Fourier's Series - Professor Raymond Flood

oh well thank you all very much for coming and happy New Year to all of you I think we'll make a start after all you're not here to enjoy yourselves are you well today I want to talk about the famous French mathematician and physicist jean-baptiste Joseph Fourier and some of the consequences of his mathematical investigations into the conduction of heat on a very cold winter's day I think talking about the conduction of heat should be quite a good topic so he derived an equation not surprisingly called the heat equation to describe the conduction of heat but more importantly really an approach to solving that equation the heat equation by use of an infinite series of trigonometric functions now called Fourier series and the ideas he introduced had major applications and other physical problems and also layered the some of the most important mathematical discoveries of the 19th century and some of them we'll be mentioning in my lecture after next and contour and infinity so let me start my overview of the lecture by talking briefly about situation in mathematics and its use in explaining and predicting the world about us at the end of the 18th century well during the 18th century many master mathematicians for example the plas and LaGrange had built on Newtonian mechanics to describe from predictable the motion of bodies on earth and in the heavens in the sky and they're the underlying equations of the motion of bodies indeed they had them in many different formulations and they investigated these equations mathematically however the behavior of heat light electricity and magnetism had yet to yield their governing equations and this lectures are part applying mathematics to one of those four areas namely heat and it was Joseph Fourier who was a distinguished mathematician physicist but also Egyptologist demographer and public servant who tackled the conduction of heat and as I've said and developed the methods for solving that were of lasting importance so let me start by outlining fury as rich and dramatic life and the events of the turbulent times in which he lived centering around the French Revolution then I'll consider his work on the conduction of heat and modus budva at his derivation of the heat equation and his use of trigonometric functions in its solution then I want to look at some case studies involving approximation approximation by trigonometric functions on tide prediction and on the magnetic compass both involving William Thomson Lord Kelvin Kelvin was also involved with the successful transmission of telegraph signals under the underwater transatlantic cable and that will bring us back to the heat equation because the heat equation approximately models and pulses of electricity going through an underwater Telegraph cable and finally I'll finish by looking at some current applications so something about ferry a we've borne burgundy 1768 the son of a master tailor his mother died when he was 9 and his father shortly afterwards however he had an artist anding school career particularly in mathematics under the age of 17 he applied to the military of war to join the artillery or the 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 mutant furrier then decided to enter the church in 1787 joined the Benedictine teaching order and to prepare for his vows and he was also a professor of mathematics for the other novices however the outbreak of the French Revolution and its constantly change Abul developments and factions was a dramatic influence in fury A's life and in the letter the leader from prison and justification of his part in the revolution and Oaks era in 1793 and 1794 Faria describes the growth of his political views he said 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 perd by night studies my health scarcely suffice for the work my position required of me another point of view I will admit frankly that I regarded these events as the customary disturbances of state in which a new server tends to pluck the scepter from his predecessor history will say to what extent this opinion was justified Republican principles still belong to an abstract theory it was not always possible to profess them openly and the partner particularly wanted to share with you after that quote about not being of royal blood even if he were a second year he finished or he continued 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 unto free from this double yolk the long usurped soil of Europe I readily became enamored of this cause in my opinion the greatest and the most beautiful which any nation has ever undertaken and that was written during one of Fury's many spells in prison and on this occasion of fear it appears that was only robbed Spears fall that sea of fury a from the guillotine subsequently Faria went to Paris first of all to study at they call normal however he was again arrested release tree arrested and then following yet another change political rulers became a teacher at they called Polytechnique were in 1797 he succeeded LaGrant in the chair of analysis and mechanics and where he gained a reputation as a night standing lecturer his burgeoning academic career was ended or at least interrupted in 1798 by government orders to join as a scientific adviser the French invasion of Egypt commanded by Napoleon fury returned to France in 1801 and accepted an offer from Napoleon to become prefect of the department of the air in southeastern France he was an effective and well-liked administrator organizing the draining of swamps and supervising the building of the roads across the Alps from Grenoble to Turin and at the same time he was helping to organize the description of Egypt a multi-volume work by the scholars and scientists who were on napoleon's exhibition expedition to Egypt furia wrote