# Eliminating the fictitious frequency problem in BEM solutions of the external Helmholtz equation

## TL;DR

This paper addresses the fictitious frequency problem in boundary element method solutions of the exterior Helmholtz equation, proposing a modified Green's function and a desingularized BEM to detect and avoid these issues.

## Contribution

It introduces a new approach using a modified Green's function and a fully desingularized BEM to eliminate fictitious frequencies in Helmholtz problem solutions.

## Key findings

- Modified Green's function reduces fictitious frequencies.
- Desingularized BEM improves numerical accuracy.
- Method effectively detects and avoids fictitious solutions.

## Abstract

The problem of the fictitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Green's function is replaced by a modified Green's function, it is shown that these fictitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these fictitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.