Diffusive series representation for the Crandall model of acoustic impedance

Abstract

Porous media used in soundproofing systems can be modeled as a set of cylindrical perforations. 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 practical interest because it replaces the costly evaluation of transcendental functions by that of rational ones. In the time domain, the impulse response of the cylindrical 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 acoustic 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.

Appendix: Poles and zeros in the non-dimensional model

In this Appendix, we are interested in the poles-zeros location in the complex plane of the non-dimensional model $\widehat{{\mathcal {Z}}}_\text {non-dim}$.

Figure 6 shows a contour plot of the principal value of the argument of $\widehat{{\mathcal {Z}}}_\text {non-dim}(s)$ within a given domain of the complex plane. The set of poles is ${\mathcal {P}}_\text {non-dim} := \{(-j^2_{2,k})_{k\in {\mathbb {N}}}\}\subset {\mathbb {R}}_-$ while the set of zeros is $\{(-j^2_{0,k})_{k\in {\mathbb {N}}}\}\subset {\mathbb {R}}_-$.

Fig. 6

Plot of the principal value of the argument for $\widehat{{\mathcal {Z}}}_\text {non-dim}(s)$, $s \in \Omega := [-80, 5] \times [-10, 10]$. Poles are highlighted by crosses whereas zeros by circles

This plot suggests that the poles and zeros of $\widehat{{\mathcal {Z}}}_\text {non-dim}$ are intertwined. Indeed, using the following relation

$$\begin{aligned} I_0(\sqrt{s}) - I_2(\sqrt{s})&= \frac{2}{\sqrt{s}}I_1(\sqrt{s}) \qquad (\sqrt{s}\in {\mathbb {C}}^*) \end{aligned}$$

(23)

$$\begin{aligned} I_0'(\sqrt{s})&= I_1(\sqrt{s}), \end{aligned}$$

(24)

(see §9.6.26 and §9.6.27 in Abramowitz and Stegun [1]), it is easy to notice the following: between two given consecutive zeros of $I_0(\sqrt{s})$, a sign change of $I_1(\sqrt{s})$ takes place thanks to Eq. (24) and to the continuity of Bessel functions; as $I_2(\sqrt{s})$ and $I_1(\sqrt{s})$ assume opposite signs at a root of $I_0(\sqrt{s})$ because of Eq. (23), one concludes that $I_2(\sqrt{s})$ undertakes a sign change in between the two consecutive zeros of $I_0(\sqrt{s})$ and hence a root of $I_2(\sqrt{s})$ (i.e., a pole of $\widehat{{\mathcal {Z}}}_\text {non-dim}$) must lie in between.

Poles-zeros interlacing seems to be a common feature for diffusive kernels with positive weights. This can be seen on the most simple rational ones: for instance, for some $\xi _1 > 0$, $\xi _2 > 0$, $0\le \mu \le 1$, the following convex combination of low-pass filters

$$\begin{aligned} {\widehat{h}}:{\mathbb {C}}\backslash \{-\xi _1,-\xi _2\}&\rightarrow {\mathbb {C}}\nonumber \\ s&\mapsto \frac{\mu }{s+\xi _1}+\frac{1-\mu }{s+\xi _2}\equiv \frac{s+\zeta }{(s+\xi _1)(s+\xi _2)}, \end{aligned}$$

(25)

where $\zeta := (1-\mu )\xi _1 + \mu \,\xi _2$, presents poles at $s=\{-\xi _1,-\xi _2\}$ and a zero in between at $s=-\zeta$.

An example of similar behavior among irrational functions is given by $H_{\varepsilon }(s):=\tanh (\sqrt{s}/\varepsilon )/\sqrt{s}$ for some $\varepsilon >0$, discussed in Mignot et al. [13]. Moreover, this paper shows a very interesting fact: the function’s intertwined poles and zeros in ${\mathbb {R}}_{-}$ become a branch cut in the limiting process leading $\varepsilon$ towards 0, the limit transfer function being $H_0(s):=\frac{1}{\sqrt{s}}$.