Thank you, Alan, and thank you for that very overly kind introduction.
So I want to tell you a little bit about waves and resonance, starting with musical instruments,
ending with vacuum cleaners and taking in some rather strange things in between.
So I want to start with my guitar and think about what happens when I play one of the strings side by a string.
You all hear the nice sound it makes. I want to think about what the string is doing, first of all.
So this is roughly what my string would be doing when I play it around about a third of the way down.
And you see, it's quite a complicated motion. It's periodic.
It's repeating,
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.
So I can describe that the motion at the top just by adding together the three disturbances that you see below.
So if I some of these three together, then I get the top motion.
This is known mathematically as an eigen mode decomposition and for guitar strings,
anybody who's used to and plays an instrument or is a been involved in music will understand
that the first mode on the left there is the fundamental mode of this guitar string,
and the one in the middle is the first harmonic and the one at the right is the the second harmonic.
And there would be a series of increasing harmonics that oscillate more and more quickly.
And but the amplitude typically decays, as I've shown it here,
that most of the energies in the first and then a little bit more energy in the second, etc.
So the first thing I want to show you is how, how you generate these.
So how did I draw those? So imagine I have a wave.
This is my you sinusoidal little wave going up and down at the top here.
And so it's a repeating sign you sort of pattern. And I've called Lambda is the wave length.
So that's the the minimal repeating unit.
So in this case, you can think of it as the distance between two peaks is the wavelength and each of those modes well.
So my string is is pinned at the end point, so it's not allowed to vibrate at the ends.
And so to generate a mode,
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.
The simplest or the shortest choice would be to choose to neighbouring zeros.
And if I do that, then I get a wave, which just has one hump, and that gives me the fundamental.
And so you can see by comparing how much of a of a wavelength that I get here
that I only got half a wavelength in in between the two end points of my string.
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.
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.
So the wavelength of this wave is half of that of the the fundamental.
And of course, the next one, I just skip another zero. So I choose.
I skipped two and I choose this segment of the wavefront that gives me the second harmonic.
And in this case, I managed to get one and a half times the wavelength in between the two end points.
So because the length of my string is fixed in order to get more and more of these waves in there,
I have to reduce the wavelength so that the wavelength of each mode is getting successively smaller.
So this one is the wavelength is half of this one.
This one is a third of the fundamental. So that's how I generate the work out, what the wavelength of these modes is.
You've probably noticed that there are all oscillating at different frequencies as well.
So the formula for the frequency of one of these waves is that it's it's inversely proportional to the wavelength.
So the smaller the wavelength, the bigger the frequency and the constant there sea is the speed at which wave propagate on the string.
So that depends on how tight the string is.
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.
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.
So from this formula, I could work out what the frequency of the fundamental is.
And then because the wavelength of the first harmonic is half that of the fundamental
that tells me that the frequency is going to be double that of the fundamental.
So whatever I got for omega one, the frequency of the first harmonic is going to be twice the frequency of the fundamental.
And for the next harmonic, the wavelength goes down again.
So it's it's a third of the fundamental and therefore the frequency is three times the fundamental.
So the string I played corresponded to a two.
So this is the musical notation in which middle C is C four.
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.
So on a piano, it would be this key here. And so the the first harmonic I double the frequency.
So that would get me to 220 hertz, and that corresponds to going up by an octave.
And so that would be this key there on the piano the same way, but an octave higher.
The for the next harmonic, I'm three times the frequency, so that would be three 30 hertz.
And that's an e e for here. The E above middle C.
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.
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,
I raise the pitch by an octave. So now it's always dangerous to do things live, but I'm going to try.
And switch. We go and show you that this is really true.
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.
You just go to the App Store and download an app for it. So this is my phone here.
Which? I decided to die.
Oh, come on. Let's try again.
Oxford Wave Research. Right, there we go.
So it's it's giving you frequency on the horizontal axis,
and it's giving you how much energy there is at that frequency on the vertical axis and decibels.
So it's a logarithmic scale on the vertical axis. So if I play by a string, you should see it.
I said it was a hundred and ten. If I stop talking so that you can see the guitar.
