DIRECT AND INVERSE PROBLEMS FOR THE THREE-DIMENSIONAL WAVE EQUATION: THEORY, MODELING, AND APPLICATIONS

Abstract

The three-dimensional wave equation plays a central role in describing the propagation of mechanical, acoustic, and electromagnetic waves in complex media. This article presents a comprehensive scientific study of both direct and inverse problems associated with the three-dimensional wave equation. The direct problem is formulated as a well-posed initial-boundary value problem whose solution describes wave evolution under known physical conditions. In contrast, inverse problems aim to reconstruct unknown sources, coefficients, or geometric features of a medium based on observed wave data and are typically ill-posed. The paper discusses theoretical foundations, mathematical properties, and computational approaches, with particular emphasis on applied modeling. Selected real-world applications are analyzed, including subsurface exploration, acoustic diagnostics, and infrastructure monitoring. Numerical modeling results and structured data tables are presented to illustrate wave behavior and reconstruction accuracy in heterogeneous environments.