The beautiful sine wave turns out to have a huge number of practical applications, from the motion of springs, to waves in the sea, to sound waves, light waves and more. It is curious that the function which defines the sine wave, sin(x), comes from comparing the lengths of sides in right-angled triangles – just about the least curvy things you could imagine.
How does that concept result in the lovely curve of the sine wave?
A lecture by Sarah Hart
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- So today, I want to talk to you about the incredible sine wave, and its uses. We've all seen a sine wave in our lives, I'm sure, but we also all learned at school that sine is opposite over hypotenuse, right? So that is associating to some angle in a right-angled triangle a ratio of lengths, a number that must be between zero and one, so how does that very un-curvy thing, a number associated to a triangle, end up being related to this very curvy thing that goes on forever in both directions and takes both positive and negative values, how is that? Well, I'm going to give you today a brief history of sine to get us from triangles to sine waves, and look at some of their huge range of uses. Along the way, we'll encounter banned books, archery, and the workings of the human ear.So let's get started:
the ancient Greeks did not have sin, cos, and tan, so how does it happen that Hipparchus, the ancient Greek astronomer and mathematician, is called the father of trigonometry? Well he didn't use sin, cos, and tan, but he did work with triangles, and I want to just briefly show you a little bit about what he did. He used chords, he worked with chords of circles, so lines joining places on the circumference of circles, in astronomical calculations. He did lots of amazing things in astronomy, he measured the procession of equinoxes, he calculated an estimated distance from the Earth to the Moon, he said it was between 59 and 67 Earth radii, which is not bad, the true value is around about 60 Earth radii. He did that using estimates for the parallax of the Sun, which I'm not going to go into, but just to give you an illustration of how chords of circles could be used in astronomy, I'll just show you how you could use them to work out the distance from the Earth to the Moon. It isn't Hipparchus' method, but it's a method that can be used, and has been used. So imagine there's the Earth, there's the Moon, I'm going to draw in some lines here, and what you can do is you and your friend can stand as far away as possible from each other on the surface of the Earth and look at the Moon, and look at its position in the sky, and you can compare it to something that's so far away that it looks like it's in the same place in the sky to everybody on Earth, so a distant star will look fixed in the same position for everyone, but the Moon will be at a slightly different inclination, a slightly different angle. So you can measure the angles, and then compare notes with your friend, and that gives you the angles in this diagram here, and you can then work out, I guess the height of that big isosceles triangle, and that tells you the distance to the Moon. You can work that out as long as you know the base, and what's the base of that triangle, that red line there? Well that's a chord of a circle. If you slice through the Earth, you get a circle, ish, and that's a chord. So knowing the length of that chord, being able to work that out, is a useful thing in astronomy. I've just thrown away the extraneous parts of that diagram, so this is the kind of diagram that we're interested in. And for astronomical applications, like the ones Hipparchus was working with, this is the kind of thing we want to know, you want to know the length of that red line, the chord, and the yellow lines there, each one of those is a radius of that circle, so this is why chord lengths were interesting. And later on, actually, it turned out that maybe half chord lengths were the things you really wanted to know about,and here's why:
if instead of being the Earth, that circle is now the whole celestial sphere, or at least a slice through it, actually, you can't see all of that at night when you're doing your astronomy because the horizon gets in the way, there's the horizon. So if you want to find out something like the height of a star above the horizon in the celestial sphere, it's actually the half chord length that you might want to know. That's a more convenient thing because lots of the calculations want the half chord length, but even if you then do want the full chord length, of course you just double, and doubling is easier than halving, so if you're going to write a table of things, maybe half chord lengths are what you want. So we can see now the link between chords and the sine function, because if I just zoom in on the important bit of that diagram, you see, look, there's that horizon, the horizontal diameter, you've got the radii of the circle, radius r, let's say, the chord is there in red, and that half chord length I've called L. We know in a right-angled triangle, which we've got here, the sine of the angle theta there is opposite over hypotenuse, so it's L over r. And what happened? So the ancient Greeks were looking at chord lengths, over in India a few centuries later, they started out looking at half chord lengths, and then they looked at the ratio of the half chord length to the radius of the circle, and that is how we came to have sin-theta So it started with chords, ended up with sin-theta, and cos-theta is where you can take the cos of the angle, that would be just the horizontal, the width there, divided by the radius. If you are trying to gather together information, I'm going to write down all the chord lengths I need to know, when you're doing that, the chord length depends on the angle, but also on the radius of the circle that you're in, whereas the sine of the angle, that's already a ratio, you don't need to know the radius of the circle ahead of time. If you then want to know the actual chord length, you multiply by the radius of the circle. So somehow, it's more convenient to have the sine function, because, then, if you were trying to tabulate and get a list of values, if you're doing chord lengths, it's implicit that somehow you have to make an assumption about the radius of your circle when you're doing it, so your list of values has a fixed radius in mind. You can then scale it up or down if you want to, but the sine function doesn't have that disadvantage, so that's probably why sine won out over chord lengths in the end. So yeah, that's why, 'cause you just have to know the angle to find out the sine of the angle there, you don't need to know the radius to begin with. This diagram also tells us how sine got its name, because, well, I'll show you this diagram superimposed on a photograph right now just to show you, this is the archery link. That picture that we just saw, if you imagine the arc