On one-dimensional Helmholtz equation

Abstract

The study of time-periodic solutions of the multidimensional wave equation on the entire 3D space is an important field of research in applied mathematics. It is known that this study leads to the Sommerfeld radiation condition at infinity. The radiation condition states that for a solution to a one-dimensional wave equation, such as the Helmholtz equation or the wave equation, to represent an outgoing wave at infinity. The Helmholtz equation in 1D, which models the propagation of electromagnetic waves in systems effectively reduced to one dimension, is equivalent to the time-independent Schrodinger equation. The one-dimensional Helmholtz potential is widely used in various areas of physics and engineering, such as electromagnetics, acoustics, and quantum mechanics. The Sommerfeld problem in the one-dimensional case requires special investigation, and the radiation conditions in the one-dimensional case differ from those in the multidimensional case. These differences are related to the peculiarities of the fundamental solutions. In this paper, we constructed the fundamental solution of the one-dimensional Helmholtz equation. Then, we found the boundary conditions for the one-dimensional Helmholtz potential. Finally, the equivalent conditions with Sommerfeld radiation conditions were found for the one-dimensional Helmholtz equation.