Diffusive series representation for the Crandall model of acoustic impedance

Porous media used in soundproofing systems can be modeled as a set of cylindrical per- forations. Starting from the classical expression of the acoustic impedance of cylindrical tubes which is proportional to a ratio of Bessel functions, this paper briefly presents the derivation of an equivalent expression depending only on the positive zeros of J 2 , the Bessel function of the first kind of order 2. In the frequency domain, such a derivation is of great prac- tical interest because it replaces the costly evaluation of transcendental functions by that of rational ones. In the time domain, the impulse response of the cylin- drical tube reduces to a Prony series of which only 10 terms prove sufficient to reliably describe it, for any values of physical parameters within the acous- tic model assumptions. The method applied in this use case from acoustics was primarily employed by Giusti and Mainardi (Meccanica 51(10): 2321–2330, 2016) in their study of pulse propagation within blood vessels and by Colombaro et al. (Meccanica 52(4–5):825–832, 2017) in their work on the Bessel models of linear viscoelasticity.