A new method for studying the vibration of non-homogeneous membranes

TL;DR

This paper introduces a collocation method using localized 'little sinc functions' to accurately solve the Helmholtz equation for non-homogeneous membranes with arbitrary shapes.

Contribution

The paper presents a novel collocation approach with little sinc functions for solving membrane vibration problems of arbitrary geometries.

Findings

The method achieves high accuracy in numerical tests.

Implementation is straightforward for general problems.

Results compare favorably with existing literature.

Abstract

We present a method to solve the Helmholtz equation for a non-homogeneous membrane with Dirichlet boundary conditions at the border of arbitrary two-dimensional domains. The method uses a collocation approach based on a set of localized functions, called "little sinc functions", which are used to discretize two-dimensional regions. We have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature. Our results show that the present method is very accurate and that its implementation for general problems is straightforward.