Monopole-dipole type resonator in a narrow pipe

Abstract

A Helmholtz resonator without dissipation loss is an efficient reflector for sound propagating in an infinite narrow pipe [1–3]. At the resonance frequency, the incident sound wave is completely reflected from the resonator, so that no travelling wave is observed behind the resonator. The Helmholtz resonator is a resonator of the monopole type. A sound wave propagating in a narrow pipe can also be reflected by a dipole-type resonator [4]. The simplest version of the latter is a rigid sphere fastened to the pipe wall via a small bar. The radius of the sphere and the length of the bar are small compared to the sound wavelength. Under the effect of the incident wave, the rigid sphere fixed on the bar vibrates and generates a scattered field of dipole type. At the resonance frequency, the scattered field completely suppresses the incident wave behind the resonator. The friction in the resonator reduces the efficiency of its operation as a wave reflector. Resonators with friction absorb sound. Previous studies showed that a single Helmholtz resonator with optimal friction absorbs no more than half the energy of the incident wave. A complete absorption of sound at the resonance frequency can be achieved using a combination of a lossless resonator and a resonator with a certain loss (the friction resistance is equal to the radiation resistance) and with the distance between the two resonators being equal to an odd number of quarter-wavelengths [5].