The solution of the scalar wave equation in the exterior of a sphere

TL;DR

This paper presents explicit solutions for the scalar wave equation outside a sphere, enabling stable, high-order, grid-free numerical evaluation with no dispersion errors, and corrects previous analytical inaccuracies.

Contribution

It introduces new explicit representations for the wave equation solution in exterior spherical domains, improving numerical stability and accuracy, and corrects errors in existing literature.

Findings

Stable, high-order numerical scheme demonstrated

Solution evaluated at arbitrary points without spatial grids

Corrections made to previous asymptotic analyses

Abstract

We derive new, explicit representations for the solution to the scalar wave equation in the exterior of a sphere, subject to either Dirichlet or Robin boundary conditions. Our formula leads to a stable and high-order numerical scheme that permits the evaluation of the solution at an arbitrary target, without the use of a spatial grid and without numerical dispersion error. In the process, we correct some errors in the analytic literature concerning the asymptotic behavior of the logarithmic derivative of the spherical modified Hankel function. We illustrate the performance of the method with several numerical examples.