Modelling physical processes based on a three-dimensional wave–diffusion model

Abstract

This paper investigates a three-dimensional wave-diffusion equation as a unified mathematical model for describing physical processes involving both wave propagation and energy dissipation. In contrast to the classical wave equation, the proposed model incorporates a damping term that accounts for diffusion and energy loss in real physical media. An analytical solution is obtained using the method of separation of variables, which reduces the problem to a spectral boundary-value problem for the Laplace operator and a damped second-order ordinary differential equation in time.

The resulting solutions describe exponentially decaying oscillations, where the damping coefficient governs the rate of energy dissipation and the spectral parameters determine the frequency characteristics of the system. The physical interpretation of the model is discussed in the context of acoustic wave attenuation, thermoelastic interactions, electromagnetic signal weakening, and other dissipative phenomena. The proposed approach provides a general analytical framework for modelling coupled wave-diffusion processes and can serve as a basis for further numerical simulations and applied studies in physics and engineering.