Using symmetry to generate solutions to the Helmholtz equation inside an equilateral triangle

TL;DR

This paper classifies solutions to the Helmholtz equation in an equilateral triangle based on symmetry, introduces methods to generate new solutions from existing ones, and links solutions across different triangular geometries.

Contribution

It provides a symmetry-based classification of solutions, a novel method for generating solutions with different energies, and establishes a correspondence between solutions in equilateral and (30,60,90) triangles.

Findings

Solutions form four symmetry classes.

New solutions can be generated using symmetry operators.

Established a correspondence between solutions in different triangular domains.

Abstract

We prove that every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes. We then show how solutions with different symmetries, or different energies, can be generated from any given solution using symmetry operators or a differential operator derived from symmetry considerations. Our method also provides a novel way of generating the ground state solution (that is, the solution with the lowest energy). Finally, we establish a correspondence between solutions in the equilateral and (30,60, 90) triangles.