of the circle is now the bow, and the bowstring would hang vertically down, but when you draw it, when you pull it back like that, I'm standing with my bow, like that, you get the radii of the circle, and then the horizontal part, which was our diameter, that's the arrow, or part of your arm. So the whole diagram looks like a bow and arrow being drawn, and the bit we're interested in, the chord length, is the bowstring, so the Sanskrit word for bowstring is what was used for sine originally. Now, when those texts got translated into Arabic, as the knowledge passed through the ancient world, they transliterated that word, they didn't try and translate bowstring, they just transliterated and wrote down the letters that would make the same sound, and so we got, in Arabic, jiba, something like this. That's not a word in Arabic, so then as these things got recopied, the scribes would sort of go to the nearest word, which was jayb, and that means cavity. And so the final step is that when, in turn, those Arabic texts got translated into Latin, they translated that word cavity into the Latin word for cavity, which is sinus, and that's how we get like, we all have sinuses, right, that's how we get this word sinus, and then in English it becomes sine. So that just tiny, little vignette shows us something about how mathematical knowledge is transferred, from the etymology of that word, I think it's fascinating. But we're not here to talk about that, we're here to talk about sine, so let's do so. We can see from the origin story of sine how it is that we can get sine of angles that are bigger than 90 degrees, that don't occur in right-angled triangles,because we can think of it like this:
what is the sine of that angle theta telling us? Well let's take a unit circle, a circle of radius one, then sin-theta is just telling us exactly the height above the horizon, of the horizontal diameter, the height of that point on the circumference. So we can imagine taking a radius, and just rotating it around the circle, and inside a circle of radius one, the height of where the radius meets the circumference will exactly be sin-theta. And so we can find sine of zero very easily, that's when the radius is just flat on the horizontal, so of course sine of zero must be zero, it's at height zero above the horizontal. And as we go round, and round, and round, we get to the vertical, and when it's vertical, so 90 degrees, the height is exactly one radius, and this is a circle of radius one, so it's exactly one, so sine of 90 is 1. And then we can keep going and going, there's nothing to stop us, and the height starts decreasing again as we get back down to the horizontal, so at 180 degrees, sine of 180 will also be 0, 'cause it's zero height above the horizontal. And we can keep going, my arm can't, but we can, and then we go underneath the horizontal, so our height above it becomes negative, we have negative heights, so that's when sine can take negative values all the way down to -1 at 270 degrees, and then back upwards again, ad we get back to zero. So this origin of how we came to use the sine function tells us how to get sine of angles other than those strictly between 0 and 90 in a very natural way. I'm talking there in degrees, you've noticed, that's because, for most of us, that's the first kind of measurement of angles that we learn, and I will mostly talk in degrees for that today, but I've used that word mundane. What I really mean by that, 360 degrees in a circle is a very Earth-centered measurement unit. Dividing circles into 360 equal parts is done on the planet Earth because early astronomers noticed there were about 360 days in a year, that's the reason we have 360 degrees in a circle. So that's not a very mathematical reason, I'm fine with it, we love all the history, but actually, mathematically speaking, there's a slightly better, a way that makes sense more mathematically of measuring angles. So I just thought I'd mention that to you because that allows me to accidentally say that instead of degrees and not get into trouble. But this is what you will start using if study mathematics at university or beyond, you tend to switch to this other way of measuring angles. The other way of measuring an angle is to say, how far around the circumference have I traveled, has this point gone, in this circle of radius one. The circumference of a circle of radius r, two-pi-r, so if the radius, r, equals one, then the whole way around the circumference is a distance of two-pi. So we will say our unit of measuring angles is something called the radian, it's just the distance traveled around the circumference in the unit circle corresponding to that angle. So if you travel the whole 360 degrees, then you've traveled a distance of two-pi, therefore that angle is two-pi radians. If you only travel a quarter of the way around, 90 degrees, a quarter of two-pi traveled, so 1/2 pi radians, that's a right angle. So that gives you a bit information, it's mathematically a nice unit to use for angles because if you have your angle in radians, then the angle itself tells you how far you've traveled around the circumference, sine of the angle tells you your height above the horizontal, and actually, cosine of the angle tells you how far to the right of the vertical line you are. So if you're plotting these as coordinates, you'd have the coordinates of that point and you'd have the distance traveled just by knowing the angle, which is quite nice. And this also gives us our sine curve, because you can draw, and we'll do it in a moment, if you take your angle, the picture in a minute has it measured in radians, so it goes from zero to two-pi, a full circle, and you just work up the angles, and the radius travels around the circle, and what we're going to plot on our graph is just the height above the horizontal, I guess it will now be the x axis, the height of that point above the x axis, the horizontal. And you'll see when we do that, there's it going from zero all the way up to two-pi, that's exactly what happened. Just follow it around, it'll happen again in a moment, we just see that point going around the circle, it's almost like we're unrolling the circle onto a graph. It's going to go again now, we're just plotting the height, and there we are, so that is our sine curve. And actually, once you start doing it like this, you don't have to stop when you get back to the beginning, you could keep going if you like, and that would mean the graph can carry on going, and going, and going, and you can get just these nice, infinite undulations of the sine curve. You could even change direction and go backwards, and that would give you negative angles, so it can carry on infinitely in both directions. So that is where we get the sine curve from, and it's to do with the way of thinking of sine as being a height in a circle. So we've said the beginnings of the study of what we now call trigonometry were really chord