Helmholtz Resonator and other Maintained Oscillators

Alright, so we are going to do today a series of demonstrations of maintained oscillators. They are systems that do intrinsically oscillate, however the way they are maintained is by an external mechanism, a forcing, that has no oscillatory behavior in itself. So we have worked out examples of two oscillatory systems that are driven by forces that oscillate in time. The parametric oscillator was one example, and an oscillator driven by an external drive was another example. So we have developed the theory for those systems. Now what I want to do is show you qualitatively how to represent other oscillators that are not driven by oscillatory forces but by DC forcing. So we have an external mechanism, some kind of source, that has zero frequency and interacts with an oscillator. And there is somehow in the interaction mechanism between the source and the oscillator, there is a modulation of the energy flow going from the source to the oscillator as the oscillator feeds back into the source. A very simple example of this is what is called the Helmholtz resonance, the Helmholtz resonator. A Helmholtz resonator is a fixed volume object that has also a neck, in this case that's the neck of the resonator. The volume for this particular Helmholtz resonator is the volume within that column. I have put water there, so I have made, water is seen stiff, like a rigid termination right there. That's if I had the bottom of the wall, of the bottle. And so my volume is the volume in that particular enclosure. And I can excite sound, which is clearly an oscillation, by blowing into the bottle, which is, and the blowing part is clearly not an oscillation. I'm not making the flow of air from my mouth through the neck of the resonator, or the But the interaction between the flow and the natural frequency of the oscillator modulates the energy flow to the oscillator as the oscillator feeds back into the source that is producing the flow, which in this case is my mouth. So there is an interaction, a very active interaction between a source that does not oscillate, and an oscillator that has a particular frequency of oscillation. The Helmholtz resonator, just, we're going to see this later as well, but just so we understand it, it's just a volume that has a neck, and you can change the neck, extend it, shorten it, whatever. You have a volume V, and this system mechanically is equivalent to a mass and a strength. So there is only one resonant frequency to the Helmholtz resonator. Given the volume of the resonator and the particular neck, we would have only one resonant frequency. And so this system is equivalent to a mass, the mass in this case would be the plug of air that is oscillating back and forth, and a spring, and the spring is proportional to the inverse of the volume of the resonator. So if you increase the volume of the resonator, the resonant frequency goes down. So let me show you that. I have that particular volume, now I'm going to increase it by just dumping water into the cup. So clearly there was a reduction in the frequency by the increase of the volume. In fact, the resonant frequency of the resonator decreases as the inverse of the square root of the volume. We'll spend a little more time in Helmholtz resonators later on in the course, but just for awareness, that is the relationship, that is the mechanical equivalent, and we have again an example of a maintained oscillator where I'm using a DC source, the blowing, that couples the energies modulated by the interaction between the source and the Helmholtz resonator. The Helmholtz resonator is, in this particular mechanism of excitation, is a very hard thing to understand actually. There's ongoing research on that. But just a first approach to try to understand this is that the blowing of air through the neck generates vortex shedding. There are vortices that are being shed as the airflow goes through, and those vortices are going to be, the frequency of those vortices are going to be modulated, the way they shed out is going to be modulated by the resonant frequency of the resonator. So a complicated mechanism, but fundamentally the root of the modulation of energy is coming from vortex shedding. So that is one example of a maintained oscillator. Let me show you another one. This is what is called an escapement mechanism. What is the oscillator in this case? The oscillator in this case is this pendulum going back and forth. And what is the forcing? The force is a DC force, is the falling of this weight, gravity. Gravity is acting on this weight to bring it down that moves, that spins this wheel. But this wheel is stopped at two particular places, those dents, those indentations. And so this escaping mechanism results in an oscillation driven by a DC force, zero frequency. But there is an interaction between, there is a modulation of the energy between the DC force and the oscillator, and that interaction is taking place, that modulation rather is taking place with the teeth of this wheel. So notice that there is a particular orientation that would allow that flow of energy be modulated. So I have my oscillator is the pendulum. You can actually see the oscillation also being rendered here in the escaping mechanism. So this is the mechanism that makes clocks work, okay? That's the fundamental, you know, the grandparents, the grandfather clock works through this very simple mechanism. You wind it, that means you increase, you bring the weight back up, and then you let it go on. The number of teeth on the wheel will essentially determine the duration of your winding, okay? And also the excursion. So this will go on for as long as the weight is falling. So let me stop it right there. I think we are, we got the picture, right? Are these demos decided here, or does another demo? This one was designed and fabricated here. This is a 3D printing, as you recognize a 3D printing, so this was designed here. The rate of escapement is again determined by the spacing of the teeth, so you want to several, right? So you can... In the case of the clock, how does the weight come back? You pull it, right? I don't know if you've seen those, on a grandfather's clock, there are a couple of things, a couple of weights you pull down, and that lifts a weight up. Some of the newer ones, they were huge. That too, yeah. But the real old ones, you don't wind, you just count, you have counterweights to pull the weight that does the escaping, okay? There is another interesting demo along those lines. You know, these kind of clocks were very popular back in the old days, right, in the 1700s. There was a big effort in that time to come up with, or even before then, to come up with a clock that would keep time very precisely on board a ship. And that was because by keeping time very precisely, that was a way to measure the longitude, going across different longitude lines. But you need to have a precise timing. So cartography depends very closely, I mean, it's closely dependent to having a good time. I mean, not a good time, having a good timing mechanism, a way to measure time very precisely. You want to measure the coastline, you need to know how you're moving about the various longitudes. There was, back then, a very important prize. There is actually a book that talks about the longitude problem, measuring the longitude. With fireworks, one of the ideas was to put a ship with fireworks to tell the guys how... Yeah, but then that doesn't keep time very well. So the idea was to have a clock on board the ship that would keep time without being perturbed by the motion of the ship. So you need to... There was the design, the mechanical design of the clock had to be very, very sturdy. And a lot of people tried to compete on that prize. There was a lot of money involved in winning the contest. Huygens and his father and son, they both work on clocks. The whole point of them working on clocks was to enter into the contest. One of the things that Huygens noticed is that as they were designing clocks, they would hang them on the wall. And apparently the walls of the houses back then were very thin, very easy to flex. They were not rigid walls, very easy to flex. And I think that is true in Holland. The walls of the homes are thin and easy to flex. So as he mounted the clocks on the wall, some clocks would go faster than others, right? But as he mounted the clocks on the wall, they noticed that eventually all the clocks would start syncing. Independently of each other, they would not. But mounted on the wall, they would synchronize all of them. And that was the result of coupling of the clocks through the wall. That is another way of, that is another form of a maintained oscillator. We have a third oscillator that sort of like has a very different frequency that manages to couple these two guys. So this is an example of the Huygens problem. So I have these two metronomes here, and they measure time differently. And you can hear it, they are now off. Eventually they are going to come back, after some time they are going to coincide. And then they depart. Okay, so let me now mount it on something that can couple them. Not right now, eventually it's going to couple. Okay. No, but now they are timing the same way. Okay. So that is, again, the maintained oscillators are themselves. These guys are, you know, they have a scaping mechanism, a winding that unwinds DC and makes this thing oscillate. But these two independent oscillators are now coupled through this flimsy wall or mount. And eventually, you notice the mount was oscillating at the beginning. Eventually the amplitude of the oscillations went down. The frequency also changed. And they end up synced with this thing essentially at rest. Okay, so these two maintained oscillators managed to face lock by virtue of the coupling with a third oscillator. Okay. Well, in themselves, they are driven by DC source. They are coupled through a third oscillator. But notice that the frequency of this guy is not necessarily the frequency of these guys, right? And then the coupling, while they are syncing, while they are face locking, this is wandering. This oscillator is wandering and brings them into face lock. And eventually, the amplitude of this oscillator is reduced, highly reduced, when they finally lock out of face. Okay, there's another one. This is called the Franklin's Chimes, okay? What I have here is a capacitor that I'm going to charge with a very large voltage source like the Van de Graaff. This is a lot of voltages, probably a thousand or so or more. But no current. But it charges these capacitor plates, okay? They are connected. Each plate is connected to a different element of the source. One is connected to the sphere. The other one is connected to ground. So the charging is opposite here. And in between them, I have a ball that is a ping pong ball that is being coated with metallic painting, okay? So this is a pendulum, but the frequency of the pendulum is determined by the length of the pendulum, right? And gravity. What you're going to see is not the pendulum frequency. It's just simply a ball bouncing back and forth between the two walls of the capacitor. The string there is just to maintain the ball. That's all. There is no pendulum here. It's just a ball. Imagine the ball is free. Or gravity is zero. You have two capacitor plates. And you have a conductor, conducting ball that can undergo displacement in only one line, okay? Which is what the string is doing. Just restricting that. So notice that if I have this ball place a little bit off center, okay? Then what is going to happen is that there is going to be, this side is going to be more positively charged than, or the density, the charge density here is going to be, positive charge density here is going to be bigger than the negative charge density on the other side because of the proximity. You know, in a conductor, electrons can move freely, or charges can move freely. So they are going to move such that the density, the positive charge density is bigger here than compared to the magnitude of the negative charge density on the opposite side. So this is going to feel an attraction here, okay? So in the next picture, then touches the capacitor wall, right? This guy has, and then what happens? So this is a conductor. When it touches the charge wall, what happens to the conductor? It gets, charges go in, right? It's a conductor. These positive charges want to move in. So there is not going to be a negative charge there that would tend to repel it. And so this just shoots to the other side. And as it goes to the other side, there is going to be a buildup of more positive charge density on this side, more so than the negative charge density on that side. Or, excuse me, there's going to be a buildup of negative charge density more so here than there would be positive charge density on that side. So this guy is going to be pushed that way. And so the subsequent motion is going to be an oscillation back and forth. So we have an oscillator, not spring, not spring. It's just an oscillator. And what defines this oscillator is the fact that this guy has a period, right? It's moving back and forth with some periodicity. The periodicity is going to be determined by the strength of the electric field in the capacitor, okay? So I increase my electric field and I can also increase the corresponding frequency of oscillation. So I'm going to charge it through here. Can you see all the ball? Can you all see the ball? There are other, there's other motion there because there is the modes of a string there. But ignore that. Well, in itself it's interesting, but ignore that. So notice that I reduced the electric field so the frequency of oscillation went down. Okay, that is called the Franklin chimes, Benjamin Franklin, okay? You know, along the lines of a Helmholtz resonator is also this corrugated tube, along the lines in the sense that DC flow is going to excite the modes of this pipe. There are several modes of the pipe. This is an open, open pipe. So the modes of the pipe are such that the pressure, the pressure at the ends, the acoustic pressure at the end has to match the ambient pressure, which is, so the pressure swing has to be zero at, right at the ends, so that at the ends you have atmospheric pressure. So if you look at the, then for a pipe, that open, open pipe, I have the modes of an open, open pipe for pressure are, have nodes at the ends, et cetera, right? So you can feed integer multiples of half wavelengths in the pipe. And so where is the airflow coming from? Well, it's coming from, this guy's been spun. Higher flow, higher flow excites higher modes. So the higher flow would, would have the ability to shed vortices at a rate that is, that is faster, okay? Kids love this. Parents hate it. Kids love it. There's another interesting demonstration of also a maintained oscillator, which is, there's the consequences and something that, that oscillates. There is a timing, you know, there is a on and off, on and off with a periodicity. And the timing is determined by, by, by elements that themselves do not oscillate, okay? So let's take, let's take a look at this circuit. I have, I have a voltage source, DC voltage source, switch, capacitor, resistor, look at me, okay? Capacitance R resistance, a capacitance C resistance R. And then this capacitor is in parallel with, with a, with a neon, neon light, light bulb, okay? So when I turn the switch on, what happens? This capacitor is charged, okay? And then the, when the capacitor, the voltage across the capacitor reaches the breakdown voltage of the light bulb, then the capacitor discharges on that, okay? So to, to get there, right? To get there, remember to charge a capacitor to a given voltage, the voltage in the capacitor, you have a source, a capacitor, and a resistor. The, the voltage of the capacitor is going to be, as a function of time, is going to be V times 1 minus E to the minus T, can you help me, over RC, okay? The time constant, the capacity time constant, okay? So it takes some time to get, to get to a voltage that eventually matches the voltage of, first of all, I have to have a voltage source that, whose voltage is bigger than the breakdown voltage of the neon light bulb. And I would be charging the capacitor with that source, and when that voltage matches, when the voltage across the capacitor matches that breakdown voltage, then the capacitor discharges right there. So this is a system that does not oscillate, and that was another, right? This is a system that does not oscillate, clearly, right? But the consequence of, of this coupling, coupling these elements, electric elements in the circuit, is that I have something that, that has a periodicity, repeats with some timing. And so the timing that it should repeat with is about the RC constant, okay? So I don't know if you can see it from there. I have, I'm driving right here, this is my source. There is a resistor right there, okay? There's only one resistor, I can put more. And there is one capacitor, and that is my neon bulb. Can you see that? I'm going to turn the lights off, so you can see the lightning of the capacitor, of the neon light bulb. Okay, can you see that? Now it's blinking with a periodicity that is determined by the RC constant of, of the, of the system, okay? So I can reduce C by putting another capacitor in series, right? So let me, let me take this off. I'm going to put another capacitor, reduce C, and okay, so RC comes down, so the, the period between flashes, the time between flashes is reduced. Similarly, I can reduce the resistance, right? By putting a resistor in parallel. Let me see, did I, oh, screwed up, okay. So I doubled the resistance and doubled the, I mean, half the resistance and half the capacitance is a quarter of the timing, okay? So this is another example of a maintained oscillator, where there is a DC source, clearly this is a DC voltage source. The time, the periodicity is determined by, by something that does not oscillate, but has an intrinsic time constant. RC. Okay, I don't think I need power for this, so let me take, so let me go back again to an open, open pipe. In an open, open pipe, the acoustic pressure is, say, let's take the fundamental now, it's such that the pressure, the pressure is, is, is a null, the acoustic pressure is, is a null at the ends of the pipe, so that it matches the, the atmospheric pressure, right? And why is that? Because if there is a pressure discontinuity at a, at a, at a surface, the surface would have infinite acceleration, right? So a pressure difference is force per unit volume. And so it's a, and, and if you have no mass in, no volume mass in a surface, the, the surface would have an infinite acceleration if there is a pressure difference across. So the pressure has to match, the acoustic pressure has to match the, the, or rather the, the pressure at the ends has to match atmospheric, so that's why the acoustic pressure has to be zero at the ends. This thing oscillates back and forth, and it would have a maximum, maximum in the center. So this is pressure, and it oscillates, that is half a cycle later. Corresponding to the pressure, corresponding to the pressure, there is a, there is a particle velocity, you know, there. So the, the fluid particles, the fluid at the end is, is moving, it's oscillating, okay? And it's most vigorous at the end, and it has a null right at the, at the center. So I have that, that situation right there. Velocity is oscillating everywhere, the fluid velocity, the gas velocity, the gas is oscillating everywhere in the pipe, and, and there is a phase relationship between the two ends. Obviously, the, the phase in here is 180 relative to that one, okay? So, what else? Yeah, I mean, if I pick a spot on the pipe, somewhere there, I have both pressure and velocity swings. Pressure and velocity swings, they are oscillating, both of them back and forth. So as a function of time, I would have, I would have this situation. As a function of time, I have the pressure, say it's doing that, okay? It's a function of time. So this is the pressure swing, the acoustic pressure swing, and the velocity, as a function of time, is, if I look at that end, it's doing something, something like that. So when the, when the pressure is a null at that place, the velocity is, say, a maximum right there. And when the pressure is maximum, the velocity is a null right here, I would have a minimum in the velocity, a null right there, and so on, okay? So that's how, that's how the velocity is doing. Let's call that V, as a function of time. Now, if I place right there, somewhere right there, an element that, and I heat it up, okay? Then, another thing to notice is that when the pressure increases, yeah, in a compression, the gas gets hotter. In a rarefaction, the gas gets cooler. So this could also represent the temperature swings in the pipe. So, where's my, let me take the following setup. Ah, okay. Can you all see that? Really, the important thing is this, not so much the flame. There's going to be a flame there. I have a pipe here, and right somewhere around the quarter, quarter distance from the end, I have a gauze, a metallic gauze, gauze, like a mesh, a mesh, metallic mesh. Can you see that? All right, okay? So it's metallic, it has to be metallic so I can really heat it, I can dump heat into it. And I'm going to dump the heat of the flame, okay? So let me turn the flame on. Ignore, ignore the sound that you're going to hear first. That is not, I'm not interested in that one. I'm interested in a different aspect of the demo. And when I put this in, there's going to be some sound. Ignore that one. I'm going to remove it. That's what I, that's what I want you to pay attention to. Ignore that one. They say I'm pouring out the sound. There's no sound left. Only one orientation can do that. I love that equation. So what happens when I'm moving? There is air flow, okay? There is, ah, it doesn't come anymore. Oops, I don't like this. So let me try again. Only one direction. Not the opposite direction, okay? So when I move or when I stand it up, there is flow from moving up, moving across the metallic mesh that I have right there, okay? So, so on, not only do I have particle velocity oscillating, but also have a net flow when I stand this upright and the mesh is right there, okay? Quarter, quarter wavelength, okay? What happens to build this pressure? I need to have particle velocity rushing in, right? So I have to have, to get to that pressure, as say, imagine this is going to the, to that spot to where the pressure builds up, okay? Then for that to happen, gas has to converge so I can have a region of high pressure, okay? And so the particle velocity at this end is, say, is moving in the positive direction, say, or if I put it, have it upward, it's moving upward, okay? But over here, it's not. It's moving downward. Now, on top of the particle velocity, I have a flow because I have convection. I'm forcing convection by the heat, by having a hot spot right there. So air is rushing from under upwardly, right? Hot air rises. And so this, that means that I have to superimpose to this velocity a DC flow. So when I superimpose a DC flow, that is what is happening to the velocity, right? It's not, the nodes of the velocity don't happen here anymore. Perhaps they happen lower because now I have a superimposed DC flow. So if you add this cosine function, say, with, you add that, a flow, right? Constant flow in the same direction as the velocity swing, then you are raising the overall thing relative to the zero line, right? So the nodes, the velocity nodes are there, but they are not there anymore, okay? So you raise that. That clear? Okay. So what happens then? If you have a net upward velocity, then a gas, a fluid gas, a gas particle, a fluid particle is going to be ascending up the pipe. But as it ascends up the pipe, it's going to reach, you know, as it comes to the gas, it's going to heat up, okay? And it's going to heat up, say, if I place, looking at the gas in this particular location, right? The pipe upside up. The fluid particle is going to be ascending with some oscillation. If you integrate this as a function of time, then you would have the displacement of a gas particle as a function of time, which is, I don't know, it looks like this, okay? It's ascending. And I missed my periodicity, of course. So it has the following. Let me, I have my drawing here, so let me pick it up. If I look at the y-coordinate as a function of time of a fluid particle, so this is what is doing as I integrate this guy, okay? There's a net displacement. So this would correspond, this line here would correspond to the line y equals v naught t plus a constant. So imagine that I have a constant flow, okay? Which is okay. We can approximate that. We have a constant flow, so this would be, this would be the y equals v naught plus a constant. What is the constant? Well, the place where I am at, okay, in the pipe. So a fluid particle is going to be, say, undergoing excursions like so. One that is higher up is going to be undergoing with the same periodicity, but is now going like that. And now imagine that at some point you have the wire. That's where the wire mesh is. So a fluid particle comes from a region that is, say, room temperature, goes through the gauze, goes through the mesh, the metallic mesh, and then gets a temperature increase. And if you are in the right place of the pipe, as I've drawn, the pressure would be this if the velocity is like that, when the gauze is a quarter of the distance. Then I'm going to bring a fluid particle that has an excess amount of temperature into, at the time where the sound is compressing. If I dump temperature, if I dump heat on during a compression, then the amplitude of my compression is going to increase. If this represents delta T, if I add heat when delta T is positive, then delta T is going to become bigger. And bigger delta T has also bigger delta P. So the fluid particle, as it moves up the pipe, goes through the gauze, and happens to have the right phase to dump heat at the time there is a compression. Most of the time. There are times where it dumps heat at the time that there is a rarefaction. But most of the time, it dumps the heat at the time there is a compression. It's more likely that the heat is going to be transferred during compressions than during rarefactions. Because as a function of time, there is going to be rarefaction. So it's just the timing is right if the gauze is in the bottom part. If the gauze is in the top part, the timing is not right. Because now I'm inverting the relative phase between velocity and pressure. And so I'm dumping heat most of the time while there is a rarefaction. So that's why turning the pipe upside up doesn't work. I mean, upside down doesn't work. Having it side up works. OK. Moving when the gauze is in this place, moving in that direction makes it produce sound. Moving in the opposite direction doesn't. OK. So it's, again, another example of maintain oscillator. What is the energy? Where is the energy coming from the heat on the metallic mesh? That's where the energy is coming. It's oscillating. No, it's not oscillating. It's a DC source. But the way energy is being modulated is by having that convection flow that is rising, that is bringing gas particles through the metallic mesh and dumping the heat that they get during the compression cycle. So we'll have some more demos tomorrow of maintain oscillators. And then next week we'll start, we'll spend a couple of days on chaos. And then we'll start another topic after that. OK.