the general introduction survey of Egyptian history up to modern times and this worked it much to arise Egyptian interest in Egypt European interest in Egypt and helped establish the subject of Egyptology the other important lasting consequence of the expedition was the discovery of the Rosetta Stone the decree on the stones described three times in hieroglyphic suitable for priestly decree and demotic the native script used for daily purposes and in Greek the language of the administration and it was to provide the key for the deciphering of the hieroglyphics the reason I mention it is the shampo neo who eventually deciphered the rosetta stone was encouraged by Ferrier to take up Egyptology he took him under his wing and in fact protected him from conscription into the army it also it was also while he was prefect that furia began his work at heat conduction FIRREA was unusual among mathematicians very unusual and having a distinguished non mathematical career this work on heat conduction was to fulfill his youthful ambition which he hinted out at a letter written much earlier in 1789 when he wrote yesterday was my 21st birthday at that age Newton and Pascal have already acquired many claims to immortality and I think I write that at every birthday actually but nah in his mid-30s Furai started on the work that was to bring him in more to him his immortality so during three remarkable years 1804 to 1807 he discovered the underlying equations for heat conduction discovered new mathematical methods and techniques for solving these equations and applied his results to various situations and problems and then 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 fury a successful submission for an eighteen eleven prize by the Paris Academy of Sciences the report stated referees report the manner in which the author arrives at these equations is not exempt of difficulties and others analysis to integrate them still leave something to be desired on the score of generality and even rigor and a ludus report and the attitude of the referees Laplace and Lagrange to furious submission it's been severely criticized by many recent mathematicians I think I agree roller with Tom corner when he says in his really brilliant book on Fourier analysis the court from coroner is Laplace and LaGrone which the referees could not see into the future and their doubts are surely more attributed to the originality of fury A's methods than a reproach to mathematicians who fury a greatly respected and Lagrange's case admired so furious main results were published in his 1822 analytic theory of heat this contained the first large-scale mathematize ation of a physical process that was not part of mechanics to one of its major importance in history mathematics finally years of fury a's life were spent in paris where he became part of the academic mainstream by them to the extent of becoming secretary to the mathematical section of the paris academy of sciences and I'll just end this be brief biographical sketch of fury a with a comment by the distinguished historian of mathematics ivor grattan-guinness who died last month and that's a picture of Iver on the right and one of Ivers most important works was early in his career when he wrote on fury a and what he wrote about fury a was the following which does capture the essence of the man because he survived a lot of the political turmoil by the support he had and for the people who risk their own life to speak up for him rather than by political manoeuvring I rewrote he preserved his honor in difficult times and when he died he left behind him a memory of gratitude of those who'd been under his care as well as important problems for scientific colleagues so let me now turn to some of fury A's work in particular his Mian publication the very first words in his analytic theory of heat of 1822 were fundamental causes are not known to us fundamental causes are not known to us but they are subject to simple and constant laws which one can discover by observation and her study is the object of natural philosophy and I want to pick out three things very a achieved first is derivation for the fundamental equation for the deductive heat and the resulting temperature distribution and here we see on the bottom right this is from on the right hand side from a translation English translation of 1878 we see the general equation for the prop of the propagation of heat and the interior of solids and there it's given in three dimensions I'm going to come back and talk a little bit about it in 1 dimensions on the top right hand side is the first example of the use of trigonometric series in the theory of heat and again I'll come back to that later on so that's the first thing he did was they hate equation the second thing that he did was the fury a series that we see in the top right and the third thing was the use of those Furious series in order to be able to solve the heat equation ok so let me first think about temperature in the flow of heat and illustrated with an example as I promised in my last lecture we returned to a beverage base theme last time it was about coffee killing this time is to where to store all that wine that you didn't drink over Christmas so we want to build a cellar and we need to decide on what depth it should be now ideally who to wish that the cellar the temperature should be constant throughout the year or whether it's summer whether it's winter or perhaps more realistically that the temperature fluctuation should be as small as possible as the seasons change now I want to give you first of all some qualitative arguments to show that that might be possible and it's got to do with the passing of time as time passes heat will propagate through the ground in summer when the surface is hot heat will propagate downward and in winter