 So it's a pleasure, a great pleasure for me to introduce our next speaker, so Professor John Chapman from University Oxford, who's going to present the second of our two public talks. This is accompanying our thematic program on exact ascent projects from fluid dynamics, the quantum geometry. So this is a program that's going to extend to the end of October. I'll provide a little introduction to embarrass John here. So he completed his D fill at Oxford, and then held postdoctoral positions at Stanford and Caltech before returning to Oxford, and he's been, he was appointed the chair of mathematics applications in 98 and has held that position since He was awarded the Siam Richard the Prima Prize in 94. In 98 he was awarded London mathematical society, Whitehead Prize, the Siam Julien Cole Prize in 2002, the nailer prize from the London mathematical society in 2015. His research lies on on topics and mathematical modeling and asymptotic analysis with applications, the physical sciences, biological sciences, and a lot of industry applications as well. And to me he exemplifies the very best of this long standing tradition of British applied mathematics of scientists like James Lighthill and GI Taylor and Gabrielle Stokes who were able to discover deep mathematics in applications all over the place. So John has made many contributions in areas such as superconductivity, acoustics, fluid mechanics, mathematical finance, mathematical biology. And most recently he's, he's had a lot of exciting developments in modeling of lithium ion batteries. The last thing I'll say about john is that one of the easiest ways to solve your problems to get him interested in your problem. He'll be here for the next two weeks so I hope you can bring some very interesting discussions to him during that time. And otherwise, thank you for coming to OIST and we're looking forward to hearing your talk. Thank you very much for let's, it's a great pleasure to be here and to give this talk and I will look forward to interesting discussions over the next couple of weeks. So I'm going to tell you about a story that starts with musical instruments and I'm going to get to vacuum cleaners and along the way we're going to talk a little bit about invisibility cloaks and metamaterials. So I want to start with the guitar string musical instrument what happens when you pluck the string of a guitar is a simulation that I did, I plucked it about a third of the way along. And you can see you get quite a complicated motion now that this it's periodic in time it repeats time, but spatially it's quite complicated. So mathematically we find it convenient to decompose this motion into a set of simpler motions. So I can get the complicated motion at the top here by adding together in this case these three motions at the bottom each of which is much simpler to describe. So mathematically this is known as an eigen mode decomposition, but any musician in the audience will recognize these vibrations of this string as this one is corresponding to the fundamental mode of vibration. And then this one corresponds to the first harmonic, this corresponds to the second harmonic. In general that'd be an infinite series is I've drawn the first three but this these harmonics would carry on you'd get infinite series of higher and higher harmonics with the motion of this string, but the energy tends to decay so most of the energy as I've drawn it here is in the fundamental, there's a bit of energy in the first harmonic and a bit less than the second, and it decays as you go down. So, let me start by showing you how I generated those how do I, how do I know what the fundamental looks like what the first harmonic what the second harmonic look like. At the top here I've just drawn a regular oscillating sinusoidal pattern. So a wave, and I've marked the wavelength lambda so that's the shortest repeating unit so you can think of it as the difference between one crest and the next crest. That's this wavelength that I've called lambda. And my string is is pinned at either end so it's not allowed to vibrate at the ends. And so to, to get a wave on the string, I have to look at this, this wave here and have to choose the segment of this wave that lies between two zeros. So the zeros will correspond to the ends of the string. The shortest or simplest one I can do is just to choose neighboring zeros. So if I choose neighboring zeros of this wave, that gives me this shape, and that gives me the fundamental. And so you can see how much of a wavelength did I get in into the string here I only got half a wavelength I got the peak here but I didn't get the corresponding trough. So the fundamental the wavelength is, I get half a wavelength in so the wavelength is twice the length of the string. The next thing I could do is I could skip one zero, and choose the next one. So look at the difference between this zero and this zero. And that would correspond to this picture here, and that gives me the first harmonic. And now you can see I got a whole wavelength in between the two zeros. So the wavelength of this wave is equal to the length of the string. So I'm going to skip another zero, and get, get three up down up in here and that gives me his which is the second harmonic. And now you can see that I got one and a half wavelengths into that between the two zeros. So the wavelength of this wave is two thirds of the length of the string. And to get to get, as I go up through the higher and higher harmonics, the length of the string is fixed. So to get more and more of these peaks and drops in I have to reduce the wavelength of the wave. So the wavelength of this is this fundamental is twice the wavelength of the first harmonic and three times the wavelength of the second harmonic etc etc. So the second harmonic has a shorter and shorter wavelength. So you probably also notice that my, my fundamental and different harmonics we're also learning at different frequencies. And so the relationship between the wavelength and the frequency is is given here so