Wave approach for the resonances of rectangular and triangular membranes

TL;DR

This paper introduces a ray-based method to calculate resonance frequencies of rectangular and triangular membranes, comparing results with exact solutions and using closed path analysis for approximation.

Contribution

It presents a novel ray technique for membrane resonance calculation, applicable to different shapes, and relates resonance frequencies to closed ray paths within the membrane.

Findings

The method accurately predicts resonance frequencies for rectangular and triangular membranes.

Results show good agreement with exact calculations where available.

The approach provides a new perspective on membrane resonances using ray paths.

Abstract

This study develops a ray technique for determining the resonance frequencies of triangular membranes. The technique is demonstrated for homogenous rectangular and triangular membranes with fixed boundaries. Where possible, the results are compared with exact calculations. The membrane resonances are calculated using an equivalent string whose length is proportional to the reciprocal of the length of closed paths starting from an arbitrary point within the membrane. Closed paths are ray paths which arrive back at the starting point going in the same direction.