when the surfaces cool heat will propagate upwards but the heat needs time to propagate so that for example when it's the hottest day of the year at the surface it will not be the hottest day in the cellar because there would be a delay as the heat propagates downwards through the grind to the cellar we will see that the temperature in the ground under the cellar there are fears that is they peak at different times if the depth is chosen correctly we might hope then there will be six months so two fears that is peak six months apart meaning then when its hottest out the grind in somewhere at the surface its coolest in the cellar and when it's coolest in the surface at winter its warmest in the cellar so that's due to the delay caused by the time it takes heat to propagate there's another effect which is that when the heat or cold eventually reaches the cellar they were just a smaller change in temperature they're attenuated or they're dumped that's because some heat is absorbed in the grind as a passage three so the temperature difference in the cellar during the year will be less than the temperature difference at the surface during the year right so their quality of arguments to say that the task should be possible but harder you decide at what depth you're going to construct the cellar to achieve your objectives and furious techniques allow is to quantify that and we'll see or a lil astray tour indicate but it varies of course with the type of soil whether it's clay or sand or whatever but various techniques that with a particular type of soil which corresponds to choosing the value of a particular parameter the depth of about four and a half meters the change of fears is six months so when it's somewhere at the surface it's winter in the cellar and vice versa under this step the temperature fluctuation during the entire year in the cellar is 1/16 of what it is at the surface so the surface yearly fluctuation is 32 degrees then in the cellar the temperature fluctuation will only be two degrees so relatively small all right so let's go to work and start by finding the heat equation we're going to do one dimension but the same observational approach that I'm going to make applies in more than not so we need some notation first of all because it impresses people looking at the lecture so we're going to denote I'm going to have one dimension we're just going to move from the ground dine vertically through the ground and what we're going to denote the temperature at a point vertically dying on the ground at depth X at time T by U of X comma T so here is the the notation for temperature now the fundamental observation that we're going to use for getting an equation involving the temperature is the following and it's not actually that's surprising it says that the rate of change of the temperature the rid of change of the temperature our depth x over time is the rate of change is proportional to the flow of heat into our out of X so the rid of change of temperature at a particular point is proportional to the rate of flow of heat into the point more heat going in the faster the Tempur changes and less heat going in the less slowly the temperature changes so we can express this in symbols as follows at the top on the right on the left hand side you again is the temperature and this D you by DT that expression up there that's just the rate of change of temperature with time at a point X so that's the rate of change of day of the temperature over time and what we've got on the right hand side the K is a constant of proportionality depending upon the soil so we'll have a value for it it's got to do the conductive properties of the soy whether it's clay or sand or whatever the other term d 2 u by DX squared well that's the flow of heat into the point and the reason is the flow of heat into the point is that d u by DX tells us as they cheer the flow of heat to do with them distance and what we're looking at is the change of flow of heat so change whenever you see the word change you know that you have to differentiate again which is why you've got the second derivative there okay so that's the equation that we want to solve we want to be able to find an expression for U of X T in terms of X in terms of T that will tell us the temperature at X a depth X at time T and that's the first idea we know how to the heat equation and the observation that I mentioned would show you you know har together than with appropriate notation in two and three dimensions so now the second main thing and that's this idea of Fourier series and what I'm going to do here is give an illustration in fact the very first one that furia gave himself in his analytic theory of heat of being able to represent a particular wave form the square wave form which is in blue here it's the one that square it only oscillates periodically between two fixed values and it's also important these days and digital engineering and I've given the first approximation to it here in fury a series I'll be coming to say later on how we calculate at the fury a stage but at the moment I just want to show you how they work and how are effective they can work with only a few terms so that's the first approximation cosine of U and that's the cosine curve there and it's in red and now I come to the second term ii approximation rather which is cosine of u minus the third the cosine of three years and you can see about the way the approximation is getting better drops died getting better that's just after two terms i always find this very surprising this is after three terms cosine of u minus the third cosine of three year plus a fifth cosine of five years and really the approximation is getting very good indeed after just three trigonometric terms and i only did one more and to bring up the fourth term here and you can see