the frequency is inversely proportional to the wavelength. The frequency goes up the frequency goes down the wavelength goes down the frequency goes up. And this constant see this constant of proportionality is the speed of which waves would travel on the guitar string. So that depends on what the string is made of is it nylon or is it steel depends on how thick the string is. It depends on what the tension is. That's how you tune the musical instrument you vary the tension that will vary this number C, and that will change the frequency. So the fundamental is the longest wavelength so it's the lowest frequency and from this formula I could work out what it is, and the formula is given there. And then for the first harmonic, I reduce the wavelength. And that means so the wavelength is half and that means I double the frequency, because this constant C is the same for all these modes that depends on the string but doesn't depend on the mode. So the first harmonic has twice the frequency of the fundamental. So the second harmonic, the wavelength is a third of that of the fundamental so the frequency must be three times that of the fundamental. The frequencies of each higher harmonic is going up in integer multiples of the fundamental of omega one to omega one three omega one, etc. If I had my guitar here I would play you an A string, and I'm using the musical notation here in which C four is middle C on a piano. And so that would be about here. So my A string on my guitar is in corresponds to a to which oscillates at about 110 hertz. And so if the fundamental is that 110 hertz that tells you that the first harmonic should be 220 hertz double the frequency, and doubling the frequency corresponds to going up by an octave. So that string would, the first harmonic would be at a three so this shows you the corresponding key on the piano so this is the a to and this would be a three. So the second harmonic, I would multiply the fundamental by three. So if I, my string was 110 hertz that would be 330 hertz, and that corresponds to e four on the piano so you've gone up by a fifth. And the reason that stringed instruments sound so pleasant to the western ear that the integer multiples of the fundamental frequency correspond to what we perceive as melodic intervals on a piano. So the notes that are an octave apart sound nice and then notes notes that are a fifth apart sound nice as well. Okay, so from this you should take out home the fact that long strings have low frequencies short strings of high frequencies. Higher modes are integer multiples of the fundamental. And if I double the frequency then I raise the pitch by an octave. Now, if I had my guitar with me I would now do you a live demonstration but unfortunately I don't. So I'm going to show you a video of me doing a live demonstration. So, here is me with my guitar. I'm just going to explain what you're seeing here so this is a spectrogram so it analyzes the frequency of the sound that you're hearing. So the horizontal axis here is frequency. And on the vertical axis is how much energy there is in that frequency and it's a logarithmic scale it's in decibels. Let me play this movie. So it's, it's giving you frequency on horizontal axis and it's giving you how much energy there is at that frequency on the vertical axis in decibels it's a logarithmic scale on vertical axis. So if you see my A string you should see it, I said it was 110 by stop talking so you can see the guitar. So you see the peak at the far left 110 would be about here. You see that the fundamental is at 110 and then you see a lot of harmonics over the top of it. Now if I, if I touch the string in the middle when I'm playing this that kills the fundamental. It wants to vibrate in the middle but it doesn't kill any even harmonic. So all the even harmonics had a zero. So if I play the string and touch it in the middle I get every second harmonic. And I can do the same thing if I touch it a third of the way down then I get every third harmonic. It doesn't matter whether I do it a third here or I could do it a third down here. The tone that you get out of it depends on the ratio of the harmonic. How much is in the fundamental and how much is in the other mode. So if I play the string somewhere in the middle I get a lot of the fundamental not much of the harmonic. And you can see that, soon as I talk you can't see, but you can see that there's a large peak. It's a logarithmic scale remember so there's a large peak on the left and it decays quite rapidly as I go down. That's a nice round note. If I play down this end I generate a lot more of the harmonics much less of the fundamental. See it's a much tinier sound. Trying to generate the same amount of energy. Okay, so that's, I'm sorry I couldn't do the demonstration live, but that's sort of by way of warm up that everybody's pretty familiar with stringed instruments. But what I really want to tell you about is pipes. So, here's a picture of pipes and pipes behave in roughly the same way. So I told you for the guitar string that the constant see the wave speed depends on what the string is made of. For a pipe, the wave speed depends on what the stuff inside the pipe is made of, and usually the stuff inside the pipe is there. The wave speed depends on the compressibility of that material the compressibility that it depends on the density of air. And this is just write this up because we're going to come back to this formula a little bit later so it's not so important now but we will come back to it. Okay, so how do pipes work. It's a little bit harder to draw the oscillations of a pipe so for a string if I stretch the string in this direction. The movement of the string is is an orthogonal direction is vertical so it's known as a transverse wave. The string is this way and the motion is this way for a pipe. It's a longitudinal wave. So the motion of the, the air particles inside the pipe is along the pipe, and that makes it a little bit harder to draw. So let me explain to you what these pictures are. So the