Fourier's Series

20 January 2015

Fourier’s Series

Professor Raymond Flood

Slide: Title

Thank you for coming to my first lecture of 2015 and a Happy New Year to you all!

Today I want to talk about the French mathematician and physicist Jean Baptiste Joseph Fourier and the consequences of his mathematical investigations into the conduction of heat. He derived an equation, not surprisingly now called the heat equation, to describe the conduction of heat but more importantly an approach to solving the heat equation by using an infinite series of trigonometric functions, now called Fourier series. The ideas he introduced had major applications in other physical problems and also led to many of the most important mathematical discoveries of the nineteenth century.

Slide: Overview

Let me start my overview of the lecture by talking briefly about the situation in mathematics and its use in explaining and predicting the world about us at the end of the eighteenth century. During the eighteenth century many master mathematicians, for example Laplace and Lagrange, had built on Newtonian mechanics to describe and predict the motion of bodies on the earth and in the sky. They had the underlying equations for the motion of bodies, indeed they had them in various formulations and they investigated these equations mathematically.

But the behaviour of heat, light, electricity and magnetism had yet to yield their governing equations and this lecture is about applying mathematics to one of these four areas namely heat.  It was Joseph Fourier, the distinguished French mathematician, physicist, Egyptologist, demographer and public servant who tackled the conduction of heat and found the fundamental equations its conduction but as I’ve mentioned, just as importantly found new mathematical methods for solving these equations.

I will start by outlining Fourier’s rich and dramatic life and some of the events of the turbulent times in which he lived.

Then I will consider his work on the conduction of heat and motivate his derivation of the heat equation and his use of trigonometric functions in its solution. Then I will look at some case studies involving approximation by trigonometric functions: on tide prediction and the magnetic compass both involving William Thomson, later Lord Kelvin. Kelvin was also involved with the successful transmission of telegraph signals over the underwater transatlantic cable where current flow was also governed, at least approximately, by the heat equation.

Finally I will finish by mentioning some current applications.

Slide: Images of Fourier

Fourier was born in Burgundy in 1768, the son of a master tailor. His mother died when he was nine and his father shortly afterwards. However he had an outstanding school career, particularly in mathematics, and at about the age of seventeen he applied to the Military of War to join the artillery or engineers. But his application met with the crushing reply that as he was not of royal blood he was not acceptable “even if he were a second Newton”.

Fourier then decided to enter the Church and in 1787 joined the Benedictine teaching order to prepare for his vows while serving as professor of mathematics for the other novices.

However the outbreak of the French revolution and its constantly changeable developments and factions was a dramatic influence on Fourier’s life.

In a letter written later from prison, in justification of his part in the Revolution in Auxerre in 1793 and 1794, Fourier describes the growth of his political views:

The first events of the Revolution did not change my way of life. Because of my age I was still unable to speak in public; and impaired by night studies my health scarcely sufficed for the work my position required of me.

From another point of view I will admit frankly that I regarded these events as the customary disturbances of a state in which a new usurper tends to pluck the sceptre from his predecessor. History will say to what extent this opinion was justified. Republican principles still belonged to an abstract theory. It was not always possible to profess them openly. 

And he continues.

Slide: letter from prison

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 the most beautiful which any nation has ever undertaken.

This quotation is from a letter written by Fourier during one of his spells in prison and it appears that only the fall of Robespierre saved Fourier from the guillotine.

Subsequently Fourier went to Paris first of all to study at the Ecole Normale. However he was again arrested, released, rearrested and then following yet another change in political rulers became a teacher at the Ecole Polytechnique where in 1797 he succeeded Lagrange in the chair of analysis and mechanics and where he gained a reputation as an outstanding lecturer.

His burgeoning academic career was ended or at least interrupted in 1798 by government orders to join as a scientific advisor the French armed forces, commanded by Napoleon, for the invasion of Egypt.

