view to optimizing sound absorption, some effort has gone into designing liners that generate exceptional points (EPs) at specified frequencies. However, analytic modelling of acoustic scattering in the presence of an EP is not well developed, with most authors relying on standard methods applied close to (but not at) EP conditions. Indeed, exact treatment requires care since the mathematical system under-pinning the scattering process is degenerate. This article presents an analytic mode-matching approach to modelling the scattering of a plane wave travelling towards the junction of a rigid duct with a lined duct at EP conditions. Both EP2 and EP3 (coalescence of two and three modes respectively) are considered. The enhanced mode-matching scheme is shown to be valid and numerically robust, and it is anticipated that it will be straightforward to adapt to a wide range of applications involving complex symmetric operators.

Edge resonance: exact results and fresh insights

“…Another avenue for potential investigation is the double zeros arising in the characteristic equation (2.9). It is anticipated that the analysis presented by Lawrie et al [18] can be extended, enabling a better understanding of this situation.…”

“…Although expression (3.1) is well known, the relationship between E n and the derivative of the characteristic equation (see (2.9)) given by (3.2) is less so. This result, which is crucial to the following analysis, can be established using a fairly standard approach (although the algebra is somewhat tedious) and for the sake of brevity the reader is referred to the appendices in [13,18] for details. In this section, following [13][14][15], a number of identities are proved.…”

“…The integrand is chosen such that its numerator is equal, or closely related, to the numerator of (3.3) when s = s n (in this case, to aid convergence, the numerator is equal after differentiation with respect to w) and the denominator gives rise to E n through (3.2) when the relevant residues are calculated. The method is the same as that used in [13,18]. The appropriate integral is The path of integration is indented above (below) any poles on the negative (positive) real axis.…”

“…It is worth noting that when two modes coalesce the eigensystem is degenerate (since En=0 for at least one value of n) and eigenfunction expansions of the form (2.10) do not fully represent the displacements or stresses. In general, a further mode is required to restore completeness, see [17,18]. It follows that, Detfalse[Nfalse]=0 is not a valid resonance condition for parameter combinations that give rise to a repeated root of (2.9).…”

“…Expressions (3.3) and(3.14), which are used to derive the Parseval identity, do not allow for mode coalescence. However, it is anticipated that f (y) and g(y) are continuous at parameter values that induce mode-coalesence[18] and thus (4.8) should still be valid.…”