curve on the top. I'm showing you what the pressure is what the pressure inside the pipe. And so this has got high pressure in the middle and low pressure at the ends. So here I'm showing you what the color scheme down here is showing you the density of the air in the pipe and pressure is proportional to density. And so if you have high pressure you have high density and vice versa. And so the dark blue regions correspond to the high density regions, and the white regions correspond to the low density regions. So here I have a pressure high in the middle so I have density high in the middle. And then I'm also showing you here. A few representative air molecules by the blue dots. And there are more molecules where the material is more dense and so there are more molecules in the, in the dark blue regions that are in the white regions. So now that you know what we're seeing I can, it's a little video so let me start the video. I believe that as each mode oscillates, the air molecules shittle back and forwards along the pipe, and the pressure and the density go up and down corresponding to where the air molecules are. Now let me explain what these pictures are so for a, for a pipe, or any wind instrument you have to have some way of generating the, the disturbance in the first place so the for the guitar, that's your finger on the string. The left here is the mouthpiece which is generating the disturbance, so you can more or less ignore that the pipe basically starts from this whole to the end here. And this is a pipe which is, which is open at both ends. And that means the particles are free to move in and out to the ends, and it means the pressure at the end should be the same as atmospheric pressure. So this is the perturbation to the pressure so that must be zero at the ends, as pressure is atmospheric so there's no perturbation of the ends. So in fact the modes you get for this open ended pipe are exactly the same as the modes that you get for a catastrophe that the pressure has got to be zero at the ends is corresponds exactly to me saying that the, the string is pinned at the ends there's no displacement at the ends. And so you recognize these pictures for the pressure doing exactly what the string vibration was. So here you have a half wavelength, here you have a full wavelength and here you have one and a half wavelengths. But for a pipe, you could also put a stop in at the end and close the end of the pipe. And that corresponds to not allowing the particles to move out of the end of the pipe here. And that corresponds to instead of having a zero of the pressure you have either a maximum or a minimum. And so instead of cutting this, this waiver at a zero I couldn't add either a peak or a trough. And so you can see the effect of that the fundamental now I have only a quarter of a wavelength in there. This is half of that and that was already half a wavelength. So here I have a quarter. And for the next one I've got three quarters of a wavelength in here. And then for this one I've got five quarters. Okay, so this one is going up a half one, one and a half. This is going a quarter, three quarters, five quarters. If I convert that to frequency. So let me suppose that my original pipe was was an a three if it's an open pipe, and say that has a fundamental, as we've seen of 220 Hertz. And then the first harmonic where we know just like the guitar string integer multiples of the fundamental. So if this is 220 then the first harmonic would be 440 that corresponds to going up by an octave. And the second harmonic would be 663 times this corresponds to going up by another fifth. What about if I have a close pipe. Yeah, the first of all the fundamental would be half the fundamental for an open pipe, because I've reduced the wavelength by half. So I've increased the wavelength by a factor of two so I reduce the frequency by a factor of two. That close pipe would have half the fundamental frequency of an open pipe and that drops it by an octave. Then the first fundamental is corresponds to the wavelength. The wavelength getting three quarters. And that means that it's got three times the frequency of the fundamental. So in fact I jumped from a to up to 330 Hertz which is the fall. And then the second harmonic would be five times the fundamental. And that gets me up to C sharp five. You get a different series of harmonics for an open pipe than you do for a close pipe. And now I have another little demonstration of that for you to watch. This was the closest I could get to a pipe really is still in. I've, it's a Irish in whistle. I've sell it up all the holes. I couldn't do all the whole with the same time. So if I blow it, it really is just an open pipe of this length. It says on it that it's a D and I think that means not to blow too hard. If you blow too hard, you end up getting the higher. So I think it's around about 600 Hertz. And you can see that the harmonics are at 1200 and 18. You're really getting one, two, three. Now I'm going to put my finger up there. So again, now it's designed to be an open ended pipe. So as soon as I put my finger over there, mouthpiece is not really right for this. And it's quite hard to get it to go at the fundamental. If I just blow without thinking about it, I'm going to hit the first time on. Sounds like it went out, right? I told you it should go down because I'm hitting the first time on. You can see that I am because I'm getting 900. If I blow really gently, I can get 300. Okay, so you can just see that you get 300 and then you get 900. And then you get 1500, right? One, three, five. More or less convincing. I spent the last two days trying to make pipes out of straws to give you a live demonstration, but I haven't been able to so I had to show you the video of that. And this open close pipe thing explains a little conundrum to do with flute versus clarinet. So the two instruments that are roughly piped yet roughly cylindrical. And they're both about the same length. So I looked it up a flute typically about 66 centimeters long