Eigenvalue Problem in Two Dimensions for an Irregular Boundary II: Neumann Condition

TL;DR

This paper introduces a systematic perturbative method for calculating eigenvalues of the 2D Helmholtz equation with Neumann boundary conditions on irregular boundaries, effectively handling degeneracies without additional formalism.

Contribution

It presents a novel perturbation scheme using a real boundary-derived parameter, simplifying eigenvalue calculations for irregular shapes and degeneracies.

Findings

Accurately computed eigenvalues for elliptical and supercircular boundaries.

Developed a semi-empirical formula matching numerical results for supercircular boundaries.

Demonstrated the method's effectiveness across various low-lying modes.

Abstract

We formulate a systematic elegant perturbative scheme for determining the eigenvalues of the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions when the normal derivative of {\psi} vanishes on an irregular closed curve. Unique feature of this method, unlike other perturbation schemes, is that it does not require a separate formalism to treat degeneracies. Degenerate states are handled equally elegantly as the non-degenerate ones. A real parameter, extracted from the parameters defining the irregular boundary, serves as a perturbation parameter in this scheme as opposed to earlier schemes where the perturbation parameter is an artificial one. The efficacy of the proposed scheme is gauged by calculating the eigenvalues for elliptical and supercircular boundaries and comparing with the results obtained numerically. We also present a simple and interesting semi-empirical formula, determining the eigenspectrum of the 2D Helmholtz equation with the Dirichlet or the Neumann condition for a supercircular boundary. A comparison of the eigenspectrum for several low-lying modes obtained by employing the formula with the corresponding numerical estimates shows good agreement for a wide range of the supercircular exponent.