Fourier returned to France in 1801 and accepted an offer from Napoleon to become Prefect of the Department of the Isèrein south-eastern France. He was an effective and well liked administrator organizing the draining of swamps and supervising the building of the road across the Alps from Grenoble to Turin.

Slide: Description of Egypt and Rosetta Stone

At the same time he was helping to organise the Description of Egypt, a multi-volume work by the scholars and scientists who were on Napoleon’s expedition to Egypt. Fourier wrote the general introduction – a survey of Egyptian history up to modern times - and this work did much to arouse European interest in Egypt and help establish the subject of Egyptology. The other important lasting consequence of the expedition was the discovery of the Rosetta stone. The decree on the stone is inscribed three times, in hieroglyphic - suitable for a priestly decree, in demotic - the native script used for daily purposes, and in Greek - the language of the administration. It was to provide the key to the deciphering of hieroglyphics. Champollion who eventually deciphered the Rosetta stone was encouraged by Fourier to take up Egyptology and Fourier also protected him from conscription into the army.

It was also while he was Prefect of Isère that Fourier began his work on heat conduction – Fourier was unusual among mathematicians in having a distinguished nonmathematical career. This work was to fulfil his youthful ambitions which he hinted at in a letter written much earlier in 1789:

Slide: Immortality quote

Yesterday was my 21st birthday, at that age Newton and Pascal had [already] acquired many claims to immortality.

Now in his mid-thirties Fourier started on the work that was to bring him his immortality.

Slide: Remarkable years 1804 - 1807

During three remarkable years from 1804 to 1807 he:

·         Discovered the underlying equations for  heat conduction

·         Discovered new mathematical methods and techniques for solving these equations

·         Applied his results to various situations and problems

·         Used experimental evidence to test and check his results

The resulting memoir on his researches, On the propagation of heat in solid bodies, was initially not well received and even though it formed the basis for Fourier’s successful submission for an 1811 prize offered by the Paris Academy of Sciences their report stated:

Slide: Referee’s quote

…the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

Although this report and the attitude of the referees Laplace and Lagrange to Fourier’s submission has been criticised by many I agree rather with Tom Körner when he says in his brilliant book on Fourier Analysis:

Slide: Quote from Körner

Laplace and Lagrange [the referees] could not see into the future and their doubts are surely more a tribute to the originality of Fourier’s methods than a reproach to mathematicians who Fourier greatly respected (and, in Lagrange’s case, admired).

Slide: Analytic theory of heat

Fourier’s main results were published in his 1822 Théorie  Analytique de la Chaleur (Analytic Theory of Heat). This contained the first large scale mathematisation of a physical process, heat, that was not part of mechanics.

The final years of Fourier’s life were spent in Paris where he became part of the academic mainstream to the extent of becoming Secretary to the mathematical section of the Paris Academy of Sciences.

I’ll end this brief biographical sketch of Fourier with a comment by the distinguished historian of mathematics, Ivor Grattan-Guinness who died last month.

Slide: Grattan-Guinness quote

He preserved his honour in difficult times, and when he died he left behind him a memory of gratitude of those who had been under his care as well as important problems for his scientific colleagues.

Now let me turn again to Fourier’s main publication.

Slide: Analytic theory of heat with fundamental causes quote

In his 1822 Théorie  Analytique de la Chaleur (Analytic Theory of Heat), Fourier wrote:

Fundamental causes are not known to us; but they are subject to simple and constant laws, which one can discover by observation and whose study is the object of natural philosophy.

I want to pick out three things Fourier achieved. First is his derivation of the fundamental equation for the conduction of heat and the resulting temperature distribution.

Slide: Analytic theory of heat with excerpts

On the slide we see at bottom right the section on the General Equation for the Propagation of Heat in the Interior of Solids. This is known as the heat equation and here it is in three dimensions but I will illustrate it in a moment in one dimension. The excerpts on this slide are from a translation of 1878.

Then he developed the marvellous and powerful technique of representing a function or a waveform in terms of a sum of trigonometric terms – we just need to know how much of each trigonometric term to throw into the mix.  There is Fourier’s first example on the top right of the slide and I will come back to it later.

The third thing is the use of these trigonometric functions to solve the heat equation.

First let us think about temperature and the flow of heat and illustrate it with an example.

Slide: Fourier’s cellar

We want to build a cellar and need to decide on what depth it should be. Ideally we would wish that in the cellar the temperature should be constant throughout the year whether it is summer or winter or perhaps more realistically that the temperature fluctuations should be as small as possible as the seasons change.

We can expect that this is possible from the following qualitative arguments.

As time passes heat will propagate throughout the ground. In summer when the surface is hot heat will propagate downwards and in winter when the surface is cold heat will propagate upwards. But heat needs time to propagate so that for example when it is the hottest day of the year at the surface it will not be the hottest day in the cellar because there will be a delay as the summer heat propagates downwards through the ground to the cellar. We will see that the temperature at the ground and the cellar are out of phase, that is, peak at different times. If the depth is chosen correctly then they will be six months out of phase, that is peak six months apart, meaning that when it is hottest at the surface (in summer) it is coolest in the cellar and when it is coolest at the surface (in winter) it is warmest in the cellar.

This is due to the delay caused by the time it takes heat to propagate.

There is another effect which is that when the heat or cold eventually reach the cellar they produce a smaller change in temperature: they are attenuated or damped. So the temperature difference in the cellar during the year will be less than the temperature difference at the surface during the year.

Fourier’s techniques allow us to make this qualitative approach quantitative.

Of course it varies with the type of soil but Fourier’s techniques show that with a depth of 4.5 metres the change of phase is six months – when it is summer at the surface it is winter in the cellar and vice versa. Also at this depth the temperature fluctuation during the entire year in the cellar is one sixteenth of the temperature fluctuation during the entire year at the surface. So if the surface yearly temperature difference is 32 degrees the cellar yearly temperature difference will be 2 degrees.

Now let us find the heat equation – we will do it in one dimension but the approach applies to two and three dimensions.

Slide: Fourier’s cellar – one-dimensional heat equation.

We need some notation first and we will denote the temperature at depth x at time t by u(x , t).

Slide: Notation for temperature

The fundamental observation we are going to use to describe the change in temperature at depth x over time is that the rate of change of temperature with time at depth x is proportional to the flow of heat into or out of depth x.

Slide: fundamental observation

We can express this in symbols

Slide: Heat equation

•       u(x , t) is the temperature at depth x at time t.

•       The left hand side is the change of temperature over time at depth x.

•       The right hand side is the flow of heat into the point at depth x.

•       K is a constant of proportionality depending on the soil.

Let us now look at the second of Fourier’s achievements: Fourier series.

Slide: Fourier series 1

Now I will illustrate the idea of Fourier series using an example from Fourier himself and this was representing a square wave form using a sum of cosines.

The square wave form arose in his investigations of heat flow. It is in blue here and it is a function that just takes two values and oscillates regularly between these two values. It is also important nowadays in digital engineering.

The first approximation is

cos u 

Slide: Fourier series 2

He calculated the next term in the approximation as

cos u  - 1/3 cos 3u 

Slide: Fourier series 3

He calculated the next term in the approximation as

cos u  - 1/3 cos 3u  + 1/5 cos 5u 

The last one I will show is

Slide: Fourier series 4

cos u  - 1/3 cos 3u  + 1/5 cos 5u  -  1/7 cos 7u

The approximation becomes better the more terms we take. Fourier gave explicit formula to allow him to calculate the coefficients, 1, -1/3, 1/5, -1/7, etc. These coefficients are called the Fourier coefficients of the Fourier series and tell us how much of each trigonometric function is needed to give the original function.

In fact:

cos u  - 1/3 cos 3u  + 1/5 cos 5u  -  1/7 cos 7u +  … ,

equals:

0 when u = π/2,

π /4 when u lies between - π /2 and π /2,

- π /4 when u lies between π /2 and 3π /2.

Fourier wrote about this surprising outcome:

As these results appear to depart from the ordinary … it is necessary to examine them with care, and to interpret them in their true sense.

Fourier then considered the more general question of which functions can be represented by Fourier series in this way. He also derived formulas (involving integrals) for the coefficients in the Fourier series of the function.

In fact, the question of what conditions should be imposed on a function so as to ensure that its Fourier series does indeed converge to the original function generated much new mathematically activity.

The third thing that Fourier used was the idea of linearity to solve the heat equation using Fourier series.

Slide: Linearity

Much of Fourier’s success arose from the observation that the Heat Equation and indeed many of the laws of mathematical physics were linear. This just means that if you have found two solutions of the underlying governing equations then you have in fact found many others because, for example, their sum is also a solution.

Water waves on the surface of a pond satisfy the wave equation which is a linear equation. At the top right is a picture of the superposition of waves produced by the ducks – the disturbance of the surface of the water is obtained by adding the disturbances produced by each of the ducks. Below are one dimensional examples of adding waves together.

Slide: Linearity and Fourier’s cellar

As I have said the heat equation is linear which just means that if you have found two solutions of the underlying governing equation then you have in fact found many others because, for example, their sum is also a solution as is their difference.  In fact, if u1 and u2 are solutions then so is α u1 + β u2 for any constants α and β.

Fourier then represented the temperature distribution at depth x at time t as a Fourier series and used the idea of linearity to obtain a simple equation for each of its Fourier coefficients. He then solved these equations. There were still some constants to be determined but the temperature variation at the surface where x is 0  is known and so can be written as a Fourier series which let him find the constants and so the complete solution.

From this, as I have said, he was able to show that at about 4.5 metres below the surface the temperature fluctuates very little over the course of the year. Of course, this must have been known empirically for a long time but the truly innovative thing was that Fourier had derived the heat equation and solved it using Fourier series and this could be applied in many other situations where objects of different shapes were heated and we wanted to know the subsequent distribution of heat.

Fourier’s work was to be very influential.

We can think of Fourier series as using the trigonometric functions of sine and cosine to produce better and better approximations to a function.

I now want to illustrate this approximation approach by the work done by William Thomson, Lord Kelvin on predicting the tides. This example will also show how we can calculate the coefficients, i.e. how much of each sine and cosine to use, when we are approximating a function using sines and cosines.

First I will give a little background about William Thomson, Lord Kelvin

Slide: William Thomson portrait

William Thomson was born in 1824 and died in 1907. Later ennobled as Lord Kelvin, he was a dominant figure in Victorian science with contributions to mathematics, physics and engineering, particularly in the areas of electricity and magnetism. A leading figure in the creation of thermodynamics, the area of physics concerned with heat and energy, he was also instrumental in the laying of the first transatlantic telegraph cable.

Thomson was born in Belfast, Ireland. He was educated at Glasgow and Cambridge Universities, and was appointed Professor of Natural Philosophy at Glasgow University at the age of 22. He remained at Glasgow until his death and was buried in Westminster Abbey alongside Isaac Newton.

William Thomson was always a great admirer of Fourier, an admiration  which went back to his youth. Thomson told the story of when he was sixteen and on a family holiday to Germany:

Going that summer to Germany with my father and my brothers and sisters, I took Fourier with me. My father took us to Germany and insisted that all work should be left behind, so that the whole of our time could be given to learning German. We went to Frankfort, where my father took a house for two months… Now just two days before leaving Glasgow I had got Kelland’s book (Theory of Heat, 1837), and was shocked to be told that Fourier was mostly wrong. So I put Fourier in my box, and used in Frankfort to go down to the cellar surreptitiously every day to read a bit of Fourier. When my father discovered it he was not very severe upon me.

I like the idea of Fourier being secretly read in a cellar!

Let me give an example of Fourier’s influence on Thomson. It is in the area of tidal prediction.  Modern tidal analysis and prediction in all its mathematical and mechanical detail is due to William Thomson from around 1867.

He designed a machine for predicting tides which computed the depth of water over a period of years, for any port for which the tidal constituents have been found from harmonic analysis of tide-gauge observations

 — that is, from the coefficients of the trigonometric series representing the rise and fall of the tide.

Thomson talked of substituting “brass for brain” when discussing the mechanization of tidal prediction.

Slide: Tide Prediction

I want to briefly:

•           Show how to describe the tide.

•           Introduce the beautiful idea that allows us to calculate the tide theoretically.

•           show the practical developments that allow us to calculate the tide practically.

The tide is caused by the pull, the gravitational pull, of the sun and the moon on the oceans, and the rotation of the earth, but its exact pattern at any particular spot on the coast depends on the shape of the coastline and on the profile of the sea floor nearby. So even though the forces that move the tide are completely understood, the tide at any one spot is essentially impossible to calculate theoretically. What we can do is record the height of the tide at that spot over a certain period of time, and use these measurements to predict the tide at that spot in the future, and I want to show you how that can be done.

Slide: astronomical periodicities

The tidal force is governed by a small number of astronomical motions which are themselves periodic, but since the various frequencies have no whole-number ratios between them, the whole configuration never repeats itself exactly. There is an almost exact 19-year cycle in the joint pattern of equinoxes and solstices, and phases of the moon, so the tidal record repeats almost exactly if you wait long enough: this fact had been used to prepare useful tables for European ports. But a major motivation for Kelvin’s work was predicting the tides in India where they did want to wait nineteen years for accurate tide predictions.

Because the earth, moon and sun are all in relative motion the gravitational pull at a point of the ocean is constantly changing. The five main astronomical periodicities are:

•           Length of the year

•           Length of the day

•           The lunar month

•           The rate of precession of the axis of the moon’s orbit

•           The rate of precession of the plane of the moon’s orbit

When you look at the geometry you can see that the gravitational pull can be described using a collection of sine waves.

Slide: sine functions

Here is a collection of sine waves where the frequency is increasing as we move down the page.

Slide: Height of the tide at a given place is of the form

A0 + A1cos(v1t) + B1sin(v1t) + A2cos(v2t) + B2sin(v2t) + ... another 120 similar terms

Here t denotes time and the expression tells you how the height of the tide changes with time.

There are two important points.

First the analysis of the geometry to get this expression tells us what the frequencies, the terms v1, v2 and so on are, so we know the components making up the tide. Each of the terms v1, v2 and so on is a combination of the five astronomical frequencies. However we do not know the coefficients or how much of each is in the mixture. The coefficients depend on the location and geography of the place.

In other words the frequencies v1, v2, etc. are all known – they are combinations of the astronomical frequencies.

We do not know the coefficients A0 , A1  ,A2 , B1  , B2  ,…- these numbers depend on the place.

Slide: tide at Glasgow

Here is the record of the tide at Glasgow over a week

What we need to do is to use this record to find the coefficients and this builds on the work of Fourier.

Slide: First Fourier Analysis

As is often the case in mathematics we will look at something simpler, not 120 terms added together, not even 25 but 2!

Here we have a pretend tidal record made up of two sin waves but we do not know how much of each is in the record. But there is a beautiful thing we can do, which makes use of the fact that the long term average of a sine wave is zero.

Slide: Second Fourier Analysis

If we take our pretend tidal record and multiple it by sin(t) and take the long term average we will get the  coefficient  A. This is because the long term average of sin(t) times sin(21/2 t) is zero. This is because the product can be written as a sum of two other sine waves and the average of each of them is zero.

Similarly to find B multiple by sin(21/2t) and calculate twice the long term average.

Here we do it to get A as 2 and B as -3

Slide: The method followed in the sample problem can be extended to the complete calculation.

This tells us how to calculate the coefficients theoretically but there is a lot of practical work involved in multiplying the tidal record by each of the sin curves and calculating the long term average. William Thomson used an invention of his brother, James, to mechanize this process and obtain the coefficients. They restricted themselves to the 11 most significant tidal components.

Slide: Kelvin Tide Predictor.

We are now in the position that if we know the tidal record for say a year at a particular spot we will be able to determine the amount of each tidal component at that spot. How to use that to predict the tide in the future!

Kelvin came up with his idea of the tidal predictor on a railway journey to attend the British Association meeting in Brighton in 1872.

A wire is fixed at the right and passes alternately over and under 15 movable pulleys, after which it suspends a weight in this image: in practice, an ink bottle with a pen. Each of the movable pulleys is driven in a simple sine wave, as follows.

•           Turning the crank drives eleven gear assemblies. The gear ratios are chosen so that the speeds of the output gears in each assembly are proportional to the speeds/frequencies of the tidal constituents being summed.

•           The amplitude of the vertical motion can be set to match the coefficient of the corresponding constituent at the port in question

•           The crank also moves a strip of paper horizontally in front of the pen (this is not shown) to record the predicted tidal curve.

The machine could predict the tide a year ahead in a matter of hours.

Slide: D Day landings

Here we see Kelvin’s tide predictor still in use responding to a request from the Admiralty in Bath for a prediction of the tide at position Z during the D-day landings in the Second World War.

Another problem to which Kelvin applied approximation by the trigonometric functions, sine and cosine, was the construction of the magnetic compass and correcting for the influence on a magnetic compass when it is used in a ship made of a large amount of iron or steel.

Slide: Kelvin’s magnetic compass

We will assume that the true compass bearing and the one actually shown on the compass are related in the following way.

True compass heading = displayed heading, b,  + error term

The main contribution to the error term is the permanent magnetism of the ship due to heating and hammering of the iron and steel plates during construction when the ship is lying in a fixed direction.

We can assume that the error term is a combination of trigonometric functions in the displayed heading, b, of the form:

Error term = a0 + a1 cos b+ a2 cos 2b+ b1 sin b+ b2 sin 2b

If this is the case and theoretical modelling and practical measurement showed that it was reasonable approximation for the error term then it was possible to determine the coefficients a0, a1, a2, b1, and b2 as follows:

While in port, point the ship in various known directions (say five of them since we have five unknowns to determine) and for each known direction read of the corresponding value of the displayed compass heading, b.

For each direction putting this into the equation

True compass heading = displayed heading, b,  + error term

gives five equations from which we can determine the unknown coefficients a0, a1, a2, b1, and b2. Now it was a matter of neutralising the error term by introducing other magnets to the compass.

Slide: Kelvin’s compass card

Kelvin took on this engineering challenge of how to mount small permanent magnets, in the form of magnetic needles, and soft iron globes to compensate for the error term. He wrote:

These magnetised needles are symmetrically disposed about the NS [North – South] axis of the [compass] card and parallel to it. The small size of the needles allows the magnetism of the ship to be completely compensated for by soft iron globes

of an acceptable size.

The last Kelvin related topic I want to mention is to do with how to transmit signals over the transatlantic cable.

Slide: Route of the transatlantic cable

In its day the laying of the first transatlantic cable was thought of as the eighth wonder of the world. It is hard to imagine, from our point of view, so used are we to instantaneous communication and, since the Web, instant access to information, what a dramatic difference the cable made. Before it was laid news was carried by ships crossing the Atlantic and this could and did take many weeks. It could take months to send a message to someone in America and receive a reply. So, we can understand the enthusiasm with which the cable was greeted and the benefits it could bring.

Although the first 1858 cable broke down completely shortly after it was laid the benefits were obvious and lessons were learnt on how to construct and use underwater telegraph cables with successful laying and transmission over a cable in 1866.

The shortest direct route from Newfoundland to Ireland was over 1800 miles and, fortunately, oceanographic surveys showed it was also suitable in terms of depth, profile and composition of the seabed. 