and the clarinet's about 60 centimeters long. And a flute you blow across the mouthpiece. So it's an open ended pipe it's open at both ends. So if you if you work out that if the if it's 66 centimeters long, then for the fundamental the wavelength should be twice that two times 66 speed of sound in air is about 340 meters per second. So that gives you about 257 hertz is the lowest note this flute should be able to play. And that is that turns out to be about C4, which I think is around about the lowest note that a flute can play I don't know if there are any flutist notes. And the clarinet is roughly the same length slightly shorter. But it's closed ended so you blow through a read at this end. There's no way for the air to escape at this end so it's only open at this end. So the clarinet is a closed ended pipe and that means the fundamental has a wavelength which is four times the length of the clarinet not just twice. And so if I do four times 60 centimeters, and again use 340 is the speed of sound in there you get to a frequency which is about 141 hertz, which is about a D3 so it's about an octave lower. And despite being the same length, the difference between being an open ended flute and the closed ended clarinet gets you a fundamental which is about an octave lower. All right, that's musical instruments. How does it relate to vacuum cleaners. When you, you have these pipes that will resonate a certain frequency so if I stimulate them, I can resonate them and get sound out that's what a musical instrument is. If I have sound already, then I can use the instrument to take that sound away. So the resonance there is is taking away sound instead of making sound. So that's a way that you can use these resonance. So, in particular if you have sound of a particular known frequency, and typically that comes from when you have a fan and something so if you have a fan, or this sort of thing that you might have in a vacuum cleaner it's going to oscillate at a certain speed and that's going to generate noise and you don't want that noise. So I would like to extract it. So imagine you have noise of a certain wavelength that's coming along your exit of your vacuum cleaner here. And you want to stick the equivalent of a clarinet on the side that will resonate at exactly the right frequency to get rid of that noise. So we've seen that if the if the wave has a wavelength of lambda, then I could generate sound with that wavelength by having a length of pipe stuck on the end which is any of these frequencies. So we've seen that the fundamental would have a wavelength which is four times the length of the side chute. The first harmonic would be four L over three. The second harmonic would be four L over five. So if I if I stick this on the side it will suck energy out of the pipe here at any of these frequencies. Because you don't really want to have large clarinet stuck on the end of your vacuum cleaner so you'd like this side branch to be as small as possible. And so the smallest way the way to get as small as possible is to choose the fundamental down here. And in that case I need the length of this pipe to be a quarter of the wavelength of the wave that's coming along here. And that's why these things are known as quarter wavelength resonators. So let me just give you a little schematic of how this works, or one way to think about how it works so I have my wave which is coming along the top here. As it gets to this point, some of that wave is going to go down into the resonator. So as I go along I'm again showing you the peaks and the troughs, the blue and the white. It gets to the bottom and it bounces off the bottom. And goes back up again and then merges back with the wave which is coming along here. So, by the time it's gone down and back up, it's gone half a wavelength was quarter of a wavelength down and it was quarter of a way back. And so by the time it gets back to the, to join with the main pipe, it's exactly out of phase with the way that's going along the pipe if this one's got a peak this has got a trough and vice versa. And that's how the energy then cancels out that you don't get any wave coming out of the end here. So this is, it's quite a common to put these quarter wavelength resonators in to suck energy out when you want to quieten things down like vacuum cleaners. And it's very easy to do for high frequency noise, you've got high frequency noise you need little quarter wavelength resonators and it's easy to take the energy out. But if you've got low frequency noise, then you, your quarter wavelength might be quite long. And that's, you don't want these, your vacuum cleaner to have a meter long resonator on the edge of it. So, the problem is, is how do you get rid of low frequency noise, which requires a long side branch. And I want to tell you about an idea that some colleagues in Manchester and Cambridge had for for solving this problem of reducing the length of the quarter wavelength resonator. And in order to explain their idea to you that's how I have to take you around and talk about things like invisibility cloaks and metamaterials. And I'm going to, I'm going to come back to that idea of reducing the length of the quarter wavelength resonator, but first I'm going to take a side branch into invisibility cloaks. I realize in making this slide, it's quite hard to draw an invisibility cloak. Fortunately, you all know what I mean when I say invisibility cloak thanks to Hollywood. This is Hollywood's idea of what an invisibility cloak is. And automatically, this is what you might mean by an invisibility cloak. So you imagine, you have a wave which is incident on some object. My object is going to be here in the white. And what I want. So my, the, this light blue region is supposed to represent my invisibility cloak. And what I want to happen is that when I come to the back of the object here, the wave is going along as though nothing happened. Okay, as far as somebody observing or listening from this point would be concerned, this wave just passed straight