Eigenmodes of fractal drums: A numerical student experiment

• DOI: 10.48550/arxiv.2309.13613
• Publication Date: 2024-02-01
• Authors: Simonsen, Veronic...; Hale, Nathan; Simonsen, Ingve
• Keywords: Physics Education (physics.ed-ph); Physics - Physics Education; FOS: Physical sciences; 0101 mathematics; 01 natural sciences

“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.