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Thank you, Alan, and thank you for that very overly kind introduction.

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So I want to tell you a little bit about waves and resonance, starting with musical instruments,

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ending with vacuum cleaners and taking in some rather strange things in between.

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So I want to start with my guitar and think about what happens when I play one of the strings side by a string.

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You all hear the nice sound it makes. I want to think about what the string is doing, first of all.

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So this is roughly what my string would be doing when I play it around about a third of the way down.

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And you see, it's quite a complicated motion. It's periodic.

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It's repeating,

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but we find it very convenient mathematically to try and break up this complicated motion into a series of things which are a bit simpler.

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So I can describe that the motion at the top just by adding together the three disturbances that you see below.

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So if I some of these three together, then I get the top motion.

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This is known mathematically as an eigen mode decomposition and for guitar strings,

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anybody who's used to and plays an instrument or is a been involved in music will understand

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that the first mode on the left there is the fundamental mode of this guitar string,

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and the one in the middle is the first harmonic and the one at the right is the the second harmonic.

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And there would be a series of increasing harmonics that oscillate more and more quickly.

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And but the amplitude typically decays, as I've shown it here,

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that most of the energies in the first and then a little bit more energy in the second, etc.

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So the first thing I want to show you is how, how you generate these.

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So how did I draw those? So imagine I have a wave.

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This is my you sinusoidal little wave going up and down at the top here.

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And so it's a repeating sign you sort of pattern. And I've called Lambda is the wave length.

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So that's the the minimal repeating unit.

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So in this case, you can think of it as the distance between two peaks is the wavelength and each of those modes well.

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So my string is is pinned at the end point, so it's not allowed to vibrate at the ends.

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And so to generate a mode,

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I have to choose a segment of this wave in which the the I go through zero at each end so I can choose any two zeros to do that.

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The simplest or the shortest choice would be to choose to neighbouring zeros.

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And if I do that, then I get a wave, which just has one hump, and that gives me the fundamental.

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And so you can see by comparing how much of a of a wavelength that I get here

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that I only got half a wavelength in in between the two end points of my string.

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The next thing I could do is I could skip a zero so I could choose not the next zero, but the one after that.

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And that would give me the first harmonic. And in that case, I managed to get a full wavelength in between the two end points of my string.

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So the wavelength of this wave is half of that of the the fundamental.

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And of course, the next one, I just skip another zero. So I choose.

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I skipped two and I choose this segment of the wavefront that gives me the second harmonic.

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And in this case, I managed to get one and a half times the wavelength in between the two end points.

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So because the length of my string is fixed in order to get more and more of these waves in there,

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I have to reduce the wavelength so that the wavelength of each mode is getting successively smaller.

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So this one is the wavelength is half of this one.

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This one is a third of the fundamental. So that's how I generate the work out, what the wavelength of these modes is.

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You've probably noticed that there are all oscillating at different frequencies as well.

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So the formula for the frequency of one of these waves is that it's it's inversely proportional to the wavelength.

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So the smaller the wavelength, the bigger the frequency and the constant there sea is the speed at which wave propagate on the string.

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So that depends on how tight the string is.

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If I change, the tension I don't see depends on what it's made of, whether it's metal or nylon, and depends on how thick it is, for example.

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But for each of these modes, C would be a constant. So the way the frequency changes between modes is all down to what how the wavelength is changing.

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So from this formula, I could work out what the frequency of the fundamental is.

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And then because the wavelength of the first harmonic is half that of the fundamental

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that tells me that the frequency is going to be double that of the fundamental.

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So whatever I got for omega one, the frequency of the first harmonic is going to be twice the frequency of the fundamental.

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And for the next harmonic, the wavelength goes down again.

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So it's it's a third of the fundamental and therefore the frequency is three times the fundamental.

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So the string I played corresponded to a two.

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So this is the musical notation in which middle C is C four.

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And so the two is sort of two octaves below that, and it's the edge of that scale and that it's got a frequency of 110 hertz that string.

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So on a piano, it would be this key here. And so the the first harmonic I double the frequency.

