## A historical interlude Jean Baptiste Joseph Fourier 17681830

Jean Baptiste Joseph Fourier was born in Auxerre, a small cathedral town in the Burgundy region of central France. He was educated at a local school run by the music master of the cathedral and later at the Ecole Royale Militaire.

Fourier had intended to take the tradition route into the army after finishing his studies but his application was refused because of his low social status. In 1787, he entered the Benedictine Abbey at St Benoit sur Loire, but left after two years to embark on a career in mathematics. He returned to Auxerre, becoming a teacher at the school where he had been a student. His intention to make significant contributions to mathematics may be seen in this extract from a letter written to Bonard, a professor of mathematics in Auxerre: 'Yesterday was my 21st birthday; at that age Newton and Pascal had already acquired many claims to immortality.'

In the turbulent years following the French revolution in 1789, Fourier developed an active interest in politics and joined the local revolutionary committee in 1793. A political idealist, he wrote: 'As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken.'

French politics at that time were extremely complex. There were numerous small groups of 'liberators' who fought bitterly amongst themselves, despite having the same general aims. 'Citizen' Fourier was arrested and imprisoned in 1794 because of his support for victims of terror. At this time, during the 'Reign of Terror' imposed by Robespierre, the interval between imprisonment and being guillotined was often very short. Fortunately (for Fourier), Robespierre himself was guillotined shortly after Fourier was imprisoned and the political climate changed. Fourier was freed and went to the Ecole Normale in Paris, where he studied

under the eminent mathematicians Lagrange, Laplace and Monge. Fourier was appointed to a teaching post at the newly established Ecole Polytechnique. He continued his research until he was arrested and imprisoned once more (for the same offences), but he was soon released and was back at work before the end of 1795. Within two years, Fourier succeeded Lagrange as professor of analysis and mechanics at the Ecole Polytechnique.

In post-revolutionary France, Fourier was no longer debarred from a military career as he had been in his youth. In 1798, he was drafted and joined Napoleon's expeditionary force to Egypt as a savant(scientific advisor). The campaign began well. The occupation of Malta in June 1798 was rapidly followed by the fall of Alexandria and the occupation of the Nile delta some three weeks later. Napoleon's fleet was decimated by the English fleet, under the command of Lord Nelson, at the battle of the Nile and he was cut off from France by the English fleet. Nevertheless Napoleon proceeded to set up an administrative structure modelled on the French system. Fourier was appointed governor of Lower Egypt and was one of the founders of the Institute d'Egypte in Cairo. He began to develop ideas on heat, not only in the mathematical sense but also as a source of healing.

Napoleon left about 30,000 troops stranded in Egypt and returned secretly to France. Fourier himself returned in 1801 hoping to continue his work as a professor at the Ecole Polytechnique. He was not allowed to remain there for long. Napoleon, having noticed his remarkable administrative abilities in Egypt, appointed him as prefect of the Departement of Isere, where he was responsible for the drainage of a huge area of marshland and the partial construction of a road from Grenoble to Turin.

Whilst in Grenoble, Fourier found the time not only to further his mathematical studies but also to write a book on Egypt. He was made a count and appointed prefect of the Rhone shortly before Napoleon's final defeat in the Battle of Waterloo in 1815. At that point, Fourier resigned the title and the position and returned to Paris, where he had a small pension but no job and a rather undesirable political pedigree. One of his former students secured him an undemanding administrative job. Fourier finalised his most important mathematical work during this time. His major achievement was to describe the conduction of heat in terms of a differential equation (similar to the wave equation) without trying to figure out what heat actually was. The conclusion of his work was that these equations could be solved in terms of periodic functions which themselves could be constructed from simple sinusoidal functions. Fourier died in 1830 in Paris.