lengths to do with astronomical calculations, and there are loads and loads of applications throughout the centuries of these trigonometrical functions. It took a while for things to settle down for them to be fixed on sin, cos, and tan, and one over sine, and all over those, but throughout this time, many, many applications. This is, for example, this is a picture of astronomers working at the famous Istanbul Observatory at the time of the Ottoman Empire, all of their mathematical instruments, they were needing to know about sines and coses. Another application, another use of trigonometrical functions is in navigation. This ship, by the way, it's a lovely picture, it hasn't got any particular mathematical importance on its own, this ship, it's just a nice picture of a ship whose name I like, because the name of this ship is the Peter Pomegranate, which I think is a great name for a ship, it was built in 1510 for Henry Tudor's navy, but this is just a picture to illustrate navigation requires the use of triangles, and therefore trigonometry, and sines, and cosines, and things like this. I should say, the sine curve and the cosine curve are the same curve, just one is shifted along a bit, because just as you can measure the height of that radius above the horizontal, you could equally measure the width of it from the vertical, and you'd get exactly the same graph, but just shifted by 90 degrees, if you just think about rotating that whole picture 90 degrees, that's what you'd get, so sine curves and cosine curves are the same things, just shifted a bit. Why might you use trigonometry in navigation? I'll just give you a silly, little example. Suppose you set off from your home port, and you have to decide on a direction. You've got some maps, so you're heading off somewhere, so you set your bearing, you've got an angle, and off you go in that direction, and then at some point, there's a storm or something, your ship gets damaged, and you have to say, we're going to have to pull in at the nearest port, so where is that? So you look at your charts, and you say, there's an island nearby, let's head there, what direction, what bearing are you going to go in? And to do that calculation, you know the initial direction you set out in, and you know what direction you would have set out from home in order to reach this island going directly, so you've got two sides of this triangle, and the angle there, so you've got two sides of a triangle and the angle between them, and there's a law of trigonometry called the cosine rule that tells you how, then, to calculate the rest of the triangle. So you can work out the other angles, the other side, and then you've got your bearing. So that's one use of trigonometry in navigation, we've got uses in astronomy, another really important application is in surveying, the technique called triangulation. It sounds like it will involve triangles, and it does. In that situation, suppose you want to make a map of your country, we haven't got a nice, flat, I could make a map of this stage very easily because I could just pace it out, or measure it, lay flat things out and measure my distances, all easy, but countries tend to have annoying things like mountain ranges, and forests, and rivers, and lakes, and swamps, and all sorts of things that get in the way of you just pacing out nicely your distances, so what you do in that case is you use triangulation. You measure out very, very accurately an initial line, which you do have to use with chains, or something like this, measure it out very carefully. So you've got this initial line, which is going to be the base of a triangle, and you know that length as exactly as you can. Then you send your associate a few miles away, maybe they go and stand on the top of a nearby hill, or something, so then they are standing at the third apex, the third vertex of a triangle, and you measure the angles then very carefully in that triangle, so you don't have to be trudging over your great, big theodolite up and down hills and through marshes, you're just looking and measuring the angles. So now, in this triangle, you've got your base line you know, and you've got two angles, well you can work out the third for free 'cause angles in a triangle add up to 180 degrees, or pi radians, and then you use a magic bit of trigonometry, the sine rule, and this tells you there's a lovely relationship between these ratios of side length to sine of opposite angle. And in our situation, we know all the angles, so we can look up the sines of those angles, it's pre-calculators, or anything, you have to have a little table, and you'd look them up, and then you can use that relationship, the sine rule, to work out the missing lengths. So now you've got all the lengths of this triangle, that's great, you've got those distances, those two distances that you now know can now in turn be the bases of new triangles, and so you can put little triangles all over your country and you can measure out all the lengths, and that's how you do your surveying and map making. You may have noticed I said angles in a triangle add up to 180 degrees, well they do, not disagreeing with that, but they only do on a flat surface, on a plane. When your triangles are small enough compared to the vast size of the planet Earth, then that's absolutely fine, but when your triangles get very big, there starts to be an issue. This picture here, at the bottom of it, if you can read the writing, it says "Calcutta base line." This is a drawing representing the Great Trigonometric Survey of India, which was carried out in the 19th century. It took over 70 years, I think, it finished in about 1878, surveying and mapping the whole of India very accurately. The triangles there were so big that the curvature of the Earth had to be taken into account, the angles in a spherical triangle add up to more than 180 degrees. Luckily, that's not an issue, there's a modified version of the sine rule, we know by how much the angle will exceed 180 degrees the sum of the angles, it's proportional to the area of the triangle, so that's not a problem, but you just have to take that into account. Spherical trigonometry is another kind of trigonometry made with triangles on the surface of spheres. So these many, many applications, and others I haven't mentioned, all mean that we have needed the technology to be able to work with these triangles that we encounter in all these applications, and that means it became very, very important to get good lists, or tables, of, given an angle, what's the sine of it, cosine of it, and vice versa, given a sine or cosine, can we go backwards and say what the angle was. This is absolutely pressing, it's vitally important, life-saving work to be able to get these things correctly right, because if you're navigating, and you look at your table, and you get the wrong answer, lives could be lost, so it's important. I just want to mention a couple of people in this