Localized mode of sound in a waveguide with Helmholtz resonators

Abstract

This paper examines the existence of a localized mode of sound in a planar waveguide between two parallel rigid walls with a pair of identical Helmholtz resonators connected to the upper and lower walls. The localized mode means that linear free oscillations at a particular frequency are trapped only in the vicinity of the resonators and decay exponentially away from them. Assuming the waveguide extends infinitely, a two-dimensional problem is solved fully within lossless theory. It is revealed that, in a waveguide with the resonators connected exactly opposite to each other, the localized mode can exist at a frequency lower than the lowest cutoff frequency of the waveguide and the natural frequency of the resonator. Then the pressure field is antisymmetric spanwise with respect to the centreline of the waveguide, and symmetric axially with respect to the resonators. The localized mode is represented by the superposition of an infinite number of anti-symmetric evanescent modes and no plane-wave mode is involved. The absence of the latter mode makes it possible to localize sound without it being accompanied by radiation damping. This explains why no symmetric localized mode over the width exists. In a waveguide with the resonators offset, no localized mode is shown to exist generally except for a particular spacing. Generalization to cases with multiple pairs of resonators is also considered.