through this object. So how might you do that. One idea is imagine now say light rays. As you're coming in, you don't want to hit the object and bounce off so you want to bend them to go around the object and back down the other side. Okay. I have to have to bend the light or the sound to go around the object and then bring it back and go forwards again as though nothing was there. Okay, how might we bend light rays well, so that sounds like a crazy idea but you're all familiar already with the fact that light rays can bend. Sure everybody has seen this sort of image where you see the mirage of a puddle of water on the road. So, what is happening there. Well so here's an example showing it in the desert with the camel. And as the ground gets hot, the air near the ground is hotter than the air away from the ground. And the as light moves into a region which has got a different density so the hot air has got a lower density that bends the light. And so you see the camel going straight forward but the light that's coming from the camel down here gets bent and then hits your eye coming up from the ground. And your brain doesn't know how to cope with that because your brain is used to light traveling in straight lines. So your brain sees this as though it came from a straight line down here and sees the image of the camel upside down. And your brain has to try and make sense of that and the only way it can make sense of that is to imagine that there was a mirror there because it's used to light bending, reflecting off mirrors. So your brain imagines there was a mirror that reflected the light of the camel. But hold on there's no mirror on the road. But if there was a puddle of water it would do the same thing and so your brain does its best to interpret what you're seeing and tells you that there must be a puddle of water. So if we, we need to change the material properties in the vicinity of this object in the clock around the outside in order to get light or sound to bend around and we can do that by changing the density for example. Okay, so let me show you one way that you might work out how to do that. This is known as transformational acoustics. So I'm going to start with just a grid that I've laid down Cartesian grid that I've laid down over space and this point here is the. I'm going to expand this point now to try and hide something inside it. I expand that point and I deform my grid in the vicinity, but I didn't I don't deform it out here I just to form it in the vicinity of this thing. And the idea is that I'm going to hide something inside here. Okay, so this is my undeformed space. And then this is my deform space but as I say I didn't deform everything. You can see the only the lines that are inside this light blue region have been moved have been stretched, and I expanded this point out into this whole, and I stretched all the other ones here. So, that's, that's given by some mapping a map some point here to some other point here stretching it out and making a hole in the middle, but I, I don't do anything outside this light blue region which is going to be my invisibility cloak. Okay, so that's, that's the first step. Then if I, if I imagine what happens to a wave which is coming along and going passing through this cloak. So over here, I am my under form grid so my object here is just a point so it's not going to do anything at all to the way. So on this side this wave is just going to carry along and go all the way through. That's what I want to happen. So what I have to do is look to see right what when I do this, this stretching what happens to that way. And that's what this picture over here is showing you. So inside the light blue region the wave is distorted a lot, because I did this stretching and expanded this whole inside the whole the wave doesn't get there at all because the, it's the whole is just a point on this side. So inside this cloaking region the wave is unaffected because I did. There was no deformation outside here. So if I can arrange that this is what the wave should look like, then I can hide anything I like in the middle here. And by the time I get around the back and observer here would would see just this plane wave going straight through and it would not be aware that there was an object in the middle. That there's been no disturbance to the field here. Okay, so I had to show you some equations. What I did here was I said I take this solution take this wave, and then map it over here and see what it looks like. I'm going to do step two which is to say, instead of taking the answer over here, and then, and then using this mapping to map it over here. I'm going to take the equation that the wave satisfies over here, and I'm going to transform that equation to see what it would satisfy over here. Okay, so instead of, instead of taking the solution and then mapping it I'm going to take the equation and then map it. It's not so important what this equation is you don't have to understand that at all. You just have to know that there is some equation that the wave satisfies over there. And I can do this change of variable, and I would get some more complicated equation over here. Okay, here it looks fairly simple. Here it's much more complicated. All you need to know is that is that that that thing as possible. And then what I want to do is solve this equation and see what the solution looks like and I should get the same as, as if I sold this equation and then do the mapping it shouldn't matter if I solve and then map, or if I map and then solve. I want to show you that this is true this is the group in Valencia Spain so this is the wave that is going. And on this side there's not there's no, nothing to disturb it because there's just this point that in the middle which doesn't have any effect. This is showing you what would happen if I just took this wave and just put the circle in there, and reflected the wave off the circle. And this is just to show you that you would get something very different you've got reflected sound