Why is the head of a drum usually shaped like a circle? How would it sound if it were shaped like a square instead? Or a triangle? If you closed your eyes and listened, could you tell the difference? The mathematics used to prove that “one can hear the corners of a drum” are founded on the study of two everyday phenomena: vibrations and heat conduction. These phenomena can be described by two mathematical equations, in the sense that if one can solve these equations, then one can predict the behavior of vibrations and heat conduction.
At the heart of both of these equations is the Laplace operator, ∆, also known as the Laplacian, named for Pierre-Simon, marquis de Laplace (1749-1827). It turns out that if one can solve the Laplace equation: ∆f = λf, then one can solve both the wave and heat equations.
It is not necessary to understand these mathematical symbols and jargon because they are connected to something everyone understands: music. The sound produced by a stringed instrument is made by the vibration of the strings. The note one hears, such as an A, C, or B-flat, depends on how long the string is, and of what material it is composed. This note is also referred to as the fundamental tone or fundamental frequency.
The vibration of the string also produces overtones, known as harmonics, and these play an important role in creating the sound we hear. The collection of values obtained from solving the Laplace equation provide all these different frequencies: the fundamental frequency and all of the harmonics. Altogether these determine the sound of the string.
In the case of a string, the Laplace equation can be solved rather easily, and it turns out that all the harmonics are integer multiples of the fundamental frequency. This mathematical fact is one of the reasons that stringed instruments and pianos are so popular; it causes the sound that we hear from such instruments to have a pleasant and “clean” quality. In fact, every other instrument in a classical orchestra also has this property, with the exception of the percussion instruments.
Drums are fundamentally different. The sound created by beating a drum comes from the vibrations of the drumhead. Mathematically, this means that the Laplace equation is now in two dimensions. Acoustically, you may observe that the sound produced by vibrating drums is “messier” in a certain sense as compared to the sound produced by a vibrating string. The reason is that for drums it is no longer true that the harmonics are integer multiples of the fundamental frequency.
Although we can mathematically prove the preceding fact, we cannot, apart from a few notable exceptions, solve the Laplace equation in two dimensions. Facing this impasse, mathematicians have turned to investigate questions such as: if two drums sound the same, in the sense that their fundamental frequencies as well as all their harmonics are identical, then what geometric features do they have in common? Such features are known as geometric spectral invariants.
Hermann Weyl (1885-1955) discovered the first geometric spectral invariant: if two drums sound the same, then their drumheads have the same area. About a half century later, Åke Pleijel (1913-1989) proved that the perimeters of the drumheads must also be the same length. Shortly thereafter, M. Kac (1914-1984) wrote the now famous paper, “Can one hear the shape of a drum?” He wanted to know whether or not the drumheads must have the same shape? It took about a quarter century to solve the problem, which was achieved by Carol Gordon, David Webb, and Scott Wolpert in 1991. The answer is no.
In contrast to a nice round drumhead, the “identical sounding drums,” in Figure 1 both have corners. A natural question is therefore: can one hear the corners? This means, is it possible for two drums to sound the same, and one of them has a nicely rounded, but not necessarily circular, shape, whereas the other has at least one sharp corner? In other words, can one hear the corners of a drum? We have proven that the answer is yes. The sound produced by a drumhead with at least one sharp corner will always be different from the sound produced by any drumhead without corners. Mathematically, this marks the discovery of a new geometric spectral invariant.
Inquiring minds still have several questions to investigate. For example, if we now assume that both drums have nicely rounded, not necessarily circular shapes, and no sharp corners, is it possible that they can sound identical but be of different shapes? Can one hear the shape of a convex drum? What happens when we consider these types of problems for three-dimensional vibrating solids? We continue to work alongside our fellow mathematicians on problems such as these, and there is plenty of room for further investigation by young researchers.
Image credit: Drum by PublicDomanImages, Public Domain via Pixabay.