So you see the peak at the far left 110 would be about here,
and you see that the fundamental is that 110 and then you see a lot of harmonics over the top of that.
Now, if I if I touch the string in the middle when I'm playing this,
that kills the fundamental because the fundamental wants to vibrate in the middle,
but it doesn't kill any even harmonic because the all the even harmonics are two zero in the middle.
So if I play the string and touch it in the middle, I get every second harmonic.
So. 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 harmonics, so how much is in the fundamental and how much is in the other mode.
So if I play the string somewhere near the middle, I get a lot of the fundamental, not much of the harmonics.
And you can see that as soon as I talk, you can 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 and much less of the fundamental.
And you see, it's a much tinea sound. Trying to generate it so that the same amount of energy in the fundamental.
Right. And we see in polls that.
Perfect. So that's all sort of by way of warm up, really, that strings are nice and easy to visualise.
But what I really want to talk to you about is pipes, so pipes behave in a similar sort of way.
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.
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,
the speed of the wave depends on the compressed ability of the air and the density of the air.
And so I just write that up here because we'll come back to it later. So here is my diagram of pipes.
So let me try and explain what's going on here. So just at the moment, just think about the top row.
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,
whereas for a pipe, the air molecules shuttled back and forwards along the pipe.
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.
So this is the pressure that the air would feel the air inside the pipe.
And you can recognise the modes that I had from my guitar, but they're just in terms of the pressure now underneath.
I'm showing you the the colour here is the density, so the pressure and the density proportional to each other.
So when I have high pressure, I have high density and I'm colouring that blue.
When I have low pressure, I have low density and I'm colouring the white.
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.
So there are more dots in the dense region than there are in the nothin's region.
So now let me this is an animation now of explain what the bits are.
I can show you what waves look like in a pipe. So you can see the molecules shuttled backwards and forwards.
And as they do, the pressure goes up and down and the density goes up and down.
But the modes that you get are exactly the same at the top as I had from my guitar.
So the so first, I should explain what this picture is as well.
So you you've always got to have some mouthpiece or something to actually generate the
disturbance in a pipe that's sort of equivalent of my finger on the guitar string.
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.
And typically, there's a hole here and this is a pipe which is open at the end.
And so at this end, the pressure has got to be equal to whatever it is atmospherically.
So the pressure perturbation has to be zero here. And at this end, it also is zero.
So that's exactly like the guitar string that the pressure has to vanish at either end.
And I get exactly equivalent modes, but with pipes, I can also I could close the end of the pipe.
I could put a stop in here.
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.
Well, that stops that happening here.
And that means that instead of having the pressure equal to atmospheric instead of the pressure of perturbation being zero,
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.
And so you can see that this is as I had before,
I have half a wavelength in here and then I have a full wavelength and then I have one and a half year.
I've got a quarter of the wavelength in there only.
And then the first harmonic, the next one, I've got three quarters and then I've got five quarters.
So that tells you if I translate that into frequencies, I suppose this pipe was such that the fundamental frequency was 220 hertz.
Then we've already seen that the higher harmonics would be integer multiples of the first, so this would be double.
So that corresponds to going up an octave, and this would be times by three, which corresponds to going up another fifth.
If I take a pipe the same length and I close the end,
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.
So the wavelength is four times the length of my pipe, not just two times the length.
So that tells me the frequency goes down by half, which tells me the note goes down by an octave.
So the close by has a lower note by an octave. And the first harmonic is three times that.
So the first harmonic is actually f e four, not a three.
And then the next harmonic is five times it. OK, let me switch back and see if I can demonstrate that.
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.
And as they change the length, the note goes up. Let me change the scale on here, because this is obviously much.
It's much higher note than my guitar string, so I have to change this.
Let's go to two thousand. So this was the closest I could get to a pipe really is cylindrical IV.
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.
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.
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.
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?
You're really getting one two three. Right now, I'm going to put my finger over the end and try again.
Now it's designed to be an open ended pipe this.
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.
If I just blow without thinking about it, I'm going to hit the first harmonic. It sounds like it went out right.
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.
If I blow really gently, I can get 300. Are you convinced?
OK, so you can just see that you get three hundred and then you get nine hundred and then you get fifteen hundred right?
One three five? You try one more time. OK.
More or less convincing. And me.
Good. So this sort of explains a little conundrum, actually, that so if you think of the flute and the clarinet,
so those are the two sort of orchestral instruments that are both roughly cylindrical in which you might apply this theory.
So and they're both about the same length. So a flute, I looked it up a flute is about sixty six centimetres long.
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.
So it's open at this end. And of course, it's open at that end. So that's an open pipe.
The I have a fundamental mode, which is half a wavelength.
So that tells me that the wavelength should be twice of sixty six centimetres.
The speed of sound in air is about 30 40, so that gets me to two fifty seven hertz, which is about middle c.
And I think that is the lowest note that a flute can play any flautist in the audience.
A clarinet, on the other hand, it's about the same length, slightly shorter 60 centimetres.
But it has a reed at this end and no opening, so it's effectively a closed pipe.
At this end, all the air comes out this and. And that tells me that it's behaving like a closed pipe.
So for the fundamental, the wavelength should be four times the length, not twice the length.
And so four times the length gets me to roughly half this frequency, which is about an octave lower,
I think it gets down to D rather than C three, so clarinet is about the same length as the flute.
But it's actually an octave lower. All right.
What does this have to do with vacuum cleaners? So the these resonant cavities like my tin whistle.
So if I stimulate them, they can produce sound. But if I, if I have sound already, then they can take it away.
So they you can use them to get rid of noise as well as to generate noise.
And and they are used quite often in that way.
So imagine this is a sort of schematic of how you might want to use one.
So I have some piece of equipment which typically has a final something.
At some end fans are noisy and it generates some regular, some noise with a bit of particularly known frequency.
So I have some noise. This is my way of coming along this pipe and I'd like to get rid of it.
So one way to do that is to is to stick a little clarinet or something similar off the side of your pipe.
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,
the fundamental one is is the wavelength is four times the length of the cavity and then
the first harmonic is a third of that and then the next harmonic is the fifth of that,
et cetera. So if I if I match the wavelength, the frequency of the wavelength of the incoming sound to one of these modes,
then the cavity will resonate and it'll suck that energy out of the incoming wave.
And I won't get anything out of my outlet. And typically, you want these things to be as small as possible.
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.
So that's why the name quarter wavelength resonator comes from.
So let me just give you a schematic idea of how this thing is working.
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.
So as my wave comes along, some of it goes down. The one on the top keeps going.
The one here bounces from the bottom and comes back up again and then meet the one that was coming along the top.
Now the one that went down, it went down a quarter of a wavelength and then it came up a quarter of a wavelength.
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.
So where is the one that Cummings now got a trough and this one's got a peak and vice versa?
And so they destructively interfere and they cancel each other out and it gets rid of the noise for you.
So the problem is that one problem is very great for getting rid of high frequency noise because you
need short cavities that you don't really want a clarinet stuck on the end of your vacuum cleaner.
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
this idea to reduce the frequency of which you could get this idea to work for.
But to do that, I have to first take a turn into left field and tell you about invisibility cloaks.
I realise we're making this slide. It's really hard to illustrate an invisibility cloak.
Here's one that I drew earlier me demonstrate. Fortunately, everybody knows what I mean when I say invisibility cloak.
This is what they are. According to Hollywood. And so this is what I would mean mathematically by an invisibility cloak.
So I have some sound wave coming in the red.
Here is the low pressure. That's what was white before. So I have some wave coming in.
I have something in the white. Here is the thing that I want to shield. And the light blue here is the cloak.
That's the thing that I'm I'm going to produce in order to shield what's inside.
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.
The wave back here looks as though there was nothing in between.
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.
OK. How might you get that to happen?
Well, you want to design your cloak in such a way that if you're thinking in terms of light
that the light coming in gets to the cloak and then it bends around around your object,
comes back together and goes off again.
And if you could do that, then somebody somebody stood here would see stuff here as though there was nothing in between.
