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.