it i think it's really really nice and in fact as i do you find it quite surprising how well the square wave form only taking on two values and oscillating between them periodically same length of time it's so well approximated by the first four terms of a Fourier series okay the numbers here 1/3 1/5 what 1 minus 1/3 plus 1/5 minus the seventh there called the Fourier coefficients in other words they're just the coefficients of the fury a series representing that the periodic waveform and Fourier then went on to consider the more general question of which functions which we've forms could be represented by a Fourier series and also derive formulas involving integrals that allows you to perform dis involving integrals that allows you to calculate to the coefficients of the Fourier series alright and this generated lots of new mathematical activity some of which I'll be mentioning the lecture after next so the first thing was deriving the heat equation second thing was getting these Furi a series in terms of trigonometric functions to approximate your function for example the change of heat at the change of temperature at the surface of the grind but much of his success was really and because of the fact that the heat equation and indeed many of the laws of mathematical physics are are linear and this just means that if you find two solutions to the underlying governing equations then in fact you find many others because for example there there some is also there some is also solution and I've Illustrated this not with the heat equation but with the wave equation every lecture should cook tea and contain a picture of ducks and this is my picture here and what is saying is that the waves caused by the three ducks and each duck will cause a wave pattern on the water and if you want to find the wave pattern from all three of them you just add them together that's what I mean by linearity by superposition so that you can add the solutions together and here I've just given some sort of classic examples with a one-dimensional wave here either Don to this other one-dimensional wave down here and they're completely out of fears so they completely cancel to give you a stationary wave here we have two waves which are related to each other not way there are two fears and they add up together and to reinforce each other so the first one they cancel each other I thought the word I was looking for and here they reinforce each other and here we've got two waves that are slightly out of phase and they combine in that particular way so this principle of linearity is very important because it allows you to build up complicated things from simple component ones and just want to outline essentially how it's done here right so the cool thing here is that if you can find u1 u2 or solutions then so is any linear combination of them so alpha our mind of u1 plus B do mind of u2 will also be a solution right and the basic core ones that we're going to use are these trigonometric functions of sine and cosine so what fury a did was he represented the temperature distribution at depth X at time T in terms of these Fourier series which involves sines and cosines he plugs them into the equations and the equation magically simplifies into a whole lot of simpler equations in the Fourier coefficients which he solves to be able to get what the Fourier coefficients are so the Fourier coefficients drop right he doesn't know them completely because there's constants to be but the constants are determined by the temperature of the surface that's the boundary condition that you have and the fluctuation of temperature the surface they allow them to complete the complete solution and it's because of that that we're able to obtain the quantitative result that I told you a while ago that if you go down four and a half meters let's say then the temperature is going to flow to it very little and be completely out of fears with what it is at the surface so I won't go into the sort of details of that there but you've got the essential things to be able to be created for yourselves I hope so what I want to do now is to go on to some other examples of approximating using trigonometric functions for various other practical circumstances and by somebody who was very influenced by fury a and that is William Thompson Lord Kelvin and I want to look at some of his work and start off with his work on predicting the tides all right and this example the reason I picked it as well is it'll also show us how we can calculate the Fourier coefficients so what are some of the things that I haven't said in the last few minutes I'll now be showing to you in this other concrete example so calculating the Fourier coefficients were able to be able to show how much of each trigonometric sine and cosine to use when we're doing an approximation so first of all let me give a little background about William Thompson so he was born 1824 died 1907 he was a dominant figure in Victorian science contributions to mathematics physics engineering particularly in the areas of electricity and magnetism found in thermodynamics of branch of physics concerned with heat and energy and also instrumental in the laying of the first transatlantic Telegraph cable Thompson was born bail faster though is usually thought of Scottish but I'm always very keen to tell people he's born in Belfast very close to where I was brought up here educated classical and Cambridge University's appointed professor of natural philosophy at Glasgow University at the age of 22 remained at glasgow until his death and was buried in Westminster Abbey alongside Isaac Newton for a modest memorial well Thompson was always a great admirer of fury a a Tom raishin which backed went back to his youth and Thompson tells the following story of when he was 16 on a family holiday to Germany and I thought it was quite an appropriate one to take out he says going out somewhere to Germany with my father and my brothers and sisters I took fury a with me my father took us to Germany and insisted that all work should be left behind so the whole of our time could be given to learning German we went to Frankfurt my father took a house for two months now just two days before leaving Glasgow I got Kellen's Brook Kellan was a professor Glasgow I was shocked to be told that Faria was mostly wrong so I put Faria in my box and used to go to Frankfurt and used in Frankfurt to go down to the cellar surreptitiously everyday to read a bit of fury a when my father discovered it he was not very severe upon me not a problem many people have with teenage children I think I also like the idea of fury I've been secretly read in a cellar so let me give an example of furious influence and Thompson in the area of title prediction right what Thompson did was to design a machine for predicting tides which computed the depths of water over a period of years for any port for which quote the tidal constituents have been fined from harmonic analysis of tide gauge observation observations I'll show you what that means not at calculating the coefficients of the trigonometric series giving the rise and fall of the tide and Thompson used the lovely phrase substituting brass for Brehon when discussing the mechanization of tide protection I'll show you the machine a bit later on so what I want to do is to show you briefly hard to describe the tide introduced a beautiful idea that allows you to calculate the amount of each trigonometric term that appears in that description and then show his tide protection machine and the fundamental underlying cause of the tide is gravitation the tides caused by the gravitational pull of the Sun and the moon on the oceans are not the rotation of the earth but the exact pattern at any particular spot on the coast depends on the ship of the coastline and on the profile of the seafloor underneath nearby so even though the forces that cause the tides are completely understood the tide at any one spot is essentially impossible to calculate theoretically but what we can do is that we can record the height of the tide at that spot over a certain period of time and use these measurements then to predict the tide at that spot in the future and I want to show you how that can be done well the tidal force is governed by a small number of astronomical motions which are themselves periodic but their frequencies are not in a whole number ratio so they don't ever repeat themselves exactly after a certain period of time but there's an almost exact 19-year cycle in the joint pattern of the equinoxes and the solstices and the phases of the Moon so the tidal record repeats itself more or less exactly if you wait long enough for this period of 19 years I'mnot was used in predicting the tides at European port but a mere motivation for kelvins work was predicting the tides in India where they didn't want to wait 19 years for accurate tide productions and because the Sun the moon the earth earth moon Sun are all in relative motion the gravitational pull at a point in the ocean is constantly changed I picked out the five main astronomical periodicity here there's the length of the year the length of the day the lunar month and the fact that the moon's axis changes it processes the axis processes and the plane in which the Marine revolves also processes and when you look at it you see that the gravitational attraction can be described by means of sine functions have here written down some of my fee it's sign functions where the frequencies are increasing as you go down the screen but of course these are also cosine functions if you surf if you move slightly to the right of the left in your seat because sines and cosines are just shifted by a certain amount so these are sine or cosine functions of increasing frequency as you move down the page and what the analysis what the geometry shows you is that the height of the tide at a given place is of the following form it's a sum of 120 on 120-odd trigonometric terms and we know the frequencies v1 that's the multipliers of T we know v1 v2 v3 v4 etc what we do not know are the coefficients we don't know the easier of the a 1 the B 1 VA 2 and the beta and I want to show you how we can find those from a historical record so let me give you I think this is a historical record coming up now this is the weekly record of the tide and the river-clyde at the entrance the queen's glasgow and it's a weekly record so there are one of these curves for each of the seven days so there are seven of these plotting and the observations were obtained by another machine called the tidal gauge that thompson created now 120 things are too awkward to deal with 20 things are too awkward to deal with I want to show you how we do it with just two so here I have on the top left it's a pretend tidal record and it's made up of a composition of two trigonometric terms other point a of sine T and a might be of sine the square root of 2 times T all right and what we have to do is we have to find the a and we have to find the B that's our task we don't know as a says there are moisturise of each we do not know the coefficients a and B and we use the following I think lovely idea and here we have it so we have on the top here take our time over this here we don't know a and we don't know B but they've got some numerical values and that's what gives rise to this this curve up here now what we're going to do is we're going to multiply this expression by sine of T and when you do that you get a sine T sine T well that's the squared term sine squared term and here you have a product of two sine terms with with different frequencies one of them the square root of two the other one is one if you now calculate the long term average of this expression here the B term will average out to give you zero now the reason for that is that if you've got a product of two sine terms that are not the same they can be written as the sum of two other sine terms and if you take the average of a sine term over time let me go back to here if you take any one of these trigonometric functions and look at the average which is finding the average as you're going along because it's the same amount above and below the axis the average is going to tend towards zero so you multiply if you want to find one of the particular terms let me go forward again here we are so we want to find the a term so we multiply by this sine term here we multiply well to find the a term so we multiply across by sine of T this makes this term squared so it's not going to average out to zero and but over here we have sine of the square root of two times T multiplied by the sine of T the intermediary step I haven't written down is that this can be written as the sum of two trigonometric terms and the average of a trigonometric term is zero so doing this multiplication calculating the long-term average gives you what a is similarly you just do it the other way round to find B you multiply both the equation by sine of the square root of two times T and this time it's the a term that averages alright to give you zero and I've written this in red because that's what the red one is down here and when you do the long term average you find out that it's coming to give you the value - and the green one down here when you find when you multiply by sine square root of two times T this is coming along to give you value minus three so the actual values for a and B or a is equal to 2 and B is equal to minus 3 now if you've got the actual tidal record and you have dug beneath a trigonometric series and you're multiplying the two series together there's an awful lot of work in that and I won't show you but one of the things that Thompson with his brother James did was to devise a mechanical integrator for multiplying those two series together and calculating what the long-term average was quite one of the first mechanical integrators and they went on to to develop other mechanical machines for a solution of other very difficult problems as well really quite amazing right so what we've got is we've got a title record from where you want to be able to predict the tides we use this harmonic analyzer in order to obtain or much of each trigonometric term you have in the series and now we want to be able to predict the tide and future and this is where Thompson had the idea travelling on a railway journey to attain the British Association meeting in Brighton in 1872 and if he had been travelling down from North Hampshire today like me he would have had time to have devised many more machines apart from this one here that's just telling you that what I've done could be extended and this is the tide predictor and isn't it beautiful what we have is over here we have wire rope whatever and it's fixed and then it goes over this pulley under that one over this one under that one over this one over that one over this one over that one over this one this is very relaxing over that one over this one and I've got here wit but a natural fact and the real thing it was connected to a pan there was a rotating or a moving piece of paper that the pen was subscribing on no don't at the bottom you've got the handle to crank and port person was the person who turned the handle and the handle is connected up to a gearing system and the gear such that the frequency of oscillation of each of the pulleys the gearing is set so that it would be one of the fundamental frequencies or and corresponding to the appropriate trigonometric term they only want they only did the 11 most important one two three four five six seven eight nine well I only did that many so what you do by the gearing down here this is the frequencies so there are how hardwired into it literally and then the amplitudes how far up and dying these pulleys could move so therefore what contributions they would give they were the coefficients that have been calculated from the tidal record so you turn the handle the pulleys go up and down the end result is the movement over here are not subscribed on a piece of paper and that piece of paper can give the title record for a year ahead and a matter of hours and this was used up to 1944 here's a Kelvin tide protector slightly litter slightly modified version of it and here's the request for a most urgent 1943 note to doodsen from Ferguson the Admiralty superintendent of tides listing the 11 pers of the harmonic constants they're the coefficients of the trig terms for which the request was to prepare the early tide for April through July 1944 to prepare for the d-day landings and this is who's this here this is doodsen up here it was the superintendent of tides and at the liverpool tidal institute to predict the tides for the normandy invasion and was only well it was only when digital computers came and the gold that they and mimicked this and and software so he building upon what fury a and the point there was that wasn't a fury a series it was a you could use the geometry to find out what trigonometric terms to put in put this idea of calculating the coefficients is exactly what you do when you're calculating the coefficient of fury series only you essentially do it by integration there which is this business of finding the average alright now to another just quick one here because again it's such a nice one I've been able to solve a very important problem and it's kelvins magnetic compass and the problem here was that when the ships became made of iron and made of steel while they were being constructed when they were lying being constructed in drydock review constructions and the result of hammering going on and it would pick up the Earth's magnetic field so that they there would be a permanent magnetism within the ship and that permanent magnetism would affect the reading of any magnetic compass that you were using on it so the thing was hardly compensated for the permanent magnetism of the ship and what Kelvin did was to say quite nicely that what you can do is you can say that there is really there's just an earlier time there is a difference between the true compass reading and the displayed reading on the actual compass so what you want to get on what you're actually observing the difference between the two of them is the error term then he made an assumption which can be reasonably justified that the error term again is given by a trigonometric expression a constant term so much of a cosine so much cosine to beta where beta is the displayed having sine beta and two beta and then his resolution for that was that you take the ship when it's constructed and you point the ship in different directions and you observe in different directions that you know so you know what the correct direction is that you're pointing the ship in you read what the compass is saying so that gives you one equation because you can put that into here you know what beta is that's what's being read off the compass but you know what the error is because you know the difference between the true direction and the ER so that gives you one equation for the a 0 a 1 a 2 B 1 B 2 then you point it in another direction and you do the same again point it in other direction so you end up with five equations with five unknowns you're able to find out what the coefficients are and then the last thing was to do the compensation for it and he achieved the compensation by putting in magnets here symmetrically placed this is made of card in order that it it very very freely and the strength of these magnets and the positioning of the magnets are in order to be able to give you the compensation that will negate that error term and the compass theorem a lot of money was adopted by many navies many Admiralty's around the world except for the British which was the one to adopt at last of all I seem to be very very conservative alright so the last real topic I want to mention it's got to do with hard to transmit signals over the transatlantic cable and the laying of the first transatlantic cable underwater sort of the 8th wonder of the world in its day and it's very hard to imagine I mean from our point of view with mobile phones and for communication and with instant access to the web for for for information what a dramatic difference the cable made you know before news took weeks to travel back at well news in a response took weeks to travel back and forth to America alright and the first cable was led in 1858 and a load soon broke down completely it was able to they were able to see how successful how useful would be eventually a working version was laid in 1866 now the shortest direct route is between the south of Ireland here from Valencia Island this is a plaque here in the south west of Ireland to define London here's a sketch of of telegraph hoist at Trinity Bay in defined land unfortunately the shortest return right to be the best part sort of turned out to be suitable in terms of the profile of the sea the surface of the sea on not out there's a plateau which goes along here apparently this line underneath it is giving the depth of the sea floor as you're going along and the other lines here they're to do with the over grind and telegraph networks than in existence and the over grind the lines here the over grind tell Network than existence so we get two types telegraph wires / grind telegraph wires underwater and I now want to consider what Kelvin contributed because a loom laying over 2,000 miles of cable was that challenge and often that's a lovely story to read about perhaps this greatest contribution was to the analysis of our signals propagate over an underwater cable and hard to detect them so let me share that with you just look at the left-hand side first of all we've got a telegraph cable and we're trying to communicate over it and we're going to communicate sending pulses and there'll be some sort of coding in order to interpret the pulses you know Morse code or whatever now the behavior of pulse is going over a telegraph cable and air is very different from what happens in water under water and in earth the transmission of the pulses is approximately and described by the wave equation and the thing about it is that an electrical pulse in a telegraph cable in air will travel with a well-defined speed with no change in shape or magnitude over time there's a nice phrase I've seen describing it's like a bullet going down the wire alright stairs with the same ship and it travels with a defined speed and because of this then the main limitation to the rate at which pulses are transmitted that's going to be the limitation of you're sending straight how many pulses you can send per second and of course it's going to be dependent upon the receiving how many pulses can be distinguished at the receiving end per second but that's the main limitation on the rate at which you can send the pulses on the other hand for a cable underwater the situation is very different and data and the behavior of a pulse being sent down a cable underwater is more akin approx described by the heat equation and indeed Thompson coming back to fury a based his analysis upon the work that fury ated upon the heat equation and one of the reasons for this is that an underground cable it has to be insulated from water otherwise the signal we just leak away and the water is an earth conductor and what