It's like the old periscopes you sat with a kid where you reflect off a mirror and back, you can imagine doing it.
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.
You've all seen it at some stage.
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.
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.
And this is just it's again, it's a bending of the light.
What's happening is that the road gets very hot, so the air near the road is hotter than the air above.
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.
And so the light that's coming down here bends and actually ends up coming up into your eyes.
And so what you see is your brain always thinks of light going in straight lines, of course.
So your brain thinks that this light came from some point down here.
So you see a picture of, in this case, the camel or on the previous slide, the car.
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.
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,
that's what it does.
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.
So your brain has made quite a few connexions there to rationalise this upside-down image into something that it can make sense of.
So I have to bend the light.
So this is one way that you could imagine doing that, how can I construct something that would bend the light in that way?
So imagine here I take a grid of coordinates, just taking a grid of my X and Y directions.
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.
OK, so I've got a map now from this is my under form state.
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.
So the the bit where it's transformed here, you see you out far away here, I haven't changed anything.
I only transformed it locally. That's going to be my invisibility shield.
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.
So I have some I have some map that tells me how to expand this hole into this region.
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.
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.
So the light carries on on that side as if nothing has happened. Perfectly happy.
And then I imagine so. I use this map to say, Well, what's the wave going to look like over here?
OK. So I take the point here, I work out, where did it go to and then I'd map it down here.
And so this map, as I stretch it out, will turn this picture into this picture.
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.
I haven't moved the way back here at all. It's exactly what it was, what it would have been.
So this if I can do this, then my invisibility will be working.
So that's step one is to imagine this expansion and mapping this wave.
Step two is to think, all right, instead of just taking the answer here and using the map and working out what it becomes.
Let me work out what equation is the wave satisfying here and what happens to that equation when I do this transformation?
So, so let me. So I'm going to show you an equation now. Don't be too scared.
You don't have to understand it. You just have to know that it exists.
So there is some equation over here that looks relatively simple.
And when you do this transformation, you get some equation over here, which looks a lot less simple.
So if there are A-level maths students in the room so that the upside down triangles here are derivatives and they'll
all the extra mass you get in here is just from applying the chain rule after this transformation of variables.
It's not so important. Exactly what this looks like, just that you can do it.
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,
if or if I just solve the map first and then solve.
And in fact, that's true. So this is what the wave would look like if there was no circle.
This is what it would look like if you solve the wave hitting the circle just to show you
that you get a reflected wave and you get a shadow behind here and it looks very different.
And this is what happens if you show if you solve that transformed equation.
So you get a disturbance in the shield, but by the time you get out the back, everything is going along nicely again.
OK, so that was step two, there are three steps. The third step is to think, All right.
All I've done is mathematical manipulation. So how would I actually build it physically?
And to do that, you have to look at this equation and say, All right,
suppose I had a material in here and suppose I told you the sound speed depends on the compress ability and on the density.
So this is what the equation would look like for a material where I'm varying the compatibility and the density.
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?
If I could choose the density to be this thing here? Then that material would behave as a perfect cloak.
I've got the I've got the formula that tells me what material I should construct.
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.
This thing up here is a matrix. So what does that mean?
That means that the density of the air is different if the wave is going in that direction or that direction,
or if you if you like to think of it in terms of particles,
it means the mass of a particle is different depending on which direction it wants to go in crazy.
How can it be possible? So it is a crazy idea.
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.
And that's the idea that if if you mixed together two materials,
the resulting mixture can have some quite strange properties compared to the original material.
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.
It's about three hundred and forty three forty three metres per second. The speed of sound in water is a bit more.
It's about 1500 metres per second. What about if I have a few air bubbles in my water?
So what's the speed of sound in bubbly water? That's my mixture of materials.
So let me draw a graph. So this is the speed of sound on the vertical axis.
This is the volume fraction of air. So if I'm here, I've got no air, I'm pure water.
So it's fifteen hundred. If I'm at this end, then I'm completely air.
So it's 340. You might imagine that if I've got a 50 50 mixture of water and air,
then I should just take the average of these two and draw a straight line between them.
But you'd be very wrong. So the actual sound speed of bubbly water looks something like this.
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.
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.
Almost walking speed. So why is that happening?
How is it possible? And it's all to do with with how you average.
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.
So the first thing is that you shouldn't you should really think of this the other way up.
That it's one of the speed of sound squared is compatibility times density.
And then it turns out that it's not right to average the product.
What you should do is do the average of the individuals and then multiply them together.
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.
And these materials are actually quite different. So if you look, the density of water is about a thousand times the density of air.
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.
But the compressed ability of air is about 10000 times the compressed ability of water.
So when you average a bit of water and a bit of air as far as compatibility is concerned, it basically looks like air.
So you have a material which has the density of water, but the compressed ability of air.
And that means that each of these averages is way bigger than than you might expect,
and that means the speed of sound is way lower than you would expect.
So by mixing together two materials, you've got a material which which is not at all like either of the individual materials.
It's very different. What about anisotropy?
So I said, you have different things in different directions. Well, now you've got this idea of mixing materials.
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.
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
direction is going to behave differently to a wave propagating in that direction.
And so you can get this anisotropic behaviour by having this microstructure material where you you have some sort of non symmetric shape locally,
and that gives you this different propagation in different directions. All right, what's it got to do with vacuum cleaners?
So these colleagues up in Manchester and Cambridge will Parnell's group, they were working with Dyson on vacuum cleaners,
and they had this idea to use these same sort of transformational approach to core to wavelength resonators.
So you want to make them smaller.
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?
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,
right, what properties does the material in here have to have in order that the sound thinks it's really this shape?
And so they solve that problem. And and they came up with a microstructure.
Now it's very hard to put. You can't put bubbles and structure them, but you can't put baffles in.
So what they did was they they took this quarter wavelength resonate.
They worked out what it had to be and then they put some baffles in here.
They're very stretched ellipses and they tried two different things, one putting them in the middle and one putting them on the edges.
But the gap between them. And of course, these days you have a mathematical idea.
It's very easy to test. You just go 3D printed.
And so here's some 3D printed quarter wavelength resonators and you can see the the elliptical baffle in here.
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.
And so they can therefore get the same energy,
same wavelength of energy out of the out of the incoming sound with half the length of the effective resonator.
And then they did some experiments.
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.
And here's one that didn't have the baffles in and that this is where it's taking energy out of the system.
So it's sort of the equivalent of what's happening on my phone there.
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,
they found that this peak dropped down to around about a thousand, which is what they designed it to be.
It should take energy out about half the frequency, and they found that the green one work slightly better than the red one.
Right. I'm coming to a close, I wanted to show you one more example of.
Of waves and resonance, and it's it's sort of related going back to strings, but it's to do with metal sheets.
So if I have a sheet of metal, it will also support transverse waves.
But metal is different to a string, a string, I have to stretch tight between two end points,
so it's always sort of flat, whereas a sheet of metal I can bend.
And what you find is that if you bend it a little bit, then the waves will propagate.
But if you bend it a lot, then you can get waves on it anymore.
And so effectively, the speed of sound depends on the curvature.
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.
Then an ashtray has high curvature here and high curvature here, but very low curvature in between.
So you get a region where waves will propagate in between the two high region curvature regions.
But they won't propagate past past this point or past that point.
And now you're in a situation which is very much like a guitar string. So the wave is confined to these regions.
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.
In this case, the length of the sheet.
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.
So I've always wanted to do this, so I'm going to give you a demonstration of this.
But if it all goes horribly wrong, bear with me because I'm not a violinist, so I might get it all wrong.
I would say, don't try this at home. But in fact, I think you should try this at home.
You should just be careful when you do it. So this is a soul that I just borrowed from my garden shed.
You can see it's a little it's a bit rusty, so it's not very carefully chosen.
But. Let's hope this works. If I bend it into an s shape.
You can get a note out of it. Oh.
I'm amazed how good a note you can get out of a rusty old, so like that.
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.
Herself was a bit longer than mine. That's all for me. Thank you.