story. Regiomontanus is one, there's a picture of him looking at, that's an astrolabe, isn't it, he's looking very cleverly at an astrolabe, he wrote one of the first trigonometry textbooks in Europe, "De Triangulis Omnimodis," "Every Kind of Triangle," triangles for dummies. He says, not quite, I'm sure it was very clever and wonderful people, he says, "You who wish to study great and wonderful things, "who wonder about the movement of the stars, "must read these theorems about triangles," it's absolutely vital, you have to understand about triangles if you're going to be able to do all this stuff. He produced an early book of trigonometric tables, lists of values of these functions. In case you're worrying about the date there, it was posthumously printed, so it's not a typo, it came out in 1490. And others also produced tables like this, I want to mention Georg Rheticus. His tables were extremely precise, but even better than that, they were extremely accurate. The first version, a shorter version came out in 1551, it didn't quite use the standard sin, cos, and tan that we use today, but what it used was equivalent, you could deduce one from the other. And I just wanted to show you, I don't know if you can see here, I won't go into detail about what it all means, but the length of the numbers that are shown there, these are subdivisions of the circle into very, very tiny, little increments, and the number of places, the number of significant figures given is very high, and this is back in 1551, so this is very, very, very accurate tables. You can see by looking at this that it's extremely salacious, and it was banned by the Catholic Church.I jest:
it was banned, but it probably wasn't because trigonometry was so sinful, or anything, but actually, more likely, it's because Rheticus is famous for something else other than these amazing tables he produced, and that is that he was the only pupil, and the really important person in the popularization of the work of Copernicus, and Copernicus was not in great favor at that time with the Catholic Church, and so probably by association, Rheticus, oh he's that Copernicus guy, he's the pupil of Copernicus, he can't be up to any good. So this "Canon Doctrinae Triangulorum" was banned, but then a much longer version came out, hundreds of pages, a bit later on, and that wasn't banned, so it's all right now, we're allowed to look at this without feeling like our souls might be in danger. And these tables are so, so accurate that they were used for hundreds of years. So that's all great, we know we need these things to do our navigation, our astronomy, our surveying, how do you actually make one of these tables? We can't get our calculator out and just go, what's sine of 36.725, how on earth does Rheticus produce a table like this? In making this new technology, new mathematics was required, and tables could be produced using some rules that were found, new lamps for old, there are ways of finding out the sines and coses of angles based on things you already know. So I've just put a couple up there, so if you know sin and cos of an angle x, then there are formulae that tell you how to find those values for 1/2 x. If you know the sin and 'cause of angle a and an angle b, then there are formulae that tell you how to find the sin and cos of a plus b and a minus b, so you can do sums and differences, and there are other things as well. So that's all very well, but you need somewhere to start, and we can start with triangles we know. Here's a couple of triangles we know, collectively. There's an equilateral triangle there, you can use Pythagoras' theorem to find the height of that triangle, and that gives you a right-angled triangle with angles 60 and 30, we know all the lengths there, so we can find sin and cos of those angles. Regular pentagon, we've known ever since ancient Greek times the diagonal of a regular pentagon, if it has side one, the diagonal is precisely the golden number, phi, one plus root-five, over two, so we know the diagonal, we can then use Pythagoras again to find the height, and that's another right-angled triangle, all of whose sides we know, and we therefore can find the functions for 72, and I guess 36 is on top; maybe not 36, got to add them up to 180. So we've got 60 and 72, and we've got a difference formula, if we know sine of a and sine of b, we can find sine of a minus b, so we've got 60 and 72, that enables us to find the sine of 12, and then we have half angle formulae, so if we know the sine of 12, we can find the sine of six, and we can do that again and find the sine of three. So this kind of consideration allows you to get a long way and find a lot of these functions for smaller and smaller angles. So that's a hint of how these tables were produced, it's still an amazing feat of calculation and mathematical achievement. We're going to get onto uses of sine waves in a moment, but before we leave talking about tables, I just want to mention that there was a really clever use of this technology that had been developed. I think of these tables, this is state of the art technology, this is like having your supercomputer, here is your book that tells you all of these things, and you can calculate with them. And this technology that was already there was given an alternative use at the end of the 16th century to solve a completely different problem, and it's the problem of multiplying long numbers. This is hard to do. Adding is easy, subtracting is easy, multiplying is harder, and it takes more time. At some point, logarithms were discovered/invented, and people started using log tables, and slide rules, and things like that, and then obviously, nowadays, we don't tend to use that, we would use calculators or computers. But before logs came along, logarithms, which turn multiplication into addition in a way that I won't go into right now, a different technology was invented, and it made use of things that already existed, these tables that we already had. And it's got a brilliant name, prosthaphaeresis, which kind of means adding and taking away. And it uses this nice, little fact, which is that there's this formula that turns a product of two cosines into a sum, or I guess, strictly speaking, it's an average, isn't it, add them up and divide by two. And we know addition is easier, we can halve things, that's not a problem, adding is easier than multiplication, so this is, before logarithms, a way to turn multiplication into addition. It's a little bit more involved than what we might do later on, but I just want to show you a tiny, little toy example, because I think this is a really great way of, somehow it's using the existing infrastructure to solve a problem. So let me just give you a very tiny, little toy example, let's multiply 123 by 456. So what you do is you scale, you divide by a suitable power of 10 to create two numbers that are between zero and one, so here, A is 0.123, B is 0.456. Because they're between zero and one, both of those are cos of something, so you get out your trig tables, and you find x and y such that cos of x is A and cos of y is B. So now, that formula at the top is going to tell you what's A times B in terms of adding some things together. You've got your x and y, so you find x plus y, you find x minus y. Done, addition is easy, and then all you need to do is take the average of those things, so add them up, divide by two, again, we're just adding, very easy, and you get the answer 0.056088, and that is your A times B.All we have to do now is scale back up:
we divided by 1,000 twice, we've got to multiply by 1,000 twice, and we find that 123 times 456 is 56,088, and we've used our trig tables to get away from having to do that difficult multiplication, and that can be a really useful thing if you're multiplying very, very long numbers together. It may be that you don't get the exact answer because you might have to approximate slightly, but you get a good enough approximate answer for your calculations. So it's a very clever use of these tables before logarithms came along, which I quite like. So we've seen that sine, and its friends cosine and also tan, are useful, there are really, really useful, important practical applications, and that is why all this technology developed, and people got very, very good at working with these things. But we haven't yet used, or seen this nice curve that can go on forever, all of these angles that we're using, they're angles within the circle. But a bit after this point, we got to the stage where mathematicians and scientists could start understanding and solving some physical problems that involve oscillation and periodic motion, where you get some motion, and I'll show you actually, straightaway, you get some motion that repeats. This is very soothing, a very soothing, little animation there of a pendulum swinging backwards and forwards, and its path is being traced below, and we're moving the graph paper along underneath so you can see that the shape that's being produced is a sine wave. And those kinds of situations sometimes give you a sine wave as a solution, or a sine function as a solution. In particular, the kind of situation where you get a sine wave is where you've got a force, like a resulting force, you cancel everything out, and the net effect is a force that's pulling you towards an equilibrium position. So what I've got here is a pendulum swinging backwards and forwards, so it's got rigid rod and then a bob, or weight at the end, and it's just swinging backwards and forwards in arcs of circles in a very soothing way, and what happens there is there's a force downwards due to gravity, but the part of it that's trying to pull that bob away from the center of the circle is exactly counteracted by the tension in the rod. So the distance from the center of the circle can't change, so that bit of the force due to gravity is canceled out, and the remaining bit of the force, all it can do is pull downwards, but along the circumference of the circle. And the steeper the angle you get, the more proportion of that vertical gravity force is along that circumference. The other thing that you require in order to get a sine function out of this is that the force that's being pulled downwards towards the equilibrium position, which in this case is just the pendulum vertically downwards at rest, that force should be proportional to the distance from the equilibrium position. And if you look at the equations you get from a pendulum, then for small oscillations, for small angles of oscillation, that's basically true, you have the make the odd approximation here and there, but that's basically what happens, and so you get this sine wave coming out of it, it's all squashed into a vertical, but that's what you get. And in particular, if you set this thing going at some time, t equals zero, then the displacement after some time, t, will depend on the sine of t. There are other situations where you get precisely this same thing happening, and I want to show you just one other of them, because it was something due to a predecessor of mine as Gresham professor of geometry, Robert Hooke, and this is the picture of him that's hanging in Gresham College now. It wasn't painted at the time of his life, I'm not able to see that there are any reliably, provably accurate pictures of Hooke painted during his lifetime, but this is the one that's hanging up at Gresham now, and he's holding in his hand the thing that I want to talk about, which is a spring. And we may all remember having learned at school Hooke's Law, and here it is, I'm sure you recognize it, 1676. This is a jumble of letters. This is Hooke's Law, but at the time, people often did things like this, he actually published an anagram of his law, I'm sure you're quickly solving this right now, and getting the Latin phrase, "Ut tensio, sic vis," as the extension, so the force. This was a thing done to prevent, or try and prevent priority disputes. He's thought of this great, new law, if he just publishes it, what's to stop someone else coming up and saying, I thought of that last week, look, here's my bit of parchment with the date written on, last Tuesday, I thought of it first. If you don't want that kind of nuisance, and maybe they did think of it first, it's very hard to then prove who got there first, so what you would do would be you'd publish something, or in a letter to a friend, or something like this, you'd put some encrypted version of what you wanted to say, so Hooke's just put the letters in alphabetical order, hasn't he, there, and then later on, you wait a bit, if no one comes up and says, ah, he must mean, "Ut tensio, sic vis," then you're safe to reveal your amazing discovery. So that's Hooke's Law, what it actually means is we've got a spring now, it's going to bounce up and down quite quickly, it's very energetic, there it is oscillating away. So what happens here, and what Hooke discovered, is that, take a spring, you put a weight on it, if you just let it go, then the tension in the spring increases with the extension, with how stretched it is, the force of gravity is pulling downwards, the tension in the spring is pulling upwards, there'll be an equilibrium position where they exactly balance out. If you disturb it from that equilibrium position at some time, t, then if you pull it down below the equilibrium position, then the tension in the spring is higher than the force of gravity that's acting downwards, so it gets pulled upwards, and the amount of force is proportional to how far you are from the equilibrium position. On the other hand, if you lift it above that equilibrium position, this time, the force of gravity outweighs the tension in the spring that's pulling you up, and so you get pulled down, and again, proportionally to your distance from the equilibrium point. And so because of that, when you solve that equation that you get, the answer