27 Jan 2021

Can you hear the shape of a drum?

Mark Kac popularized the question in a famous paper of the same name. Given all of the pure frequencies, or the spectrum of the drum, can one determine its shape? Since the problem is usually the other way around– given the drum, what will its frequencies (i.e. its spectrum) be– this is known as the inverse spectral problem.

Kac’s paper was part of a great deal of twentieth century work in spectral geometry. We shall go back to the beginning of the century, to the work of Hermann Weyl. In 1910, Lorentz posed a question during some invited lectures at Göttingen to an audience that included both David Hilbert and his student Hermann Weyl. Namely, Lorentz posed the conjecture that the number of high overtones, or large frequencies, between a frequency $\nu$ and a small increase $\nu+d\nu$ should be proportional to the volume of the region. If we let our region be $\Omega$ and denote by $N(\lambda)$ the number of frequencies up to $\lambda$, the conjecture can be stated in the form:

\[N(\lambda) \sim \cfrac{|\Omega|}{2\pi}\lambda\]
Weyl would go on to prove the theorem two years later, using Hilbert's newly developed theory of integral equations. In essence, the theorem states that you can hear the size of the drum, i.e. $|\Omega|$.

In general though, you can’t fully hear the shape of a drum. As it turns out, there are drums with different shapes but the same spectra. The following distinct shapes both have the same frequencies:

Image by Jitse Niesen, from Wikimedia Commons; the example was discovered by Gordon, Webb, and Wolpert.

Further Exposition: