The use of the transfer matrix method to predict the effective fluid properties of acoustical systems

The transfer matrix method (TMM) is a common method for the modelling of acoustical systems. Traditionally, this method requires each unique layer within a system to be defined by a transfer matrix and then for each matrix to be multiplied together in the sequential order of the system. Whilst the resulting matrix can be used to find the effective material properties of the modelled system, the resulting analytical expressions of these properties are often unwieldy for use. Here, a simplified approach is proposed to obtain simple analytical expressions in the low frequency regime for the effective properties of acoustical systems based on the components of the TMM and inspired by the Champoux and Stinson model. It was shown that the proposed approximation of TMM matches the effective fluid properties of a cylindrical rigid tortuous pore derived with the Champoux and Stinson model. Using this approach, analytical expressions for the effective fluid properties of a waveguide of constant cross section, side-loaded by an arbitrary number of Helmholtz resonators, were derived. These expressions were validated against the traditional transfer matrix method and with numerical computation. The result of this work offers a validated general approach that provides simple analytical low frequency approximations of the acoustical properties of media which consist of complicated networks of pores or side-branches.