happens is that the pulse goes run to pass are the pulse goes down the cable the underwater cables not only conduct the electricity but it also stores it as its passing so an insulated underground cable has got what they call capacitance the ability to store electric charge and because of this an electric pulse spreads artists it travels and when it's received it gradually Rises the pulse gradually Rises and then gradually and decreases and the longer the cable the more smeared art the pulse will become I'd like to show you that knowledge you I find it helpful to think of its if between you and me there's a long metal bar you're going to be the receiver I'm going to be the sender it's a long metal bar I'm not going to use electric to send it I'm going to use a blue torch at the end so I'm going to play a blowtorch along my end of it and then the heat is going to go along the bar to your end and you can see that it's right at your end it's very smeared out it will arrive slowly and when will the crest of the wave actually arrive and when should I send the next pulse by playing a blowtorch against the end of it again and well part of what's happening is that the metal bar is storing heat as it goes along so the pulse of heat is getting smeared out and to a large extent that's what's a good analogy and Thompson loved analogies and so you might be proud of this one and of the analogy between electrical pulses and an underwater cable and heat traveling law bar but he analyzed it quantities quantitatively and he came up with his law of squirt so our difficulty is her orphan can we send a pulse so they can be detected so they can be deciphered at the other end distinguished at the other end and he find out it could be described by law of squares and this just shows you our difficult the challenge was facing them that if the cable length increases from 30 miles say across the channel which was a cable that was led pretty early on to 1500 miles which is not quite the distance across the Atlantic well to go from 30 to 1500 that's an increase of 50 that's why I chose it because of DVT divided and but the retardation effects don't become 50 times as bad they become the square of 50 they become 2,500 times as bad which is a really dramatic degradation in the problem of receiving the signal at the other end and crucial really to the eventual success of transmitting over nearly 2,000 miles of underwater cable was Thompson's invention of what's called the mirror galvanometer which was a very sensitive instrument for measuring currents it used the fact that the current we go through a little coil a mirror was attached to the coil and the light was shined upon the mirror whenever the coil moved that beam of light would be moved as well it was very sensitive very delicate and it was able to detect the tiny and the blurred messages coming through the cable so didn't clue then really I hope you've seen is various series can be used to represent a periodic function for example the square wave function in terms of trigonometric functions science and cosine and we've used it to help solve physical problems involving waves and the conduction of heat and to say the question of what conditions can be opposed on a function so as to ensure this various series does indeed converge generated a lot of new activity now the Furious series have mentioned we're a way of decomposing a periodic function into its component harmonics or frequencies indeed the subject sometimes called harmonic analysis now the various ways of generally at slicing this idea and there's one of which which is looking at functions which are not periodic the ones that we considered up tune up in periodic ones which are not periodic and representing and and creating something called the Fourier transform and that can be thought of as telling how much the original function has how much it oscillates at all the different frequencies but it didn't want to to go into that but it's a very powerful current mathematical idea with replications and diverse areas probability theory harmonic analysis itself number theory strangely enough and it was used for example to prove the important result that every sufficiently large odd number is the sum of three primes and if you look back to one of my other lectures number theory where I talk about this difficulty of being able to say the difficulty arises because of the very different nature in numbers the additive structure and the multiplicative structure are very hard to relate to each other so this is saying telling us something about our number can be written out in terms of primes using this idea of Fourier transform and then more practical areas acoustic signal theory optics computerized tomography nuclear magnetic resonance all of these imaging techniques and crystallography are very much into and seemed to me inherently have to use ideas from Fourier transform theory but let me leave the last word to Fourier and his view of mathematical discovery and he said 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 on which the science of analysis should always preserve these fundamental elements are those which appear repeatedly across the whole of the natural world and what I hope to do today was to give you some practical illustrations of where you can use mathematics to make that kind of difference so next time come along will be pleased do come along and we'll be talking about Mobius news bond and bring your mobius strip with you and we can all cut it up and in different ways and if you enjoy today's lectures I often say they do recommend to a friend and if you didn't then recommend to somebody you just like thank you

Channel: Gresham College

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Id: 3bQz6k2FRu4

Length: 53min 7sec (3187 seconds)

Published: Tue Jan 27 2015