Fano resonance scatterings in waveguides with impedance boundary conditions

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.