Drums That Sound Alike

The sounds of different types of drums in a marching band are easy to distinguish, even without seeing the instruments.

What makes these sounds so readily identifiable is that each drum vibrates at characteristic frequencies, depending mainly on the size, shape, tension, and composition of its sound-generating drumhead. This spectrum of frequencies—the set of pure tones, or normal modes, produced by a vibrating membrane stretched across a frame—gives a drum's sound its particular color.

Physicists and mathematicians have long recognized that the shape of the boundary enclosing a membrane plays a crucial role in determining the membrane's spectrum of normal-mode vibrations. In 1966, mathematician Mark Kac focused attention on the opposite question.

Kac asked whether knowledge of a drum's normal-mode vibrations is sufficient for unambiguously inferring its geometric shape. His paper, which proved remarkably influential, bore the playful title "Can One Hear the Shape of a Drum?"

Previously, mathematicians had established that both the area of a drum's membrane and the length of its rim leave a distinctive imprint on a drum's spectrum of normal modes. In other words, one can "hear" a drum's area and perimeter.

The question of whether one can infer a drum's geometrical shape from its normal modes remained unresolved until 1991.

That was when mathematicians Carolyn S. Gordon, David L. Webb, and Scott Wolpert came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

The original pair of soundalike, or isospectral, drums discovered by Gordon, Webb, and Wolpert.

In principle, two drums built out of these different shapes would sound exactly alike. Both would generate the same set of normal-mode frequencies.

Since the initial discovery, Gordon, Webb, and others have identified many pairs of soundalike drums. All of the known examples have at least eight corners; typically, each member of a pair consists of a set of identical "building blocks" arranged into different patterns.

However, it's one thing to prove a mathematical theorem and quite another to demonstrate its reality in a physical situation.

When physicist Srinivas Sridhar heard about the Gordon-Webb-Wolpert discovery, he decided to put it to an experimental test—and he had just the right kind of setup to do the necessary experiment.

Sridhar and his coworkers had been investigating aspects of quantum chaos by looking at the patterns created when microwaves bounce around inside thin metal enclosures of various shapes. The same technique could be used to identify normal modes, with microwaves standing in for sound waves and severely squished cavities standing in for membranes.

To test the drum theorem, the researchers constructed two cavities corresponding to one of the pairs of shapes discovered by Gordon and her colleagues. Fabricated from copper and having eight flat sides, each angular enclosure was nearly 8 centimeters long and less than 6 millimeters thick.

Sending in microwaves through a tiny opening and measuring their strength over a range of frequencies at another location enabled the researchers to establish the frequencies of the normal modes of each cavity. They could also map the standing wave patterns inside the cavities.

Remarkably, the frequencies present in both spectra were practically identical. Any discrepancies between the spectra could be attributed to slight imperfections introduced during assembly of the enclosures. See "Experiments on not 'hearing the shape' of drums."

At the time it was done, the experiment provided information that was unavailable mathematically: the shape of standing wave patterns and the actual frequencies making up the spectra in the pair of soundalike drums.

Subsequently, mathematician Toby Driscoll computed the standing wave patterns and frequencies for the same pair of shapes that Sridhar and his colleagues had tested experimentally. His computational results, reported in the paper "Eigenmodes of Isospectral Drums" in SIAM Review, closely matched those obtained by the physicists.

Computed standing wave patterns (first four normal modes) in pairs of polygons having different shapes but identical normal modes. Courtesy of Toby Driscoll.

Driscoll also applied his computational technique to other pairs of isospectral drums. Meanwhile, mathematicians continue to search for additional soundalike doubles. Are there soundalike triples? No one knows.

Originally posted April 14, 1997

See also "Fractal Drum."