you've got a shadow behind there's all sorts going on here. This is what you get. If you take that transformed equation and you solve it, and you see that you do get a whole mess going on inside the cloaking region, but once you get out the back the wave is nicely lined up again. And it's just going carrying on as though nothing happened. So in this case, I could hide something in the middle here and there would be, there was a person who was listening here would not be able to tell that there was anything in the middle. Okay, so that's step two. Step three is then to think. So far all I've done is mathematical manipulations, how would I actually do this physically. So here I have this equation but what sort of material does this equation correspond to. And so I told you that the wave speed depends on the compressibility and the density. And if I like we did with the light from the camel if I let the density vary. It will bend the waves and this is how it would appear in the equation so if the density of the material was varying it would appear in the equation here and this is the compressibility. So if I could let this be equal to that, I'd be in good shape. That's what I need to do physically. The problem is that we are used to density being just a number. Mathematician would call it a scalar, whereas this thing up here the mathematicians would call a matrix. And so what that means physically is that in this equation I need the density of the material to be different if the wave is going in this direction or that direction. Another way to say that if you think about particles. I want the mass of a particle to be different if it's going that way than if it's going that way, which is completely crazy idea. It sounds completely impossible. And it is a crazy idea, but it turns out it's not impossible. And to show you why it's not impossible. I have to go to metamaterials. Okay, so what, what do I mean by metamaterials so that metamaterials general term for what you get when you mix two different materials together. And what's interesting about that is that the mixture of two different materials can have very different properties to either of the materials individually. And you can get some quite surprising results so here this is my schematic picture we've talked about sound so far so I'm going to talk about sound waves in in a bubbly liquid so the speed of sound in air is about 343 meters per seconds. The speed of sound and water is a bit higher. It's about 1500 meters per second. And this is supposed to represent bubbly water, so the light blue is air and the dark blue is water. So you could ask the question, what is the speed of sound in bubbly water. So let me draw a graph. So here is the speed of sound on this axis. And here is how much air there is compared to water on this axis. So this, at this end, it's zero so there is no air so it's all water. So it's 1500 meters per second. And at this end it's all air. So there is no water. So it's 340 meters per second. So you might imagine that if I took some 5050 mixture of water and air, I should just average these two and draw a line between them and get some speed of sound which is in between. So you might imagine that but you would actually be very wrong. So the speed of sound in bubbly water looks something like this. And so what's remarkable is that a it gets very low, but it's, it's not even in between these two. So the region here, the speed of sound in in bubbly in bubbly water is much lower than the speed of sound even in pure air. Very surprising. In fact, so you can't really see how low this goes by zoom in a bit you see it gets down to 30 or 40 meters per second. So the 10th of the speed of sound in air alone is remarkable. So why is that happening. So let me come back to that I showed you totally we were going to come back to this formula for what the speed of sound is in terms of the compressibility and the density. It turns out that the right way to think about this is not that C squared is compressive is one of a compressibility times density, but to think of it as one of a C squared is equal to compressibility times density. And then it turns out that the right way to average is not just to do the average of the product. That's, that's what it would be if you were sort of drawing that straight line take the average of the sound speeds turns out that's not the right thing to do. The right thing to do is to average the density, average the compressibility and then multiply them together. So it's not obvious why that's the right thing to do you have to believe me but it is the right thing to do. And so, when you do this for air and water, so the density of air is about a kilogram per meter cubed. And the density of water is about 1000 times that. So if I have an average of a bit of air and a bit of water, the density is basically the same as the density of water the fact there's a bit of air in there is irrelevant. The density of water is much bigger than the density of air. But the compressibility of air is about 10,000 times the compressibility of water. So if you have a mixture of air and water. The water is irrelevant as far as compressibility is concerned the compressibility of the mixture is basically the same as the compressibility of the air. And so if you have a mixture of air and water, it has a density close to water, and as a compressibility close to air. And that's what gives you a material which is not like either of them. So each of these averages is much bigger than you might expect, because there's one material which is dominating here and a different material which is dominating here. Okay so once you start to think like this and thinking about what happens with mixtures. It's not so such a leap of faith now to think that I might get different behavior in this direction that I get in that direction. My microstructure in this material was not symmetric. So suppose my air bubbles were all little ellipses. You, it's not so hard to imagine that a wave propagating in this direction might