Appendix 6.1 The speed of transverse waves on a string

We can show that the speed of a transverse wave on a stretched string depends on the tension and the mass and length of the string. Take a single pulse which travels along a string at a constant speed v:

When the pulse arrives at a point on the string, a component of the tension suddenly acts at right angles to the string and displaces the string at that point. Owing to 'bonding', successive particles are displaced from their equilibrium positions and the pulse moves along the string. The faster the response of each particle, the larger its acceleration, the more rapidly it rises and falls and the faster the wave progresses. Increasing the tension increases the wave speed. Increasing the mass (a measure of the resistance to motion) reduces the wave speed.

Calculating the speed

We look at a very small segment of length 2L around the peak of the pulse and make a number of approximations.

segment expanded

segment expanded

Figure 6.10 The peak of the pulse passes along the string.

Figure 6.10 The peak of the pulse passes along the string.

Let t be the time for the left hand half of the pulse to move from A to C in Figure 6.10 and let the average transverse speed of the particles be u.

BC u

BD v

For small values of d

The force towards the centre of the circle F = T sin Q = Tu , where T is the tension of the string. v

Impulse Ft = ^^L = mu, the momentum transferred v

Tt m = — = mL (u is the mass per unit length, a property of v the string)

The speed of a transverse pulse on a string v =if (6-2)

The speed of the wave depends only on the characteristics of the string, and not on the frequency of the wave.

There is an equivalent expression for all mechanical waves.

The dimensionless analysis in Appendix 6.2 shows that the speed of any transverse wave in any medium is related to the ratio of the appropriate elastic constant divided by a mass-related constant.

Appendix 6.2 Dimensional analysis

A dimensional analysis will confirm that the form of Equation (6.2) for the speed of a transverse pulse on a string has the correct form.

We found that

Dimensions of the left hand side:

L T2

Dimensions of the right hand side:

ML IM L

confirming that Equation (6.2) has the correct form.

A dimensional analysis is independent of the type of wave and the nature of the material, so we can extrapolate and say that the general expression for the speed of a wave in a material will be elastic constant v -

mass-related constant

Appendix 6.3 Calculation of the natural frequencies of a string fixed at both ends

We can write the displacement due to a right-travelling wave as y1 = A sin( kx -at) and due to a left-travelling wave as y2 = A sin(kx + at). (At all places on the string, there are times when y1 + y2 = 0, which justifies using these expressions for yi and y^)

Using the superposition principle, y = y1 + y2, y = A[sin(kx - wt) + sin(kx + wt)] fi y = 2A sin(kx)cos(wt) (6.3)

using the trignometric identity

2sin |
f A + B ^ |
(A - B1 | |

cos | |||

I 2 V |
I 2 J |

There is no (kx-at) term in Equation (6.3), which means that the wave does not propagate. If we look at the particle at x = a, for example, the amplitude of its vibration is 2A sin(ka), which is constant. Each particle performs simple harmonic motion at the frequency of the waves and all the particles vibrate in phase.

In simple harmonic motion, the total energy of any particular oscillating particle is constant — a standing wave stores energy.

An oscillator attached to one end of a string generates waves of very small amplitude, so the 'oscillator' end of the string can be taken to be more or less fixed. If the other end of the string is also fixed, then standing waves can be set up. Suppose that the oscillator is situated at x = 0 and the other end of the string at x = L. The displacement will always be zero at x = 0 and x = L, i.e. y = 2A[sin(kL)cos(®t)] = 0

This means that sin(kL) = 0 and kL = 0, n, 2n = nn, where n is an integer.

By comparing the expression y = A sin(kx - (ot) with

Equation (6.2), we see that k = — , giving l = — . The value of l n the wavelength is fixed solely by the length of the string; it is a natural wavelength.

The natural wavelengths have a series of values, 1n =—.

The corresponding natural frequencies are fn = n-V-.

The frequency of the first harmonic f1 = -v (also called the

fundamental frequency). The frequency of the second harmonic f2 = V. The frequency of the third harmonic f3 = —v and so on. L 2L

Since v depends on the material itself v = J— for a wave on a string

, a vibrating string will not have the same natural frequencies as a vibrating rod, or indeed a vibrating air column of the same length.

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