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, then at Rockefeller University in New York City, 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 and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park 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 and Webb.

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 S. Sridhar of Northeastern University in Boston 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 [link to quantum chaos BOB] 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.

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.

Since then, mathematician Toby Driscoll, now at the University of Colorado, has computed the standing wave patterns and frequencies for the same pair of shapes that Sridhar and his colleagues tested experimentally. His computational results, which appear in the current issue of SIAM Review, closely match those obtained by the physicists. Indeed, whereas the experimentally obtained frequencies were accurate to about 1 percent, the computed results were accurate to about 12 digits.

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

Driscoll has 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.

References:

Chapman, S.J. 1995. Drums that sound the same. American Mathematical Monthly (February):124-138.

Cipra, Barry. 1993. You can't always hear the shape of a drum. In What's Happening in the Mathematical Sciences. Providence, R.I.: American Mathematical Society.

______. 1992. You can't hear the shape of a drum. Science 255(March 27):1642-1643.

Driscoll, Tobin A. 1997. Eigenmodes of isospectral drums. SIAM Review 39(March):1-17.

Gordon, Carolyn, and David Webb. 1996. You can't hear the shape of a drum. American Scientist 84(January-February):46-55.

Gordon, Carolyn, David L. Webb, and Scott Wolpert. 1992. One cannot hear the shape of a drum. Bulletin of the American Mathematical Society 27(July):134-138.

Kac, Mark. 1966. Can one hear the shape of a drum? American Mathematical Monthly 73:1-23.

Peterson, Ivars. 1994. Beating a fractal drum. Science News 146(Sept. 17):184-185.

Sridhar, S., and A. Kudrolli. 1994. Experiments on not "hearing the shape" of drums. Physical Review Letters 72(April 4):2175-2178.

Stewart, Ian. 1992. Beating out the shape of a drum. New Scientist (June 13):26-30.

Weidenmuller, Hans. 1994. Why different drums can sound the same. Physics World (August):22-23.

Diagrams of isospectral drums created using Mathematica 3.0 ( http://www.wolfram.com/.).

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.