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