Energy analysis and discretization of the time-domain equivalent fluid model for wave propagation in rigid porous media

Abstract

The equivalent fluid model (EFM) describes the acoustic properties of rigid porous media by defining the intra-pore fluid phase as a fluid with an effective density and an effective compressibility. Their definitions are based on the dynamic tortuosity α and the normalized dynamic compressibility β. These physical quantities are complex-valued functions depending on the frequency, and can be irrational as in the Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model. Hence, the system of equations derived from the EFM can involve fractional derivatives in the time domain. This paper presents an approach to formulate the EFM equations described by the JCAPL model in the time domain, leading to an augmented system for which a proof of stability is given. From the EFM, a model for numerical simulation is built with α and β approximated using a multipole model. Sufficient stability conditions are then provided for the multipole-based EFM. Lastly, a numerical analysis is carried out in order to illustrate the theoretical results and a simulation of the impedance tube experiment is presented.