is that the displacement from the equilibrium position as this boings up and down is going to depend on, not the time, but sine of time, t. So these are two places where you genuinely do get a sine function. There are lots of other sine waves in physical phenomena. Light is one, light waves are sine waves, so that's anything from light, X-rays, radio waves, all of these things are sine waves, and it's important to understand the sine function and the sine wave in order to be able to understand phenomena like refraction, interference patterns, those all rely on the properties of sine waves. Other kinds of waves, though, may or may not be sine waves, so you can get periodic things, repeating patterns, which might or might not themselves be sine waves. An example is water waves. When the wind blows water in the ocean and creates waves, there is a lot going on in that situation, a lot of confounding factors, the temperature of the water can have an effect, the depth of the seabed, which of course is constantly changing, can have an effect, the strength of the wind, I don't know, passing whales swimming by, all sorts of things can affect what's going on here in these water waves or ripples. And experiments have been done that say when the amplitude, the height of the wave is small in comparison with the wave length, then, actually, they are quite like sine waves, nice periodic things, but as the amplitude of the wave gets larger, and becomes big compared to the wave length, then you don't get things that quite look like sine waves, they have more crests and peaks like that, and they get closer to a curve that we've briefly mentioned in a previous lecture, actually, a trochoid, it isn't, but it looks a bit like semicircles upside-down. Those things, then, they aren't sine waves. So is all lost, is that the end of sine waves? No, no, because these things are periodic, they repeat, and because of the next guy, that means there's a link to sine waves. The next guy is Joseph Fourier, and it's exactly 200 years this year since his world-changing book on heat flow, "The Analytic Theory of Heat," it came out in 1822, 200 years ago. He was working on the heat equation which governs how waves of heat pass through objects. To do this, he did something very clever with sine waves, and before I tell you what that is, I just want to give you a few sine waves to look at, because we can take our standard sine wave, there it is, y equals sin-x, from 0 to 2-pi, aka 360 degrees, it goes from 1 to -1, that's your sine wave, but you can tweak those a little bit, you can scale them. Here's one way of scaling it, you can just double the height, you've stretched it vertically. That's the same as me starting with a circle of radius, not one, but two. So you can make one of double the height, it's a sine wave, we say it's sinusoidal, it's like a sine wave, just tweaked a bit. Another thing you could do is you can make your radius, you can stick with a unit circle and just make the radius rotate at a different speed, so then you might get something like this one, sine of two-x, so you're doubling the angle, that means you go around twice as fast, and that means you fit twice as many waves into the same space, so it gets squashed, concertinaed in the horizontal direction. So those are both fine, they're sine waves. You can then combine them if you like, you can combine sine waves, so you can add them together. Here, for example, is the red one and the blue one, I've just now made a graph of the sum of those two, so I've just added those points together, sin-2x plus 2-sin-x, and I get this thing, which is definitely not a sine wave, it's got a bit of a kink in the middle of it, but it is periodic, it does repeat every two-pi, because those things that made it up repeat as well. So this is a periodic wave that's not a sine wave, but is somehow made of sine waves. What Fourier did, it's pretty amazing, there's an asterisk, which I will come back to a second, in this statement, for the mathematicians in the audience, Fourier showed that any periodic function is made up sine waves, you can find sine waves to add together to make any periodic function, and we know how to do it, that's the crucial thing. It's all very well having a theoretical thing that says, but there is a standard technique, it involves a bit of calculus, that allows you to pull out the exact sine waves that make up your periodic function. I'll show you some examples in a second. This asterisk is for the benefit of any mathematicians who are watching, who are saying, Sarah, do not lie to the good folk in your audience. There are some really weird periodic functions where this kind of doesn't work. There's something called the Dirichlet conditions, which you can go and look up if you like, that specify exactly when it's possible. It's almost always possible for any kind of function you would encounter in your everyday scientific experiments, but there are some really weird, we call them pathological functions in mathematics, you can define things on purpose, just to be mean, that can't be treated in this way. I'm thinking of the kind of function where it's valued at one on all rational numbers and zero on all irrational numbers, something like that. So okay, small asterisk which I now propose to forget all about. Let's see an example. This thing on the top here, this is what's called a Fourier series, so it's a sum of lots of different sine waves. In theory, there's infinitely many of them, but we'll do a few terms and we'll see what we think is happening. So we're just going to start adding, and you can see the pattern here, you start with sin-x, and then a 1/3-sin-3x, 1/5-sin-5x, one, three, five, seven, nine, dot, dot, dot, all the odd numbers. The first term of the series is just sin-x, there's my sine curve, then I'm going to start adding in the other terms until we think we know what's going on. So I'll add in next 1/3-sin-3x, so I've scaled it and I've squashed it. So I've added that in, and now we get this weird, jelly-like shape. Now I'm going to add in the 1/5-sin-5x, so I've now got three terms of that series. Then I'm going to go up to five terms, and you'll be guessing what you think is going to happen. 