A wave equation perturbed by viscous terms: fast and slow times diffusion effects in a Neumann problem

TL;DR

This paper investigates how viscous perturbations affect wave propagation in a Neumann problem, revealing that waves remain undisturbed in slow time but are dominated by diffusion in fast time.

Contribution

It provides a detailed analysis of wave and diffusion interactions in a perturbed wave equation with small viscous terms, highlighting different behaviors over slow and fast times.

Findings

Waves propagate almost undisturbed in slow time regimes.

Diffusion effects dominate in fast time regimes.

The study clarifies the transition between wave and diffusion dominance.

Abstract

A Neumann problem for a wave equation perturbed by viscous terms with small parameters is considered. The interaction of waves with the diffusion effects caused by a higher-order derivative with small coefficient {\epsilon}, is investigated. Results obtained prove that for slow time {\epsilon}t < 1 waves are propagated almost undisturbed, while for fast time t > 1 {\epsilon} diffusion effects prevail.