Mathematics and Concert Halls

Across the world orchestras entertain us with wonderful performances. But to appreciate beautiful music it is necessary to have a concert hall with superior acoustics. The factors that affect the acoustics of a hall include its dimensions, and the absorptive and reflective properties of the walls and other surfaces. These determine the rapidity with which the sound dies down (technically measured by the reverberation time), which is crucially important, the way in which the sound is dispersed, and so on.

Amongst many other requirements for good acoustics, a fundamental one is that the signals received at the two ears of a listener should be as uncorrelated as possible. This is known as binaural dissimilarity. The need for binaural dissimilarity may not be very obvious but it is one of the reasons why surround sound is acoustically pleasing: it gives the listener the impression of being immersed in the sound. Through extensive experimentation (primarily by M.R. Schroeder of the University of Göttingen), it has been found that binaural dissimilarity is at least as important a factor as reverberation time for good acoustics. So a major aim in the design of concert halls is to maximize binaural dissimilarity.

If sound arrives in the symmetry plane through the head (in the front-back direction) it results in similar sound pressure at the two ears, which is undesirable. Direct reflections from the ceiling of the hall, called specular reflection, result in such sound. On the other hand, sound which arrives from the sides gives a better effect. So for good acoustics direct reflections from the ceiling must be minimized and laterally travelling waves, which arrive after reflection from the side walls, must be maximized. Take a look at the computer demo.

How can this be achieved? We could increase the height of the ceiling, as in older concert halls. But today this would lead to prohibitively high air conditioning costs. Another possibility is to make the ceiling absorptive, but that would result in a loss of sound energy. So these are not good solutions. On the other hand, if we are able to design the ceiling in such a way that a large part of the sound is scattered towards the side walls then our goal is achieved.

This can be done by constructing a sequence of grooves of periodically varying depths, or a reflection phase grating, on the ceiling. But the scattering must not distort the sound, and should scatter a large range of frequencies in the same way. So the grating cannot be any arbitrary sequence of groove depths.

This is where we find an interesting link between number theory and concert hall acoustics. There are number theoretic sequences that can be used to construct reflection gratings with the desired properties.

A suitably designed ceiling can achieve the scattering of sound towards the side walls because sound is a wave phenomenon, and like any other wave, can be diffracted.

When a body vibrates in a medium (say air), it causes a compression of the surrounding air, resulting in a change in pressure, which in turn pushes on the air beyond. The atmosphere exerts a pressure on all objects. The vibration of a body results in slight disturbances in this ambient atmospheric pressure, i.e. a series of compressions and rarefactions. These oscillating variations in pressure propagate in the form of a sound wave. The ear reacts to sound pressure variations and that is why we hear sounds.

The distance between successive compressions (or rarefactions) is the wavelength of the sound. The intensity of sound is defined as the square of the sound pressure.

If the wave were to just carry on beyond the apertures along the straight paths, it would not reach the red regions. But this is not what happens.

Along the path of the wave, each point behaves as a new source that re-emits the wave in all directions. The intensity at any point in the path then depends on the way the numerous waves add up or interfere. Remarkably in the absence of an aperture this addition happens in such a way that the direction of propagation is in fact along a straight line. But if the wave encounters the apertures, at points beyond them only the contributions from the two apertures will be received. The resultant addition causes the wave to bend or diffract. So the wave in fact does reach the red regions. But how exactly do the waves from the two slits interfere?

The two slits behave as independent sources emitting waves in all directions. At some points on the screen, such as P, the two wave trains will arrive in such a way that their crests (or troughs) overlap. This happens when the path difference between the two is an integral multiple of the wavelength (l ). Then they will be in phase and will reinforce each other resulting in relatively high intensity (maxima). Where the crest of one train overlaps with the trough of the other (the path difference is an odd integral multiple of l /2), the wave trains will be out of phase. They will then cancel each other resulting in zero intensity (minima). These maxima and minima form an interference pattern.

The argument for two slits can be extended to any number of slits arranged in any way you like. A diffraction grating is an arrangement of apertures that is meant to produce a particular interference pattern.

Gratings can be of two types. A reflection grating is one where diffraction is observed after reflection from the system of obstacles. On the other hand, in a transmission grating the wave is allowed to pass through a system of apertures.

The system of apertures could also be modified in such a way that they introduce a phase change (each different from the other) in the incident wave. In fact, it is this kind of a phase grating that is used to achieve good scattering from the ceiling of a concert hall.

The change in phase brought about by an obstacle is described technically by the response function. For example in the case of light, a perfectly transparent sheet of glass would have a response function of 1 which means that all the light gets through, and there is no difference in amplitude or phase between the incident and emergent wave. However an opaque body would have a response function of 0.

For a phase grating this response function plays a crucial role in determining the kind of interference pattern obtained. The response function for a grating is defined as the amplitude of the wave as a function of position, as it leaves the grating.

When the source and observer are far from the aperture, certain approximations can be made, resulting in a class of diffraction problems called Fraunhofer diffraction. For Fraunhofer diffraction the interference pattern is approximated by the Fourier transform of the response function.

where n is a positive integer and p is a non-negative prime. For example, with p=17 the following sequence is generated:

where, as before, n is a positive integer and p is a prime, and g is a primitive root of p.

g is a primitive root of p if each of the numbers g,g²,&g^n-1 leaves a different remainder when divided by p. So 3 is a primitive root of 7 but 2 is not.

Suppose a reflection phase grating is constructed on the ceiling, with grooves whose depths are in proportion to either of these sequences. What happens to the incident wave?

Upon reflection from the grating the phase of the incident wave is changed by 4d_np /l , where l is the design wavelength and d_n is the depth of the groove. If d_n is chosen as an appropriate number theoretic function,

then the change in phase is 2p s_n/p. So the amplitude of the incident wave is altered by a factor

This is the response of the grating. As a result of diffraction the incident wave is scattered in different directions. Due to interference between these scattered waves, there will be particular angles at which the intensity would be a maximum. The objective is that these angles should be such that maximum sound is sent to the side walls.

It can easily be shown that the response of both kinds of gratings have a broad flat Fourier spectrum. (see the demo)

We see that the gratings work for the design frequency. But what about frequencies other than the design frequency? In this context a very interesting property of the sequences comes through. A frequency which is an integral multiple of the design frequency, behaves as if the sequence has been multiplied by the integer. Say, a frequency thrice the design frequency is used.

This is exactly the old sequence shifted cyclically to the left by two places. Hence it retains all the properties of the original sequence. This also holds for the Quadratic Residue sequence. In addition, our calculations show that even for non-integral multiples of the design frequency the gratings scatter well. So we see that the gratings actually do work for a large number of frequencies. But the two gratings are different in a number of ways. It is worthwhile to compare the two.

This connection between number theory and acoustics is not just a mathematical fancy. Quadratic Residue Diffusor Panels, used to improve room acoustics, are in fact available in the market.

WHAT WE WANT	WHAT THIS MEANS
The incident wave should be scattered into a number of angles so we have more laterally travelling waves.	The Fourier spectrum of the response function should be broad.
The scattered wave should have roughly equal magnitudes in different directions.	The Fourier spectrum of the response function should be flat.
There should as little specular reflection as possible.	The zeroth component of the spectrum should have a small magnitude
The grating should work similarly over the range of audible frequencies	The response should not change appreciably over the range of audible frequencies.

Galois	Quadratic Residue
Low specular reflection	High specular reflection
Small number of scattered components	Larger number of scattered components
Non-symmetric scatter pattern	Symmetric scatter pattern