Approximations of the Helmholtz equation with variable wave number in one dimension

This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two-dimensional model, we formulate a simplified one-dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features.