have different properties to a wave which is propagating in this direction. Right, so that's, that's the sort of way that you can get your, your density to depend on direction. So this is a microstructured material which is not symmetric. And so waves going in different directions see a different sort of microstructure. Okay, so now let me come back and use these ideas on the vacuum cleaner on the quarter wavelength resonator. So this is worked you to will panel and David Abraham's Manchester and Cambridge respectively and this was in collaboration with Dyson who were trying to reduce the noise of their vacuum creams. Here's my, here's my quarter wavelength resonator and what I want to do is to use this idea of transformational acoustics to make it smaller. So I'm going to work out exactly like you did for the cloak, what, what density properties do I have to have here that the sound thinks that this is really this big. And so I can make it smaller than it needs to be by having an interesting microstructure in there. So they just use this exact same idea and said this is what it's going to be but with a variable density. This is what the sound is going to think it looks like. And so they came up so you, you can't. It's very hard to make bubbles that are known shape but of course in in one of these resonators it's very easy to make baffles. So I could just put baffles in there of a certain shape and these are the baffles that they came up with these sort of stretched ellipses. And they tried to ideas one to put them in the middle of their resonator and want to put them down the walls. And of course these days, when you come up with an idea like this it's not so hard to manufacture it you just take it to the 3D printer. And so this is the 3D printed resonator which has got the baffles in the middle and this is the one which has got them stuck on the walls. And they arranged it so that this was the sound would think this was twice as long as it actually is. Okay, so they've reduced the length by a half. And then they stuck it onto a tube where they had a sound wave coming down and they look to see how much energy you can suck out. And in particular what's the frequency of the energy that you're taking out of the of the pipe. So this is the resonator which is the same size, but doesn't have any of this microstructure in it. And then they tried the two ones A is the one with the baffles in the middle and B is the one with the baffles stuck to the edges. And the color code here tells you which graph is which so that this curve here is what this one corresponds to. And you can see that it's resonating at about 2000 hertz. That's the where it's sucking the air out. So this is sort of equivalent to the plots that I was doing before when I was showing you how much energy was in the strings and the pipes. There it's transmission loss so it's how much energy you're taking out of the pipe. So this one is resonating at about 2000. And these two, which are physically the same height, but they've been arranged so the sound thinks they're twice as high, which means they should take energy out at half the frequency. And that's what they found that they do take it out at about 1000 hertz instead of 2000 hertz. They're not perfect because they don't take so much energy out as this one does. I have managed to shift the peak down in particular for this green one, the peak is not taking anything out of 2000 anymore. It's taking it out at 1000 instead. Okay, I had one more very brief topic I wanted to tell you about, which again is to do with sound and waves. But this one is to do with waves on sheets of metal instead and it goes back to transverse waves that are similar to the guitar string waves. So it turns out you can have waves that propagate on the sheet of metal. I said that the speed at which a wave propagates on a string depends on how tight it is and what it's made of and things like that. If you have a wave propagating on a sheet of metal, the speed depends on the curvature of the sheet. And in fact, if you cover too much, then you find that waves won't propagate at all. And so you can have if it's not curve too much the waves propagate but once you get a two curved, then you don't get waves. And what this means is that if you bend the sheet of metal into an S shape, then you can have a middle section which has got low curvature which you can get waves properly again, and on. But the at the ends here that's high curvature here and high coverage here and there's no waves can propagate that. You can find that you get a region in between that the waves are confined to. And as you as you bend in different places you can change the length of this region, and that will change the wavelength of the waves that are confined there, and that will change the frequencies so you can change the note that you hear. And as you as you bend the metal, you, you can essentially continuously change the pitch that this is resonating at. And so I would, again, have had a demonstration here, the classic way to do this is to take a saw and you just take your own so galvan saw from your shed and play it with a violin bow and you get quite a nice note out of it. And bring my soul with me. So I'm going to show you a little video of somebody who really can play this as a musical instrument. And I recommend you all go get your sore out of your shed and try it. It's really good fun. All right, that's everything for me. Thank you very much. Thank you for your talk. And I am taking any questions. Maybe I'll get started with the question. So I'm interested in why some instruments are harder to play than other instruments. So I play the violin, the violin sounds terrible compared to other instruments, which I feel are easier to play certain notes. I think some people play the violin very well. I don't know. So I guess a lot of that is to do with how you excite the disturbance in the first place. And so what you saw with my whistle when I blew it the first time I blew it I excited the first harmonic and said of