Fano resonance scatterings in waveguides with impedance boundary conditions

Abstract

The resonance scattering theory is used to study the sound propagation in a waveguide with a portion of its wall lined by a locally reacting material. The objective is to understand the effects of the mode coupling in the lined portion on the transmission. It is shown that a zero in the transmission is present when a real resonance frequency of the open system, i.e., the lined portion of the waveguide that is coupled to the two semi-infinite rigid ducts, is equal to the incident frequency. This transmission zero occurs as a Fano resonance-due to the excitation of a trapped mode in the open system. The trapped mode is formed by the interferences of two neighbored modes with complex resonance frequencies. It is also linked to the avoided crossing of eigenvalues of these two modes that occurs near an exceptional point (a subject that has attracted much attention in recent years in different physical domains). The real and complex resonance frequencies of the open system are determined by an equivalent eigenvalue problem of matrix Heff, which describes the eigenvalue problem defined in the finite lined portion (scattering region). With the aid of the eigenvalues and eigenfunctions of matrix Heff, the usual acoustic resonance scattering formula can be extended to describe the coupling effects between the scattering region and the rigid parts of the waveguide.

Modal analysis of the scattering coefficients of an open cavity in a waveguide

“…In general, k µ takes complex value, and is known as the complex resonance frequency of the scatterer, with Re( k µ ) being the resonance frequency and Im( k µ ) characterizing the decay of the mode. For some special cases, k µ takes a real value, corresponding to localized mode with no radiation loss , which is usually referred to as a trapped mode or bound state in the continuum [10,19,11]. As defined, the frequency-dependent EVP yields the bi-orthogonal eigenmodes for the forced response of the scatterer, whilst the frequency-independent EVP produces the modal characteristics of the free vibration of the system.…”

“…For the sake of simplicity, the rigid-wall boundary condition is assumed for the cavity and ducts. Although details of the coupled mode theory [9,10,11] have already been published, this paper provides a brief description of the derivation using coupled mode theory for the scattering coefficients of the scatterer in terms of frequency-dependent eigenmodes, for the convenience of the reader and to make the paper selfcontained. The first step is to express the sound pressure in terms of a local basis in different regions.…”

“…The open cavity also has another type of eigenmodes, known as quasi-trapped modes, whose eigenvalues have very small imaginary part and hence result in very narrow resonance peaks. Although the involvement of quasi-trapped modes in producing the sharp asymmetric Fano resonance is well known [17,12,11], the theoretical analysis in Sec. 2. will be used to explain how these quasi-trapped modes are involved in producing the Fano resonances.…”

Forced acoustical response of a cavity coupled with a semi-infinite space using coupled mode theory

“…For the present paper, all boundaries of the open cavity are assumed to be rigid, but non-rigid boundaries can be taken into account as well by referring to the treatment in Ref. 11. This has potential applications in a variety of problems, e.g., sound barriers, noise radiation from enclosures, etc.…”

“…Recent progress in modelling the sound scattering coefficients of open cavities due to an incident wave from a connected waveguide have demonstrated that the frequency-dependent eigensolutions of the effective Hamiltonian matrix of the sound field in the open cavity can be used to describe the coupling between the sound fields in the cavity and waveguides. [9][10][11] In this paper, the frequency-dependent eigenmodes is used to describe sound fields in the acoustically coupled open cavity and a semi-infinite space. The eigenvalue problem is developed at a given frequency for generating frequency-dependent eigenmodes and eigenvalues.…”