10 terms, 50 terms, 500 terms, we'll just do one more, 5,000 terms. And look at that, out of these curvy, curvy waves, we've made something that just is straight lines. People tend to call this a square wave, but really, it's rectangular, but the point is, we've taken something, this is about as far from a curve as you can think of, I mean it's got to be periodic, so it's just a step, or crenelations in a castle, or something like that, but these are straight lines, right angles, this is far from a sine wave. So if you can make that, it seems like you ought to be able to do anything, and you, asterisk, can. We'll just do one more. I had all the odd numbers there, let's just have all the numbers, one, two, three, four, and so on, so I'm adding sin-x to 1/2-sin-2x, to 1/3-sin-3x, and so on. So we'll start with one term, two terms, three terms, five, and it's getting wigglier and wigglier, 10, 100, and let's go all the way to 5,000 straightaway, 5,000 is essentially infinity if you're doing these things. So look at that, this is called a sawtooth wave, again, straight lines. And I like this one 'cause it's like we've come full circle, in all the ways you can interpret that phrase: we started off with right-angled triangles, and turned them into sine waves, and now we've taken some sine waves and turned them into right-angled triangles, so that's all very nice. So you can do this, you can make these Fourier series, anything you encounter, even if it wasn't a sine wave, you can do this, and break down using a well understood process, to decompose any periodic function into a combination of sine waves. So at the heart of all periodic things are these sine waves, and that's why the sine wave has so many uses and applications, it's because of this. Even if you don't start off with the sine wave, still, what you have can be broken into sine waves, so they're the absolute basic kind of wave that underlies everything. I've got a few minutes left, and I want to talk about just one context in which we see this kind of thing, and that is music. I did give a couple of lectures previously about mathematics and music, so if you want all the details that I can't give you in a couple of minutes, go and find those, and watch those, but just a brief, brief moment on the context here, going back thousands of years, people like Pythagoras did experiments with plucking strings, with basic musical instruments, where you would pluck a string, you would hear a sound, and then you could change the length of the string and that would produce a sound of a different pitch. If you made it shorter, the pitch would be higher, if you made it longer, the pitch would be lower. We now know there is an inverse relationship between the length of your string and the pitch, or the frequency that you hear. If you halve the length of the string, the frequency doubles, for example, so there's this inverse relationship. What Pythagoras discovered was that there are some sounds that sound pleasing together, and others don't, and he found that the ones that sound pleasing together tend to be where the length of the strings, and so the frequencies too, are in very nice, small whole number ratios. So he talked about halving the length of a string, doubling the frequency, those two sounds, if you play them together, they sound very pleasing, very harmonious together. We would now call that gap in pitch an octave, and actually, they sound so similar and pleasing together almost we think of them as the same note. If you've got a mixed choir singing, the ladies would be usually singing an octave higher than the gentlemen, and still, they sound like we're singing in unison. So that's the first thing, doubling the frequency, those two sound very pleasing together, but also, other ratios like three to two in the string length and the frequencies, that we now call a pitch difference of a perfect fifth, those sound nice together, three to four, four to three, these very simple ratios make pleasing sounds, pleasing harmonies. But why? We didn't know why. And many people worked on this over the centuries, people like Galileo, Mersenne, I want to mention Joseph Sauveur because he did a lot of really good and accurate experiments. A lot of the other people, Galileo, had mostly worked theoretically, Sauveur did detailed, accurate experiments, he studied acoustics, in fact he coined the term acoustics, that was his invention, that word, he did experiments with undulating strings, vibrating strings, looking at the frequencies, and on also what that phrase says, the nodes of undulating strings. And I think one of the reasons, perhaps, that he did the experiment he did, in the way he did it, was because, actually, in his study of acoustics, he did it in spite of, or maybe because of the fact that he himself suffered from hearing difficulties, he wasn't quite deaf, but he had hearing troubles that meant he had to use different techniques, sometimes, to study this science. The nodes of the undulating strings, he found that when you pluck a string, or you displace it in some way, he'd put little pieces of paper on the strings, and he found that there were, depending on how you set the string in motion, there were sometimes places on the string that didn't appear to move, and the pieces of paper did not fall off, so they're essentially still, those nodes. And depending on how you set the string moving, you might get a node in the middle of the string that was stationary, or you might get two nodes, each at the thirds of the way along the string, 1/3 and 2/3, or quarters, or fifths, these small whole number fractions. And he found that out, and that's a visible experiment that you can see, does the piece of paper fall off or not.So again, this is another puzzle:
we know that frequencies that are related by nice whole number ratios seem to sound nice together, we now have vibrating, undulating strings that somehow have sometimes fixed points at nice, small fractions of the way along their length, what's going on? The wave equation is what's going on, and it's a rather complicated equation, so I might just show it to you, but there's no expectation that you'll instantly understand all the symbols in it, I just want to show you it and talk you through it, and if you recognize it, and you're happy, that's all great, if you're not, don't worry about it. Imagine a string that's fixed at both ends, like a violin string, and then you're going to do something, you'll pluck it, or bow it, or move it, or something, disturb it, at an initial time, zero. And then you're interested in the vertical displacement of this, what shape is that string going to make, so we'll call that displacement y, and that's going to depend on how much time has passed, so the value of t, and also how far along the string you are, we'll call that x. So I'll give you the equation, don't worry about it at all, but just to somehow say there's some stuff in there, there's some symbols that are constants, they just depend on the string, so that big T is the tension in the string, the mu is the linear density, fine, those are just qualities of the string. The thing on the left is an expression, it's called a partial derivative, it tells you how the vertical displacement changes in terms of the time, and the thing on the right is telling you how that displacement changes in terms of the distance along the string. So what the equation says is that those are closely related to each other, one depends on the other. And the French mathematician d'Alembert found a method to solve this. Note, this is before Fourier, so he found a method to solve this, and what he said was, any solution to this equation is the sum of