APPROXIMATION OF IMPEDANCE IN SERIAL ARRAYS OF COUPLED HELMHOLTZ RESONATORS

The transfer matrix method (TMM) is a simple method to model acoustical systems. Using this it is possible to analyse the sound absorption/transmission properties of one and two port systems1 and model sound absorbing acoustic metamaterials consisting of waveguide structures side-loaded by Helmholtz resonators2,3,4 . In this paper a serial array of N identical Helmholtz resonators will be modelled using the TMM. This will be done for up to N=3. A method of simplification will then be applied to the impedance terms obtained using the TMM. This method will utilise the large impedance contrast that exists between the neck and cavity of a Helmholtz resonator. By using a ratio of this impedance contrast it is possible to create a small parameter, ε. By non-dimensionalising the rest of the terms in the impedance expressions, the Taylor series expansion can be taken for ε=0. By using the leading order term of these expansions, you then result in an expression for the impedance of a serial array of Helmholtz resonators composed of a polynomial of the same order as N. These expressions are compared against the TMM to assess validity. A resonant frequency analysis is then undertaken to ascertain whether the solution of the polynomials can be used to determines the resonant frequencies of the systems.