Trapped modes and Fano resonances in two-dimensional acoustical duct–cavity systems

Abstract Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length, we use the finite-element method to numerically compute the acoustic resonances as well as the sound transmission and reflection for an incoming fundamental duct mode. To satisfy the requirement of outgoing waves in the far field, we use two different forms of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. In general, the resonances are damped due to radiation losses, but there also exist various types of localized trapped modes with nominally zero radiation loss. The most common type of trapped mode is antisymmetric about the duct axis and becomes quasi-trapped with very low damping if the symmetry about the duct axis is broken. In this case a Fano resonance results, with resonance and antiresonance features and drastic changes in the sound transmission and reflection coefficients. Two other types of trapped modes, termed embedded trapped modes, result from the interaction of neighbouring modes or Fabry–Pérot interference in multi-cavity systems. These embedded trapped modes occur only for very particular geometry parameters and frequencies and become highly localized quasi-trapped modes as soon as the geometry is perturbed. We show that all three types of trapped modes are possible in duct–cavity systems and that embedded trapped modes continue to exist when a cavity is moved off centre. If several cavities interact, the single-cavity trapped mode splits into several trapped supermodes, which might be useful for the design of low-frequency acoustic filters.