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So that would get me to 220 hertz, and that corresponds to going up by an octave.

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And so that would be this key there on the piano the same way, but an octave higher.

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The for the next harmonic, I'm three times the frequency, so that would be three 30 hertz.

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And that's an e e for here. The E above middle C.

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So from that formerly, you can see that that long strings of lower frequencies that each higher mode is is an integer multiple of the fundamental.

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So a start with the fundamental and then I have two times the frequency and then three times etc. And whenever I double the frequency,

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I raise the pitch by an octave. So now it's always dangerous to do things live, but I'm going to try.

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And switch. We go and show you that this is really true.

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So in the old days, I'd have had to bring loads of equipment in here to show you what frequency my guitar is playing these days.

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You just go to the App Store and download an app for it. So this is my phone here.

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Which? I decided to die.

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Oh, come on. Let's try again.

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Oxford Wave Research. Right, there we go.

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So it's it's giving you frequency on the horizontal axis,

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and it's giving you how much energy there is at that frequency on the vertical axis and decibels.

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So it's a logarithmic scale on the vertical axis. So if I play by a string, you should see it.

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I said it was a hundred and ten. If I stop talking so that you can see the guitar.

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So you see the peak at the far left 110 would be about here,

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and you see that the fundamental is that 110 and then you see a lot of harmonics over the top of that.

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Now, if I if I touch the string in the middle when I'm playing this,

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that kills the fundamental because the fundamental wants to vibrate in the middle,

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but it doesn't kill any even harmonic because the all the even harmonics are two zero in the middle.

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So if I play the string and touch it in the middle, I get every second harmonic.

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So. And I can do the same thing if I touch it a third of the way down, then I get every third harmonic.

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It doesn't matter whether I do it a third here or I could do it a third down here.

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The tone that you get out of it depends on the ratio of the harmonics, so how much is in the fundamental and how much is in the other mode.

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So if I play the string somewhere near the middle, I get a lot of the fundamental, not much of the harmonics.

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And you can see that as soon as I talk, you can see, but you can see that there's a large peak,

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it's a logarithmic scale, remember, so there's a large peak on the left and it decays quite rapidly as I go down.

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That's a nice round note, if I play down this end, I generate a lot more of the harmonics and much less of the fundamental.

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And you see, it's a much tinea sound. Trying to generate it so that the same amount of energy in the fundamental.

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Right. And we see in polls that.

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Perfect. So that's all sort of by way of warm up, really, that strings are nice and easy to visualise.

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But what I really want to talk to you about is pipes, so pipes behave in a similar sort of way.

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And I still have waves that go up and down. So for the the speed of the waves depended on what the string was made of four pipes.

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It depends on what the stuff inside the pipe is made of. So in particular, if I've got air in my pipe, which is the norm,

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the speed of the wave depends on the compressed ability of the air and the density of the air.

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And so I just write that up here because we'll come back to it later. So here is my diagram of pipes.

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So let me try and explain what's going on here. So just at the moment, just think about the top row.

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So what's happening? Where is the guitar? The wave is a transverse wave, so the the displacement of the string is at right angles to the string,

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whereas for a pipe, the air molecules shuttled back and forwards along the pipe.

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And so it's a longitudinal wave, and that makes it harder to draw. So what I'm plotting at the top here is actually the pressure.

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So this is the pressure that the air would feel the air inside the pipe.

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And you can recognise the modes that I had from my guitar, but they're just in terms of the pressure now underneath.

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I'm showing you the the colour here is the density, so the pressure and the density proportional to each other.

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So when I have high pressure, I have high density and I'm colouring that blue.

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When I have low pressure, I have low density and I'm colouring the white.

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And then I'm also showing you a few representative air molecules, and because the more molecules there are, the more dense the air is.

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So there are more dots in the dense region than there are in the nothin's region.

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So now let me this is an animation now of explain what the bits are.

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I can show you what waves look like in a pipe. So you can see the molecules shuttled backwards and forwards.

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And as they do, the pressure goes up and down and the density goes up and down.

