Wednesday, 16 June 2010

Joseph Fourier and the Vuvuzela

Joseph Fourier (1768 – 1830), more than anything else, wanted his mathematics to bring usefulness to society. During his lifetime, he formulated a theory of heat, which was later applied to light and, especially recently, sound.

He postulated that there is fundamentally one form of wave, to which the same mathematical principles can be applied. And he possessed a remarkable ability to generalise results and to find some unexpected applications for his work.

He worked hard from a young age; at his boarding school, he would collect the ends of old candles and studied into the night.

He became an intellectual figurehead for the French Revolution and when the Republicans won the day, he flourished in the new society. New centres of learning emerged and nobody was denied a place in the education system because of a lack of funds.

Fourier was appointed head of mathematics at the École Polytechnique. He had been a great orator during the revolution and now practised these skills in the lecture theatre. Debate was expected between teacher and pupil and Fourier, the great performer, thrived, filling his lectures with interesting historical references and practical applications.

In May 1798, Fourier received a summons to set sail for Egypt with Napoleon Bonaparte and 30,000 soldiers and academics. There he helped with the discovery of the Rosetta Stone. He also began to develop his obsession with heat, posing himself the question “How does heat vary over time?”

Eventually he would set forth his mathematical laws of heat, and this would become one of most important branches of mathematics, through the startling revelation that almost any natural phenomenon could be described as the sum of simple sine waves. In sound, for example, the idea was that every noise could be made up from the sum of the pure sounds created by a tuning fork.
There was vigorous opposition, notably from Laplace, who described considerable difficulties in Fourier’s work and a “lack of rigour”. But the final version of Théorie Analytique De La Chaleur was published in 1822 and remains a seminal work.

Lord Kelvin described Fourier’s work as one of the most beautiful theories of modern mathematics.

One example of modern day applications of Fourier’s work is the way in which computers can break music down into its component waves. Then, these waves can be manipulated using Fast Fourier Transforms to enhance, warp or distort the overall effect, or to remove the noise on a particular frequency. And hence the application of Fourier to the very topical, sometimes maligned vuvuzela. Could Fourier analysis really help to block the sound of the monotonous football trumpets? The BBC seem to think it would detract from the erudite commentary being offered. But at least one other source seems to think it could work (if you've got a Linux machine and lots of leads). Let us know your experience.

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