two waves, so there's a wave, A, it moves along the string somehow, and when it gets to the end of the string, that string is fixed, it's fixed at both ends, so that energy that's in that wave, something has to happen to it. Basically, there has to be another wave, that's wave A inverted, and traveling in the other direction so that it cancels out, any vertical displacement in the wave A has to be canceled out by that wave B, because that end of the string can't move. So that means there must be this other wave going backwards in the other direction, and off it goes, and then when it gets to the other end, the same thing has to happen, it's going to flip, and reverse, and produce wave A again. So what this means is that the solution to wave equation has to be periodic, you go all the way along one length, say the string has length l, you go along that whole length l, you flip, and invert, and go back again length l, and then you get back to where you started from, you get that wave A again. So any solution has to be periodic, i.e. repeating, and the period is two-l, every two-l, you get the same thing again. So our eyes light up at this point, 'cause we know the future, we know that thanks to Fourier, every solution, then, of this wave equation, has to be a sum of sine waves, and in particular, ones of that period, two-l. So here are the first few, and we see exactly, look, there are those points there that Sauveur saw, his little bits of paper, sometimes, depending on what you do to begin with, you could get fixed points, or nodes, that are still halfway along, or thirds, or quarters, so that's really cool, and that's why that happens. Those waves correspond to the frequencies, an initial fundamental frequency for the longest wave, let's call it f, and then if you have two sine waves that fit in there, you double the frequency, or treble, or quadruple. Different instruments will have different combinations of those sine waves, and you can analyze exactly what. You can draw yourselves little pictures, you can analyze these sounds with modern equipment, and you can break them down using Fourier analysis to work out exactly what sine waves you have. There's an initial transient sound as well which I won't talk about. So you can break down and do analysis, break these sounds into their individual sine waves, and then if you want to synthesize them later, you precisely can by reproducing those exact combinations of sine waves. It also answers the question of why some frequencies sound nice together, because every sound will have this basic, fundamental frequency, let's say f, and then various amounts of multiples of f, two-f, three-f, and so on. So if we take, say, one string of a particular length, and another one with, say, half the length, the one of half the length, it will have that fundamental frequency that's double the frequency, but it will also have those overtones, those harmonics that are multiples of that number. And so those string lengths and frequencies that are small whole number ratios of each other will share infinitely many of the same overtones and harmonics, and that's why they sound nice together, because they do have this resonance between them, so that's the answer to that question, and the answer to Sauveur's little pieces of paper. And the amazing thing is that as you're sitting here, listening to me now, your ear is doing this kind of analysis. This is a picture of your inner ear, well not yours personally, an inner ear, and the sound comes in, and it travels to the cochlea, which is this lovely, spirally part of your ear there that's filled with fluid, and the sound comes in and vibrates that fluid, and the individual component waves, this is something that was suggested by the scientist Georg Ohm, knowing that sound could be broken up into sine waves, he suggested it, and then it was confirmed by experiment many years later, the sound comes into the cochlea, and the individual components, the individual sine waves that are all joined together, each one of those is part of the sound, and the cochlea has this thing in the middle of it, the basilar membrane, it gets narrower, and narrower, and narrower, and at some point, each individual sine wave component of the sound reaches a stage where its wavelength is too big to go any further, it gets to that point where it can't travel any further, So it will collapse at that stage. but the energy from it has to go somewhere, and the energy goes into that membrane, which has lots of tiny, little hairs on it, and they are set vibrating by that collapsing wavelength. And so your brain then receives signals from the sound, and it gets signals from each place along the path of the cochlea where a wavelength collapsed, and then it can, from that, work out what sine waves exactly were going together to produce that sound, and interpret that sound accordingly. So your brain, right now, and all the way through this talk about sine waves, has been doing Fourier analysis, and breaking down the sound into sine waves, which I think is lovely. So I will stop there with a little advert for, only 48 hours from now, very, very close, 6 o'clock, 25th of May, that's two days from now at the time that I'm speaking, you only have to wait two more days for another mathematics lecture. This one is joint from Gresham and the London Mathematical Society, it's something we do every year, it'll be given by Hugh Hunt, who is a fantastic speaker, on the mathematics of gyroscopes and boomerangs, so come back then to hear about boomerangs. All right, thank you. (audience applauds) - Thank you so much, Professor Hart, for the fascinating lecture. I've got some questions from online, which I think we should take two of first before going to the room. The first question is, "On a calculator, "you often get settings "to work in degrees, radians, or gradians." "Where did gradians come from, and who uses them?" - I believe, I don't want to say anything wrong, I believe this was a French Revolution thing. There are 400 gradians in a circle, i.e. 100, nice decimal 100 in every right angle, that's where I think that originated. I believe they still have some uses in engineering, but if you're a mathematician, you tend to use radians, although, if you're like me, you still have degrees in the back of your head. - Thank you, Professor Hart. I've got another question here, again possibly more about the sine wave, "Why is mains electricity a sine wave?" - Oh? Well everything's a sine wave, there you go, next question. (audience laughs) Anything that's electromagnetic is a sine wave, light waves, and all of those things are. - Thank you so much, Professor Hart. Please do come back here for Professor Hart's series next year, she's still with us next year, hooray, so please come back for her series next year. If you get on our mailing list, you'll get notifications about that series when it launches at the end of July. Thanks so much. (audience applauds)