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But the modes that you get are exactly the same at the top as I had from my guitar.

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So the so first, I should explain what this picture is as well.

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So you you've always got to have some mouthpiece or something to actually generate the

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disturbance in a pipe that's sort of equivalent of my finger on the guitar string.

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And so this bit at the end here is that's just the mouthpiece. You should think of the pipe as being from this hole here to the end here.

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And typically, there's a hole here and this is a pipe which is open at the end.

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And so at this end, the pressure has got to be equal to whatever it is atmospherically.

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So the pressure perturbation has to be zero here. And at this end, it also is zero.

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So that's exactly like the guitar string that the pressure has to vanish at either end.

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And I get exactly equivalent modes, but with pipes, I can also I could close the end of the pipe.

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I could put a stop in here.

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And if I put a stop in the end of the pipe, that means so you see that the the air molecules here keep shuttling in and out of the side.

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Well, that stops that happening here.

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And that means that instead of having the pressure equal to atmospheric instead of the pressure of perturbation being zero,

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then it should be a maximum or a minimum there because it says I'm not allowing the air to go, so you end up cutting this wave at its peak.

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And so you can see that this is as I had before,

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I have half a wavelength in here and then I have a full wavelength and then I have one and a half year.

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I've got a quarter of the wavelength in there only.

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And then the first harmonic, the next one, I've got three quarters and then I've got five quarters.

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So that tells you if I translate that into frequencies, I suppose this pipe was such that the fundamental frequency was 220 hertz.

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Then we've already seen that the higher harmonics would be integer multiples of the first, so this would be double.

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So that corresponds to going up an octave, and this would be times by three, which corresponds to going up another fifth.

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If I take a pipe the same length and I close the end,

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then I've changed the wavelength and I've reduced it by a half because I only got a quarter of a wavelength in now.

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So the wavelength is four times the length of my pipe, not just two times the length.

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So that tells me the frequency goes down by half, which tells me the note goes down by an octave.

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So the close by has a lower note by an octave. And the first harmonic is three times that.

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So the first harmonic is actually f e four, not a three.

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And then the next harmonic is five times it. OK, let me switch back and see if I can demonstrate that.

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We've got a few toys that I bought online here. Let me first show you that that the length of the pipe determines the note, so it's a close pipe.

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And as they change the length, the note goes up. Let me change the scale on here, because this is obviously much.

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It's much higher note than my guitar string, so I have to change this.

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Let's go to two thousand. So this was the closest I could get to a pipe really is cylindrical IV.

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It's a tin whistle. I've I. It took all the holes because I couldn't do all the holes at the same time as everything else.

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So if I blow it, it really is just an open pipe at this length. It says on it that it's a D, and I think that means.

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That's better for you not to blow too hard if you blow too hard, you end up getting the higher harmonics and sounds awful.

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So I think it's around about 600 hertz this. And you can see that the harmonics are at twelve hundred and eighteen hundred, right?

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You're really getting one two three. Right now, I'm going to put my finger over the end and try again.

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Now it's designed to be an open ended pipe this.

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So as soon as I've put my finger of bend, the mouthpiece is not really right for this, and it's it's quite hard to get it to go with the fundamental.

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If I just blow without thinking about it, I'm going to hit the first harmonic. It sounds like it went out right.

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I told you it should go down. It's because I'm hitting the first harmonic, and you can see that I am because I'm getting nine hundred.

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If I blow really gently, I can get 300. Are you convinced?

153
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OK, so you can just see that you get three hundred and then you get nine hundred and then you get fifteen hundred right?

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One three five? You try one more time. OK.

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More or less convincing. And me.

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Good. So this sort of explains a little conundrum, actually, that so if you think of the flute and the clarinet,

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so those are the two sort of orchestral instruments that are both roughly cylindrical in which you might apply this theory.

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So and they're both about the same length. So a flute, I looked it up a flute is about sixty six centimetres long.

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And so if I look at what the fundamental the lowest note that a flute can play is, so it's open at both ends because you blow across the mouthpiece.

