On linear waves with memory in a Bessel-like medium

TL;DR

This paper addresses wave propagation in a Bessel-like medium governed by a memory-bearing wave equation with memory kernel in the Laplace domain given by $\widetilde{\Phi}(s) = \frac{2}{\sqrt{s\tau}} \frac{I_1(\sqrt{s\tau})}{I_0(\sqrt{s\tau})}$, focusing on the $\nu=0$ case. The authors perform a spectral dispersion analysis, deriving the SATP dispersion relation and expressing the real and imaginary parts of $k^2(\omega)$ in terms of Kelvin functions, from which they obtain the phase velocity $v_p(\omega)$ and the attenuation $\delta_{att}(\omega)$, with $v_p(\omega) \to c$ as $\omega\tau \to \infty$ and $v_g(\omega)=[d\kappa(\omega)/d\omega]^{-1}$. They show numerically that $v_p(\omega) < v_g(\omega) \lesssim c$ for $\omega \ge 0$, indicating anomalous dispersion, and they compute the step response using the Talbot method to invert the Laplace transform of $\widetilde{Y}(s,x)=\frac{1}{s}e^{-\mu(s)x}$, thereby providing a full transient analysis beyond the wave-front approximation. The work extends previous semi-analytical approaches and offers a practical, numerically robust framework for studying memory-influenced wave propagation in Bessel-like media with potential applications in viscoelastic and hemodynamic contexts.

Abstract

We discuss the propagation of harmonic and transient waves for systems governed by a wave equation with memory whose integral kernel involves ratios of modified Bessel functions of the first kind in the Laplace domain. In particular, the investigation of transient waves is carried out by means of a fully numerical approach based on the Talbot method for the numerical inversion of Laplace transforms.

Figures (3)

• Figure 1: Non-dimensional dispersion law and attenuation factor as a function of $\omega \tau$.
• Figure 2: Normalized comparison of the phase (continuous line) and group (dashed line) velocities as a function of $\omega \tau$.
• Figure 3: Solution of the system \ref{['eq:ibvp']} at different locations $x>0$ (normalized with respect to the characteristic length $c \tau$) as a function of $(ct - x)/c \tau$.