Eigenmodes of fractal drums: A numerical student experiment

TL;DR

This paper presents a numerical experiment for advanced students to explore the eigenmodes of a fractal drum, specifically a square Koch fractal, to understand its vibrational properties and relation to shape.

Contribution

It introduces an educational exercise for calculating and visualizing eigenmodes of a fractal drum, linking vibrational analysis with fractal geometry and spectral theory.

Findings

Calculated lowest eigenmodes of the fractal drum

Visualized eigenmodes and analyzed their symmetry

Compared density of states with Weyl-Berry conjecture

Abstract

``Can one hear the shape of a drum?'' was a question posed (and made famous) by mathematician Mark Kac in the mid-1960s. It addresses whether a deeper connection exists between the resonance modes (eigenmodes) of a drum and its shape. Here we propose a numerical experiment, suitable for advanced undergraduate physics students, on the calculation of the eigenmodes of a square Koch fractal drum, for which experimental results do exist. This exercise is designed to develop the students' understanding of the vibrations of fractal drums, their eigenmodes, and potentially their integrated density of states. The students calculate the lowest order eigenmodes of the fractal drum, visualize these modes, and study their symmetry properties. As an extension, the students may investigate the integrated density of states of the fractal drum and compare their findings to the Weyl-Berry conjecture.