Using symmetry to generate solutions to the Helmholtz equation inside an equilateral triangle
arXiv:1307.1023
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.