Solving the Helmholtz equation for membranes of arbitrary shape

TL;DR

This paper introduces a collocation method using Little Sinc Functions to accurately compute vibration modes of membranes with arbitrary shapes, avoiding integral calculations and enabling high precision through extrapolation and conformal mapping.

Contribution

The paper presents a novel collocation approach for solving the Helmholtz equation on arbitrary membranes, combining Little Sinc Functions with conformal mapping for improved accuracy.

Findings

Accurate mode calculations for arbitrary-shaped membranes.

High precision results via grid size extrapolation.

Rapid convergence when combined with conformal mapping.

Abstract

I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples,part of them taken from the literature, and part of them new. When possible, I have testedthe accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the L-shaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.