the fundamental. That's all about the mouthpiece, but also control in that case. And I guess the same is true of the violin. It's, it's how you excited. I suppose high frequency compared to a cello. I mean, I guess, so when my daughter was playing the violin, I noticed that, you know, there are no threats to tell you where to put your fingers you've got to listen by ear and that takes skin as well. Your formula about the formula about where you average the compressibility and the was the other one. Yeah, that was for like, I guess, just evenly spaced bubbles of some fixed size. What if I just place like an arbitrary geometry in there. I say it's more or less the same. Thank you very much. So for this invisible the tube. So did they physically implement such a matrix density to realize it. So I don't think that group did now, but I see on the web these days that you know people have started to make things that are not the similar. In terms of invisibility for light as well even they're never, they're never quite perfect because you see a little bit of but they're a bit like the old film predator right where you see the bit of a shimmer but but they work fairly well now. They won't be too far away before. Okay, so then I can you create something like a sauna. Stay with Submarine. So I guess you, I guess you could the way that's self in in things like someone or an aircraft normally works is that you don't care about behind you so much you care about things that go back in front of you. So what you don't want to do is reflect anything, just because normally somebody sends a wave in and then they're listening for you. You don't want to send anything back and so normally you want very, you want absorbing material on any edges that you have and then lines that are going to reflect it backwards afterwards. Normally you try and absorb it rather than rather than send it behind you. So I wonder if there's any creature, any natural creature which use such kind of mechanism to avoid the free data. I mean it is the way that they do the quarter wavelength is important for absorbing materials as well. So if what you try and do is have a material which is a quarter of a wavelength so that some, some is reflected, some goes down and it's reflected and those two contributions are exactly out of phase. And so they counsel with each other so it's a similar sort of idea to the taking the taking the sound out of the vacuum cleaner is a little bit like sell back one. That was a very interesting talk. In the, in the last video of this lady playing the saw with a with a violin bow. While I could understand that the bending of the saw is leading to a change in the pitch of the sound. How does the confinement of the wave being reflected qualitatively in the sound. I didn't understand that. It means that you don't get much damping of the wave due to the due to the end points, you only get dumping due to the sound moving away through the air. That's what means that you get a nice long note from it. And the fact that you have this wave can't propagate. It gets reflected back there without losing any energy. And that's crucial. So that means that the wave lasts a long time in the saw and it only dampens down because of the sound waves that are so it's a little bit like the unique way you have a bridge on a guitar or violin. If you want a nice sharp edge there so you don't lose any too much energy there, apart from the sound. Thanks for a great talk and I would like to know if you can comment on the sound made of metal, the difference of sound made by metal and wood. So, yeah, I'm thinking about this in terms of the context of percussion instruments. So let's say if I can match the like the modulus or, and, you know, their lens and everything, and I try to match the fundamental frequency of a metal bar and a wood bar. And I would imagine that they would we would still feel like there's some subtle difference in their sound. So, I'm just thinking that can we think of this in terms of like the, you know, different microstructures in them, like, it's kind of like a metamaterial because I would imagine that the metal is more crystalline. Well, there is microstructure in the wood. Yes, there's a lot in that question. So, at a simple level it depends on things like what the propagation speed of waves in the material on that would be different for wood and metal. But, and that will depend in turn on the microstructure of the woods that different woods might have different material properties depending on how things are arranged. The sort of sound that you hear often depends on how much is in the fundamental and how much is in the harmonics as well so you have a harpsichord as a very different sound to a piano or an upright has a different sound to a ground. And a lot of that is how much energy is in the different modes to what tone or do you get from from the instrument, and then further you get different sounds depending on where you hit it of course as well. So, you know, guitars, piano strings are always hit one seventh of the way down I think because the seventh harmonic is a little bit dissonant compared to the others and so you don't want to excite it. And so then you hit that a seventh of the way down so there are all sorts of tweaks that people do and have done over the years to try and get a nice sound that. So in thin air the speed of sound is slower right so I was thinking has there ever been a concert on Mount Everest or in an airplane and did it sound weird. I don't I don't know. So it wouldn't affect the pitch of the instruments because that would be determined by the instrument, not by the air of well it would for any wind instrument I guess yes so I don't know. I don't know whether people who play wind instruments have to worry about that sort of thing I presume they tune up locally. So with that, I just want to remind you that we'll have refreshments in lab five in the TSVP room so please come and join us and ask more questions then but otherwise let's thank John for a wonderful talk.