Our hero today is Joseph Fourier, who developed what is called Fourier analysis at the beginning of the nineteenth century. He led a life rich in event, conditioned by the turmoil of the French Revolution. Among other things, he was with Napoleon in Egypt, and was the first to write a systematic scientific report on Egyptian history and culture.

Today, Fourier analysis is a basic tool of all mathematicians and engineers. The idea behind it is the simple representation of oscillatory phenomena. We will focus here on musical tones, that is, on audible vibrations. The "atoms" of a tone are sine waves of various frequencies (see Figure ). If you wish, you can hear such a tone right now. Just give a little whistle, and you are hearing what is almost a pure sine wave.

Figure 1: A sine wave.

 

The theory now tells us the various intensities of different sine waves that must be added together to obtain a prescribed waveform. For a musical tone, one begins with the sine wave of the fundamental frequency, adds a bit of the sine wave for the doubled frequency, perhaps a bit of the threefold frequency, and so on.

And this can be verified with our sense of hearing. A waveform that consists of a sine frequency and a portion of the threefold frequency is a good approximation to the so-called square wave. In order to hear the difference between a sine wave and the square wave, the threefold frequency needs to be within the audible range, which for most readers will be at about 15 kilohertz. Therefore, the difference between the two types of waveforms should be noticeable up to a fundamental frequency of five kilohertz.

In order to verify this, one needs ideally a frequency generator (perhaps you have a friend who is an engineer). Or do you have a synthesizer or some other electronic musical instrument? Then simply choose the waveforms "sine" and "square," and the experiment can begin.

Those who must be satisfied with a qualitative verification of Fourier's theory might wish to take note of the voices the next time they are at a party. It is easier to distinguish the deep voices of men from one another than the higher female voices. That is because men's voices have a large number of overtones in the audible range, which gives the ear many chances to make a differentiation.

A Black Box

There are other mathematical results that you can verify with your ears at least qualitatively. Imagine a black box in which one can input signals, which are then somehow processed in the box's internal mechanism and then output. Electronics hobbyists can imagine some wildly complex circuit into which an electric signal is introduced at some point and measured at some other point (see Figure 2).

Figure 2: The input and output of a "black box."

 

This black box should have the following properties:

• It should be "linear." That is, if the strength of the input signal is doubled, the output should be doubled as well, and if one inputs a signal that is the resultant of two partial signals, then the output is the same as what would result from the combination of the outputs of the two partial signals.
• It should be "time invariant." That is, if a wave in input and the output is logged, the output for a given input should be the same today as it was yesterday.

For the electronics hobbyist this means that transistors may not be used (they are not linear), and no settings can be changed during the experiment. One should limit oneself to resistors, inductors, and capacitors, and the currents and voltages that arise should not be too great.

Although such black boxes describe a rather general situation, they all have one property in common: Sine waves, the building blocks of Fourier analysis, pass through such a black box essentially unchanged. They can be weakened or pushed out of phase, but that is all that can be done with them.

The audible consequence is this: a filter for acoustic signals (high-pass, low-pass, band-pass, and so on) that can be described as a black box with the properties described above does not change the character of sine waves. If you whistle into such a filter (which gives a good approximation to a sine tone), a whistle of the same frequency should come out the other end. On the other hand, a sung tone can have its character changed completely; for example, it could be much duller or much shriller.

A Recipe for Periodic Waves: Fourier's Formula

Periodic oscillations are combined according to Fourier's theory in the form of sine waves. What is the exact recipe? That is, in what proportions do the various sine functions appear?

Suppose we have a function f, whose graph is shown in Figure 3.

Figure 3: A periodic function f.

 

There is a number p (the period) such that the function evaluated at x+p is always the same as at the point x. Therefore, it suffices to know the values of the function on an interval I of length p, such as the section shown in Figure 4.

Figure 4: The significant portion of f.

 

One usually normalizes the function and assumes that the period is given by p=2·π, which makes the formulas especially simple. This can be easily achieved through a change in the unit length on the x-axis.

As a final preparation, one needs to know what is meant by an integral. The idea is simple: If g is a function defined on an interval, then the integral of g over that interval represents the area between the graph of the function and the x-axis. Warning: any portion of the graph that lies below the x-axis is considered to have negative area. For example, if the area between the positive portion of the function and the x-axis is 4, and that between the negative values and the x-axis is 3, then the value of the integral is 4−3=1. And if both parts are of the same size, then the integral is equal to zero. (The graph of Figure 1.4 shows just such a function.)

Now the "ingredients" can be calculated: If f is a function with period 2·π, then one can write f  as

f(x)































= a0+a1 sin(x) +a2 sin(2x) +a3 sin(3x)+ ...+ b1 cos(x) + b2 cos(2x) + b3 cos(3x)+ ... ,

where "sin" and "cos" denote the sine and cosine functions. The "weights" a0, a1,…,b1, b2, … that are used to build up the function f  are determined as follows:

• a0 is the integral of f (on the interval from 0 to 2π) divided by 2π.
• a1 is the integral of the function f(x)cosx (on the interval from 0 to 2π) divided by π.
• a2 is the integral of the function f(x)cos(2x (on the interval from 0 to 2π) divided by π.
• and so on.
• b1 is the integral of the function f(x)sin(2x) (on the interval from 0 to 2π) divided by π.
• and so on.

In sum, if you can calculate integrals, you can determine the amounts of each of the individual components that make up the periodic function.


This is an article from the book "Five-minute mathematics" by Ehrhard Behrends which was published in 2008 by the American Mathematical Society (AMS). It is reproduced here with the kind permission of the AMS.