DOI: https://doi.org/10.36719/2663-4619/125/190-193
Gulara Sarkarova
Azerbaijan State Oil and Industry University
Master's student
https://orcid.org/0009-0002-5862-1187
gularsrkrova9@gmail.com
Asymptotic Behavior of Solutions of Helmholtz Equations
With Coefficients of Changing ign
Abstract
This paper investigates the asymptotic behavior of solutions to the Helmholtz equation with sign-changing coefficients in a double-medium structure. The study is motivated by mathematical models arising in wave propagation problems in heterogeneous and layered media. Special attention is paid to the nonstandard spectral properties and local resonance phenomena caused by the change of sign in the coefficients. Asymptotic methods with respect to a small parameter are employed, including multiscale analysis and matched asymptotic expansions. The influence of boundary layers in the double medium, localization of wave fields, and the leading terms of the asymptotic expansions of the solutions are analyzed in detail. The obtained results demonstrate that, in contrast to classical Helmholtz problems, sign-changing coefficients lead to qualitatively different asymptotic behavior of solutions. The proposed approach provides a rigorous framework for studying complex wave phenomena and can be applied to problems in acoustics, electromagnetics, and composite material modeling.
Keywords: Helmholtz equation, sign-changing coefficients, double medium, asymptotic analysis, wave propagation, boundary layers