Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
As we saw in the last section, scientists and mathematicians can analyze the frequency content of a given signal to discover important information about the origins and nature of the signal. We have, so far, concentrated on the case of one-dimensional waves, but there is no reason that the technique of analyzing frequency spectra should be limited to this domain. What we can do for a string can also be done for a higher-dimensional object, such as a membrane. All sorts of objects can, and do, create sounds; the interesting question to consider is whether, solely on the basis of knowing the frequency content of a sound, you can deduce what object made it.
To be more specific, let's think about drums. There are many factors that affect the sound of a real drum, such as the tautness, or tension, of the drum head and the shape of the resonant cavity, or body, of the drum. To understand the acoustics of a drum completely, we would have to consider a broad array of physical and phenomenological aspects, including the material with which the drum is constructed. Obviously, we will first have to make some simplifying assumptions about the situation if we ever hope to develop a quantitative understanding of how a drum "works."
Because we are focused on mathematics, we will take this idea of simplifying assumptions to the extreme and examine abstract drums. With an abstract drum, we are concerned with what is knowable in an ideal mathematical sense. For our purposes, a drum is basically a two–dimensional, flat shape that vibrates with some combination of frequencies when struck. Our analysis will have nothing to do with materials or size and shape of resonant cavities. We will be concerned solely with the frequency content of the signal produced by the various vibratory modes of our abstract drum.
The question of concern can be phrased in this way: "Can we hear the shape of a drum?" More specifically, if we determine the frequency spectrum of the sound given by a drum after it is struck, can we work backward to figure out the geometric shape that produced that spectrum? For this to be possible, every conceivable shape must have a unique frequency spectrum. If two different shapes shared the same frequency spectrum, then it would be impossible to "hear" the shape of either one—you would never know exactly which shape produced the sound.
This question was first posed by mathematician Mark Kac in a 1966 paper. Mathematicians quickly took up the challenge and soon determined that one could "hear" the area of the shape. The problem of "hearing" the exact shape, however, remained unsolved until 1991 when mathematicians Carolyn Gordon, David Webb, and Scott Wolpert determined that the shape of a drum cannot be categorically determined by its frequency spectrum. They confirmed this by finding two different shapes (drumheads) that have the same frequency spectrum.
Nevertheless, it is possible to distinguish between some shapes by using the frequency spectrum alone. For example, you can "hear" the difference between a rectangular drum and a circular drum. A more meaningful question, then, might be, "what can you tell about a drum from its frequency spectrum?" In a nutshell, some features are evident, others are not. Although the frequency domain representation of a signal may not tell us everything about its source, it can indeed provide us with some information. Using the techniques of Fourier analysis and frequency domain representation, we can find out information about the shapes of drums that we cannot see. As we've seen, these techniques have also helped us determine the chemical composition of stars millions of light years away. Clearly, breaking up a sound or other signal into its frequency components can help uncover fundamental information about origin and structure that is not otherwise evident.