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Fourier Analysis and Its Impact

Fourier Analysis and Its Impact

Overview

Jean-Baptiste Joseph Fourier, in studying the conduction of heat in solid bodies, devised a way to analyze it using an infinite series of trigonometric terms. Similar mathematical problems arise in almost every branch of physics, and Fourier's methods have been applied in many fields of science and engineering.

Background

The French Revolution (1789-99) was a dangerous time to be an intellectual. Jean-Baptiste Joseph Fourier (1768-1830) was the mathematically gifted son of a tailor. The idealistic young man had considered studying for the priesthood until he encountered the Revolution's promises of equality and the rights of man, with freedom from both the monarchy and the Church. But he was horrified by the Reign of Terror that followed and became embroiled in disputes between factions, barely escaping with his head.

He was, however, in the right place at the right time when Napoleon Bonaparte took over after the Revolution. It was easy to see the problems inherent in government by an unruly mob, so Napoleon set out to banish ignorance by establishing schools. Since the guillotine—the infamous execution device used heavily during the Revolution—had drastically reduced the supply of teachers, Napoleon founded the Ecole Normale in Paris to train new ones. Fourier was among the first students there and was on the faculty within a year after graduating. He later went with Napoleon to Egypt, becoming an expert on its antiquities. In 1802 he was appointed prefect of the French region of Isere, with headquarters in Grenoble. There he proved an able administrator, all the while continuing his studies of mathematics and Egyptology.

Napoleon fell from power in 1815, and Fourier was re-assigned to a quiet post in Paris that gave him the freedom to enjoy a scholarly life. He was elected to the Academie des Sciences, and became its permanent secretary. Most importantly, in 1822 he finally had the time to finish the work on the mathematics of heat conduction that he had begun 15 years before in Grenoble.

Fourier's "Theorie analytique de la chaleur" ("The Analytical Theory of Heat") dealt with problems such as finding the temperature in a conducting plate if the initial temperatures at the edges of the plate are known. Boundary-value problems of this type are among the most common in physics. They describe situations in which the known quantities are initial or final states, or conditions at the physical edges or boundaries. The goal is to figure out what happens in between.

To obtain a solution to the problem of heat conduction, Fourier expressed it as the sum of an infinite mathematical series with sines and cosines as terms.

These trigonometric functions can be plotted as smooth, repeating wave-like curves. Fourier's methodology had been tried during the previous century, when Leonhard Euler (1707-1783) and Daniel Bernoulli (1700-1782) had studied vibrating strings fixed at their ends. However, the eighteenth-century mathematicians had distrusted the validity of using infinite series in their solution. It was Fourier who brought this method into mathematical physics, and today it is called "Fourier analysis."

Impact

One of the reasons that infinite series solutions had disturbed previous generations of mathematicians was that it was unclear whether the sum of the series was finite or not. Even though the series had an infinite number of trigonometric terms, their coefficients, or multipliers, could be either positive or negative, causing terms to cancel out. Fourier devised rules for obtaining these coefficients so that the series would be finite, and thus converge to a useful solution.

Fourier series can be used to approximate any waveform. The more terms of the series that are used, the closer the approximation will be. Sines and cosines are "well-behaved" functions, meaning that the techniques of algebra and calculus can be easily applied to them. So a series of these terms can be used instead of a waveform that would not be convenient to work with otherwise.

For example, consider the sharp-edged "sawtooth" wave. In calculus we refer to the slope or rate of change of a function at a particular point as its derivative at that point. But a derivative requires a smooth curve. At sharp points like those on the sawtooth wave, we can define no derivative. However, if we express the sawtooth wave as the sum of a series of trigonometric terms, we can take the derivative of each term individually.

Another key point in Fourier analysis is the periodicity, or repeating nature, of the trigonometric terms in the series. There are many periodic phenomena in nature. Some, like a radio wave of a specific frequency, can be represented by a single sine or cosine wave. Others have more complex patterns.

Imagine playing a note on the piano while your friend plays the same note on the violin. The pitch is the same, and if you work at it you could get the volume to be the same, but nevertheless the sounds would be different. Different instruments playing the same note produce the same fundamental tone, a simple trigonometric waveform. But the specific sound each makes is a result of its pattern of harmonics, or overtones of frequencies that are multiples of the fundamental. The sum of the fundamental and its harmonics produce a complex waveform that is unique to the particular instrument.

Using Fourier analysis, it is possible to analyze a sound or other waveform in terms of its constituent harmonics; in fact, Fourier analysis is sometimes called "harmonic analysis." Each term of a Fourier series represents a different frequency. For example, a term in sin 2x represents a sine wave with a frequency twice that of a term in sin x. By building up these terms, the desired complex waveform can be produced. Generally it only requires the fundamental plus a few harmonics to achieve a reasonable facsimile of the original. For example, an electronic synthesizer can imitate the sound of many musical instruments by using its tone generators to produce the appropriate harmonics.

Fourier analysis also provides important tools for handling scientific data. For example, a common scientific instrument is the spectrometer. It measures the different levels of energy an object gives off over a range of frequencies. Spectrometry is a central technique of astrophysics, because the frequency distribution of the light from, say, a star, can tell scientists about its temperature and other properties.

A method of Fourier analysis called the "Fourier transform" provides a way to go back and forth between data given in terms of frequency and time. So, for example, a frequency peak in spectrometer data from a distant star or galaxy can be interpreted as a periodic fluctuation of light or energy over time. The rotating disk of matter around a black hole and the energy bursts emitted from pulsars are examples of phenomena that can be investigated in this way.

Today Fourier analysis is generally done with the aid of computers. The "fast Fourier transform" is a technique that allows drastically reducing the number of numerical operations required, making the analysis much faster and cheaper. When it was introduced in the 1960s, it revolutionized the digital processing of waveforms.

SHERRI CHASIN CALVO

Further Reading

Butzer, Paul Leo. Fourier Analysis and Approximation. New York: Academic Press, 1971.

Cartwright, Mark. Fourier Methods for Mathematicians, Scientists and Engineers. New York: Ellis Horwood, 1990.

Edwards, R. E. Fourier Series: A Modern Introduction. New York: Springer-Verlag, 1979.

Folland, Gerald B. Fourier Analysis and Its Applications. Pacific Grove, CA: Wadsworth & Brooks, 1992.

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