Analysis of the effect of a rectangular cavity resonator on acoustic wave transmission in a waveguide

The transmission of acoustic waves along a two-dimensional waveguide which is coupled through an opening in its wall to a rectangular cavity resonator is considered. The resonator acts as a classical band-stop filter, significantly reducing acoustic transmission across a range of frequencies. Assuming wave frequencies below the first waveguide cut-off, the solution for the reflected and transmitted wave amplitudes is formulated exactly within the framework of inviscid linear acoustics. The main aim of the paper is to develop an approximation in closed form for reflected and transmitted amplitudes when the gap in the thin wall separating the waveguide and the cavity resonator is assumed to be small. This approximation is shown to accurately capture the effect of all cavities resonances, not just the fundamental Helmholtz resonance. It is envisaged this formula (and more generally the mathematical approach adopted) could be used in the development of acoustic metamaterial devices containing resonator arrays.