It was decided to lay the cable from Valentia Island in southwest Ireland to Trinity Bay in Newfoundland. Here we see a map of the route and on the right the plaque at Valentia Island and on the left an image of the Telegraph house at Trinity Bay in Newfoundland.

Point out the path of the cable on the map.

Point out the depth chart at the bottom on the map.

Point out the network of telegraph cables on land on the map.

Although the manufacture and laying of nearly 2,000 miles of cable was challenging enough perhaps Kelvin’s greatest contribution to the enterprise was his analysis of how signals propagate over an underwater cable and how to detect them.

Slide: Transmission over a telegraph cable

There is a considerable difference between the behaviour of an electrical pulse sent along a telegraph cable when it is in air compared with when it is under water.

In air the behaviour of the pulse is given, at least approximately by an equation called the Wave equation. This shows that an electrical pulse travels with a well-defined speed with no change of shape or magnitude over time – the pulse travelled like a “bullet”. Because of this the main limitation to the rate at which pulses are transmitted is the limitation of the sending equipment – how many pulses can be sent per second and the limitation of the receiving equipment – how many pulses can be distinguished per second. So signals can be sent close together.

On the other hand, for a cable under water, the situation is very different and the behaviour of the pulse is similar to the flow of heat along a metal bar and is given, at least approximately, by the Heat equation. Indeed Thomson acknowledged that his analysis of the transmission of a pulse of electricity along a cable under water was based on Fourier’s work on the flow of heat along a metal bar.

One of the reasons is that an underwater cable has to be insulated from the surrounding water which of course is an earthed conducting body. Because of this an underwater cable did not only conduct electricity but also stored it.  An insulated underwater cable has capacitance, the ability to store electric charge.

Because of this an electric pulse spreads out as it travels and when received rises gradually to a maximum and then slowly decreases. The longer the cable the more “smeared out” the pulse will become.

So signals sent too close together will get mixed up. So what can we say about the gap to leave between sending pulses so that the can be distinguished at the other end?

Thomson’s analysis obtained what is called the:

Law of squares:  Maximum rate of signalling is inversely proportional to the cable length.

So if the cable length increases from 30 miles to 1500 miles retardation effects become not 50 times worse but the square of 50 namely 2,500 times worse!

Crucial to the eventual success of transmitting over nearly 2000 miles of underwater cable was Thomson’s invention of his mirror galvanometer which was a very sensitive instrument for detecting the tiny and blurred messages coming through the cable.

To finish we have seen that Fourier series can be used to represent a periodic function, for example the square wave function, in terms of the trigonometric functions sine and cosine. We can use this way of decomposing a periodic function to help solve physical problems involving waves and the conduction of heat.

Also the question of what conditions can be imposed on a function so as to ensure that its Fourier series does indeed converge generated much new mathematical activity.

The Fourier series I’ve mentioned were a way of decomposing a periodic function into its component harmonics or frequencies. Indeed the subject is often called harmonic analysis.

There are various ways of generalising this idea including being able to express a function which is not periodic in terms of another function, its Fourier transform, which can be thought of as telling how much the original function oscillates at different frequencies.

These ideas allow the proof of many results in probability, harmonic analysis and number theory. One example from number theory is the theorem that every sufficiently large odd number is the sum of three primes.

There are also many and an increasing number of important practical areas including: acoustics, signal theory, optics, computerized tomography, nuclear magnetic resonance and crystallography.

Slide: Fourier quote

But let me leave the last word to Fourier and his view of mathematical discovery:

The in-depth study of nature is the richest source of mathematical discoveries. By providing investigations with a clear purpose, this study does not only have the advantage of eliminating vague hypotheses and calculations which do not lead us to any deeper understanding; it is, in addition, an assured means of formulating Analysis itself, and of discovering those constituent elements which will make the most important contributions to our knowledge, and which this science of Analysis should always preserve: these fundamental elements are those which appear repeatedly across the whole of the natural world.

Slide: Next lecture on Möbius and his Band

Thank you.

© Professor Raymond Flood, January 2015