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# Joseph Fourier

Joseph Fourier

Jean Baptiste Joseph Fourier is a French mathematician and physicist.

## His life

Joseph Fourier is born 21 March 1768 in Auxerre, France. He loses his father and his mother early and is orphaned at the age of nine. Fourier is recommended to the bishop of Auxerre and he starts studying at the benedictine order of the convent of St. Mark; next Joseph accepts a military lectureship on mathematics. In his district, he promotes the French Revolution and serves the local revolutionary committee. During the terror, Joseph spends some time in prison. In 1795 he joins the École Normale Supérieure (ENS).

## His studies

His first school is the Pallas's school. At this school he studies Latin and French. In the school he shows a lot of promise. In 1783 he receives the first prize of his studies (for his studies about bossut, mécanique en général). Just after the Revolution he integrates the École Polytechnique.

## His main work

In 1798, Fourier joins Napoleon's Army as a scientific adviser with lot of other great scientists. During this travel, he is very concerned about the heat.

When he goes back to France a few month later, he decides to study it. In 1807, he finds the equation of heat diffusion in solid bodies.

This is a very strange formula, which is for advanced mathematicians : (∂T/ ∂t)*(x, t) = D*( ∂2T/ ∂x2) (x, t )

with T for temperature ( in °Celsius, or °Fahrenheit or °Kelvin ) ; x is the position along the metallic bar ( in meter ), t, the elapsed time ( in second ). D is the coefficient of thermal diffusion of the metal ( unit : m2/s. For instance, the copper is a good conducteur of heat : D ( for copper ) = 111 mm2/s ).

This is an another typography of the equation of diffusion  :

${\displaystyle {\frac {\partial T}{\partial t}}=D{\frac {\partial ^{2}T}{\partial x^{2}}}}$

He manages to solve it with success! ( Of course, he already knew the former D'Alembert's work on the vibrating string, about the famous equation of sound propagation ).

He finds that strange but real phenomena: the diffusion is the same, further away, at a later time, along the celebrated strange equation of diffusion :

${\displaystyle x^{2}=2Dt}$

That is to say: if the heat blob spreads to 15 mm in one second, it spreads to 30mm (=15·2) in 4s, then 45mm (=15·3) in 9s, then 60mm (=15·4) in 16s, etc. It is not propagation, there is no celerity! it is diffusion: to go further twice, you wait four times. If a pan handle ten centimers long is hot after one second, one is more comfortable with a twenty cm long handle because the heat will not be felt before 2² seconds. That is the rule to remember. It is not propagation, it is diffusion. ( as an exercice, a teaser: how long does it take to cook in a microwave oven a duck egg, twice as big as an hen egg? -four . NO! never cook an egg in a microwave, it explodes ! )

In the same theory, Fourier understands this known fact: when the sun heats the atmosphere during the day (and not in the night of course ), periodically, the temperature rises in the day ans goes down at night. But when is it maximum? at midday? no! the answer is: after of course, the time to accumulate heat. And Fourier gives the answer: a quarter of demi-period after (T/8 if you prefer). Therefore, at 3 pm. And, of course, it is cold at 3 and 4 am in the night. Now, by the same argument, the weather is hot after one year/8 = 6 weeks after the summer solstice; and the weather is cold 6 weeks after the winter solstice.

Now, a more intelligent teaser: combine the two arguments, the diffusion and the T/8 time shift. Inside a cellar, is there a six-month shift between the outside temperature and the temperature in the cellar? Yes, indeed ! but the variation in temperature is very reduced; and Fourier enables to work out the most appropriate thickness of the walls.

Incidently, Fourier's equation of diffusion was used by the great physicist, Lord Kelvin, when he decided to build the first telegraphic submarine line between USA and England: exactly as Kelvin had said, queen Victoria received the first electric telegraphic signal from USA in 1858, after a 16 hours delay. Question ? if America has been twice as near? Answer: 16/2² = 4 hours! Nevertheless, one cannot have a proper conversation in those conditions! it was just a message of congratulation. Propagation of electromagnetic waves is a huge progress for nowadays communications !

To solve his equation, Fourier invents a famous method, since named Fourier's coefficients method. All mathematicians in the world use it to solve all sorts of problems. It is the main reason of Fourier's glory.

For the theory of sound and music, the main result of Fourier's theory is the following:

Fourier's Theorem: if a temporal signal s(t) is indefinitely periodic, just as a musical note of a certain frequency, it is possible, whatever the timber of the note and whatever the strength of the note, to reproduct this note with the sum of certain pure tones, multiple of the note frequency. (It is the principle of all the modern synthetizers of music).

For instance : the note "D" is composed of all the harmonics of 440 Hz : 440, 880, 1320, 1760,...: if the piano or the trumpet plays a 440 Hz note "D", we hear a different sound, but nevertheless of the same frequency, and different timber. And a synthetizer can recompose the sound of piano with a sum of pure tones of 440, 880, 1320, 1760,... Hz ; and also the sound of trumpet with another sum of the same harmonic pure tones.

Fourier was a friend of Laplace, also a great French mathematician and physicist. Laplace often used Fourier's coefficients theory for his own works. Laplace and Fourier are considered, in nineteenth century France, as the advocates of mathematical physics just as Euler in Switzerland, in the eighteenth century, and Newton, in England, in the seventeenth century.

Note: during this work on heat, Fourier realises that the Earth should be colder than it actually is, if the Sun would be its sole source of heat. To explain the actual temperature of the Earth, one suggestion made by Fourier is that the Earth atmosphere could be a kind of insulator. This is considered the first time that what we now know as greenhouse effect is envisaged (although not yet named as such nor explained).

## Anecdote

Joseph Fourier start to be sick when he has 62. He develops a problem of heart when he was in Egypt. He choose to return in Paris. Because he has cold, Joseph Fourier warm him. But he was dead for that because he warm him too much. That's a shame because he worked about what it cause his death. Because he was a great mathematician, his name is write in the Eiffel Tower, like 71 others persons.

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