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So it's open at this end. And of course, it's open at that end. So that's an open pipe.

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The I have a fundamental mode, which is half a wavelength.

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So that tells me that the wavelength should be twice of sixty six centimetres.

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The speed of sound in air is about 30 40, so that gets me to two fifty seven hertz, which is about middle c.

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And I think that is the lowest note that a flute can play any flautist in the audience.

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A clarinet, on the other hand, it's about the same length, slightly shorter 60 centimetres.

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But it has a reed at this end and no opening, so it's effectively a closed pipe.

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At this end, all the air comes out this and. And that tells me that it's behaving like a closed pipe.

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So for the fundamental, the wavelength should be four times the length, not twice the length.

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And so four times the length gets me to roughly half this frequency, which is about an octave lower,

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I think it gets down to D rather than C three, so clarinet is about the same length as the flute.

171
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But it's actually an octave lower. All right.

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What does this have to do with vacuum cleaners? So the these resonant cavities like my tin whistle.

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So if I stimulate them, they can produce sound. But if I, if I have sound already, then they can take it away.

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So they you can use them to get rid of noise as well as to generate noise.

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And and they are used quite often in that way.

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So imagine this is a sort of schematic of how you might want to use one.

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So I have some piece of equipment which typically has a final something.

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At some end fans are noisy and it generates some regular, some noise with a bit of particularly known frequency.

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So I have some noise. This is my way of coming along this pipe and I'd like to get rid of it.

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So one way to do that is to is to stick a little clarinet or something similar off the side of your pipe.

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And so this cavity here will have resonant modes, and we've seen that the the resonant, the mode, the wavelengths at which they will resonate well,

182
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the fundamental one is is the wavelength is four times the length of the cavity and then

183
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the first harmonic is a third of that and then the next harmonic is the fifth of that,

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et cetera. So if I if I match the wavelength, the frequency of the wavelength of the incoming sound to one of these modes,

185
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then the cavity will resonate and it'll suck that energy out of the incoming wave.

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And I won't get anything out of my outlet. And typically, you want these things to be as small as possible.

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And so you choose the fundamental. And so for the fundamental, the lens should be a quarter of the incoming wave length, and that tells you that.

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So that's why the name quarter wavelength resonator comes from.

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So let me just give you a schematic idea of how this thing is working.

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One way to think about it. So you imagine you have I my way of coming along here and when it hits this cavity, some of it is going to go down.

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So as my wave comes along, some of it goes down. The one on the top keeps going.

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The one here bounces from the bottom and comes back up again and then meet the one that was coming along the top.

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Now the one that went down, it went down a quarter of a wavelength and then it came up a quarter of a wavelength.

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So it's gone a half a wavelength altogether. And so when it gets back to the top, it's exactly out of phase with the one that was coming.

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So where is the one that Cummings now got a trough and this one's got a peak and vice versa?

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And so they destructively interfere and they cancel each other out and it gets rid of the noise for you.

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So the problem is that one problem is very great for getting rid of high frequency noise because you

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need short cavities that you don't really want a clarinet stuck on the end of your vacuum cleaner.

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So it's not so good for getting rid of low frequency noise. And so I want to tell you about an idea that some colleagues had in order to use

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this idea to reduce the frequency of which you could get this idea to work for.

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But to do that, I have to first take a turn into left field and tell you about invisibility cloaks.

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I realise we're making this slide. It's really hard to illustrate an invisibility cloak.

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Here's one that I drew earlier me demonstrate. Fortunately, everybody knows what I mean when I say invisibility cloak.

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This is what they are. According to Hollywood. And so this is what I would mean mathematically by an invisibility cloak.

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So I have some sound wave coming in the red.

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Here is the low pressure. That's what was white before. So I have some wave coming in.

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I have something in the white. Here is the thing that I want to shield. And the light blue here is the cloak.

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That's the thing that I'm I'm going to produce in order to shield what's inside.

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And what I would like to happen is that when the wave has come all the way through here, it looks like it looks like nothing happened.

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The wave back here looks as though there was nothing in between.

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And if that's true, then any person listening or if it was still if it was light looking here would see it as though it just passed right through.

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OK. How might you get that to happen?

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Well, you want to design your cloak in such a way that if you're thinking in terms of light

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that the light coming in gets to the cloak and then it bends around around your object,

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comes back together and goes off again.

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And if you could do that, then somebody somebody stood here would see stuff here as though there was nothing in between.

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It's like the old periscopes you sat with a kid where you reflect off a mirror and back, you can imagine doing it.

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This is a similar sort of idea. So but bending light sounds like a strange phenomenon, but in fact, we're all used to light bending.

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You've all seen it at some stage.

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So this is this is a hot day with a tarmac road and you you see this mirage here where it looks like there's water on the road.

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Of course, you know there isn't any water because you've all been up there and there's no puddle by the time you get there.

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And this is just it's again, it's a bending of the light.

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What's happening is that the road gets very hot, so the air near the road is hotter than the air above.

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And and so the hot air gets less dense and as the light goes from a high density to a low density region that causes it to bend.

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And so the light that's coming down here bends and actually ends up coming up into your eyes.

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And so what you see is your brain always thinks of light going in straight lines, of course.

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So your brain thinks that this light came from some point down here.

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So you see a picture of, in this case, the camel or on the previous slide, the car.

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You see it upside down on the road and then your brain thinks, I know there's not a car upside down on the road there.

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It thinks the only the only way I can make sense of that is if there was some mirror there because I know that when light bounces off a mirror,

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that's what it does.

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And then your brain says, I know there's not a mirror on the road, but if there was water on the road, it would behave in the same way.

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So your brain has made quite a few connexions there to rationalise this upside-down image into something that it can make sense of.

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So I have to bend the light.

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So this is one way that you could imagine doing that, how can I construct something that would bend the light in that way?

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So imagine here I take a grid of coordinates, just taking a grid of my X and Y directions.

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I've chosen a point in the middle and I'm going to stretch this out. I'm going to stretch my coordinates and expand that point until it's a circle.

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OK, so I've got a map now from this is my under form state.

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And this is my stretch state where I've taken a point and I've opened it out so I can hide something inside it.

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So the the bit where it's transformed here, you see you out far away here, I haven't changed anything.

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I only transformed it locally. That's going to be my invisibility shield.

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So I've shaded the region here where the defamation is happening and that corresponds to this region over here before I did the defamation.

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So I have some I have some map that tells me how to expand this hole into this region.

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And this blue region here becomes this region here. And then I imagine if I had my soundwave on my light wave propagating on this side.

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Well, there is nothing on this side. There's just this point that I'm going to expand, but that's not going to do anything.

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So the light carries on on that side as if nothing has happened. Perfectly happy.

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And then I imagine so. I use this map to say, Well, what's the wave going to look like over here?

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OK. So I take the point here, I work out, where did it go to and then I'd map it down here.

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And so this map, as I stretch it out, will turn this picture into this picture.

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So if I could arrange for this picture to be what happens, then I'd be in good shape because the the wave is coming along here.

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I haven't moved the way back here at all. It's exactly what it was, what it would have been.

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So this if I can do this, then my invisibility will be working.

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So that's step one is to imagine this expansion and mapping this wave.

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Step two is to think, all right, instead of just taking the answer here and using the map and working out what it becomes.

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Let me work out what equation is the wave satisfying here and what happens to that equation when I do this transformation?

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So, so let me. So I'm going to show you an equation now. Don't be too scared.

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You don't have to understand it. You just have to know that it exists.

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So there is some equation over here that looks relatively simple.

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And when you do this transformation, you get some equation over here, which looks a lot less simple.

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So if there are A-level maths students in the room so that the upside down triangles here are derivatives and they'll

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all the extra mass you get in here is just from applying the chain rule after this transformation of variables.

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It's not so important. Exactly what this looks like, just that you can do it.

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And so in principle, if I now solve this equation, I should get the same answer because it doesn't matter if I solve and then map or if I maths,

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if or if I just solve the map first and then solve.

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And in fact, that's true. So this is what the wave would look like if there was no circle.

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This is what it would look like if you solve the wave hitting the circle just to show you

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that you get a reflected wave and you get a shadow behind here and it looks very different.

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And this is what happens if you show if you solve that transformed equation.

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So you get a disturbance in the shield, but by the time you get out the back, everything is going along nicely again.

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OK, so that was step two, there are three steps. The third step is to think, All right.

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All I've done is mathematical manipulation. So how would I actually build it physically?

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And to do that, you have to look at this equation and say, All right,

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suppose I had a material in here and suppose I told you the sound speed depends on the compress ability and on the density.

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So this is what the equation would look like for a material where I'm varying the compatibility and the density.

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So this is the reason I had to show you the equations, all you have to do now is say, how do I make this into this or how do I make this into this?

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If I could choose the density to be this thing here? Then that material would behave as a perfect cloak.

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I've got the I've got the formula that tells me what material I should construct.

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Unfortunately, it's not so easy to do that because we're used to thinking of density as just being a number and not for the mathematicians.

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This thing up here is a matrix. So what does that mean?

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That means that the density of the air is different if the wave is going in that direction or that direction,

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or if you if you like to think of it in terms of particles,

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it means the mass of a particle is different depending on which direction it wants to go in crazy.

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How can it be possible? So it is a crazy idea.

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And but but it's not impossible. And to explain why it's not impossible, I have to go to my other subject, which was metamaterials.

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And that's the idea that if if you mixed together two materials,

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the resulting mixture can have some quite strange properties compared to the original material.

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So let me give you an example of that. So we're talking about a sound wave, so let me think about I told you the speed of sound in air.

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It's about three hundred and forty three forty three metres per second. The speed of sound in water is a bit more.

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It's about 1500 metres per second. What about if I have a few air bubbles in my water?

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So what's the speed of sound in bubbly water? That's my mixture of materials.

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So let me draw a graph. So this is the speed of sound on the vertical axis.

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This is the volume fraction of air. So if I'm here, I've got no air, I'm pure water.

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So it's fifteen hundred. If I'm at this end, then I'm completely air.

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So it's 340. You might imagine that if I've got a 50 50 mixture of water and air,

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then I should just take the average of these two and draw a straight line between them.

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But you'd be very wrong. So the actual sound speed of bubbly water looks something like this.

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So not only is it not the average for almost all volume fractions, it's much less than either the pure water, all the pure air.

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It's it's I mean, it gets down, it gets down, so you can't really see what no, I've got to if I zoom in a bit, it gets down to about 30.

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Almost walking speed. So why is that happening?

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How is it possible? And it's all to do with with how you average.

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So let me I told you that this the speed of sound was the square of the speed of sound as one over the compress ability times the density.

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So the first thing is that you shouldn't you should really think of this the other way up.

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That it's one of the speed of sound squared is compatibility times density.

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And then it turns out that it's not right to average the product.

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What you should do is do the average of the individuals and then multiply them together.

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And so it's not obvious why that's the right thing to do, but it turns out it is the right thing to do.

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And these materials are actually quite different. So if you look, the density of water is about a thousand times the density of air.

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So when you are averaging a bit of water and a bit of air, it basically looks like water as far as the density is concerned.

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But the compressed ability of air is about 10000 times the compressed ability of water.

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So when you average a bit of water and a bit of air as far as compatibility is concerned, it basically looks like air.

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So you have a material which has the density of water, but the compressed ability of air.

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And that means that each of these averages is way bigger than than you might expect,

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and that means the speed of sound is way lower than you would expect.

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So by mixing together two materials, you've got a material which which is not at all like either of the individual materials.

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It's very different. What about anisotropy?

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So I said, you have different things in different directions. Well, now you've got this idea of mixing materials.

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It's not so hard to think, well, if my if my air bubbles were not little circles or spheres, but they were stretched in some way.

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So in particular, I might have these ellipses here. It's not such a such a wild thing to think that a wave propagating in this

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direction is going to behave differently to a wave propagating in that direction.

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And so you can get this anisotropic behaviour by having this microstructure material where you you have some sort of non symmetric shape locally,

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and that gives you this different propagation in different directions. All right, what's it got to do with vacuum cleaners?

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So these colleagues up in Manchester and Cambridge will Parnell's group, they were working with Dyson on vacuum cleaners,

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and they had this idea to use these same sort of transformational approach to core to wavelength resonators.

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So you want to make them smaller.

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So what about if you you want something that the Soundwave thinks it's this big, but you only want to make it this big?

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So you could use the same idea you said, I'm going to map this shape into this shape and then I'm going to say,

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right, what properties does the material in here have to have in order that the sound thinks it's really this shape?

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And so they solve that problem. And and they came up with a microstructure.

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Now it's very hard to put. You can't put bubbles and structure them, but you can't put baffles in.

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So what they did was they they took this quarter wavelength resonate.

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They worked out what it had to be and then they put some baffles in here.

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They're very stretched ellipses and they tried two different things, one putting them in the middle and one putting them on the edges.

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But the gap between them. And of course, these days you have a mathematical idea.

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It's very easy to test. You just go 3D printed.

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And so here's some 3D printed quarter wavelength resonators and you can see the the elliptical baffle in here.

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And likewise, those in there. And they'd designed this microstructure so that it the sound would think it was twice as long as it was.

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And so they can therefore get the same energy,

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same wavelength of energy out of the out of the incoming sound with half the length of the effective resonator.

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And then they did some experiments.

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So this is this is the channel along which the sound wave is going, and here's they're they're called a wavelength resonator stuck on the top.

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And here's one that didn't have the baffles in and that this is where it's taking energy out of the system.

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So it's sort of the equivalent of what's happening on my phone there.

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And this one was taking it out around about two thousand hertz. If you didn't have any baffles in there, but when they put the baffles in,

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they found that this peak dropped down to around about a thousand, which is what they designed it to be.

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It should take energy out about half the frequency, and they found that the green one work slightly better than the red one.

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Right. I'm coming to a close, I wanted to show you one more example of.

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Of waves and resonance, and it's it's sort of related going back to strings, but it's to do with metal sheets.

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So if I have a sheet of metal, it will also support transverse waves.

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But metal is different to a string, a string, I have to stretch tight between two end points,

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so it's always sort of flat, whereas a sheet of metal I can bend.

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And what you find is that if you bend it a little bit, then the waves will propagate.

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But if you bend it a lot, then you can get waves on it anymore.

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And so effectively, the speed of sound depends on the curvature.

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How much you bend the way and what that means is if you take a piece of metal and you bend it into an s shape.

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Then an ashtray has high curvature here and high curvature here, but very low curvature in between.

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So you get a region where waves will propagate in between the two high region curvature regions.

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But they won't propagate past past this point or past that point.

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And now you're in a situation which is very much like a guitar string. So the wave is confined to these regions.

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It will shuttle back and forwards effectively. I'll set up my EIGEN modes with these two things at the end, as that is the length of the string.

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In this case, the length of the sheet.

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And as I change how much it's bent, I can move these points around and that will change the frequency of the note that I get out of it.

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So I've always wanted to do this, so I'm going to give you a demonstration of this.

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But if it all goes horribly wrong, bear with me because I'm not a violinist, so I might get it all wrong.

364
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I would say, don't try this at home. But in fact, I think you should try this at home.

365
00:41:19,670 --> 00:41:26,420
You should just be careful when you do it. So this is a soul that I just borrowed from my garden shed.

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You can see it's a little it's a bit rusty, so it's not very carefully chosen.

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But. Let's hope this works. If I bend it into an s shape.

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You can get a note out of it. Oh.

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00:42:10,590 --> 00:42:16,620
I'm amazed how good a note you can get out of a rusty old, so like that.

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Let me close by saying you shown you somebody who can really play this thing. So this is this is something I got from the internet.

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Herself was a bit longer than mine. That's all for me. Thank you.