Physics: Longitudinal Standing Waves

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When I think about a traveling wave, I tend to think about a transverse wave on a string. It's just a nice model. But there are lots of other kinds of waves in the world, and there are longitudinal waves. Longitudinal waves are, for example, sound waves. What's going on in a sound wave? You're applying pressure to the air. You're squeezing it a little. And then that squeezed air pushes on the air in front of it, which pushes on the air in front of it. And so there is this wave propagating. It's a pressure wave.
A nice analogy of sound waves, another longitudinal wave, is this pulse on a Slinky. You can send this wave traveling down the Slinky, and I can even make it a continuous wave, a sinusoidal pressure wave. And you can just see a high pressure region followed by a low pressure region, followed by a high pressure region, followed by a low pressure region, marching its way down. That's what a sinusoidal longitudinal pressure wave looks like.
So if you can send a sinusoidal pressure wave this way and you sent another one this way, what would happen? When you send two waves through one another and they've got the same wavelengths or the same frequency, then you get a standing wave. That's what happens with transverse waves and it happens with longitudinal waves exactly the same. So imagine you have some chamber and there's air in there, and you send a sound wave this way and a sound wave that way. You could create standing waves of air pressure in there. That's exactly what's going on in wind instruments. So it's really nice and practical. And of course, there are lots of other applications of this, but it's interesting to think about the physics. Can we figure out the frequencies of the standing waves inside of wind chambers?
So let me begin with the simplest kind of chamber that I can think of. Just imagine a tube that's open at both ends, just a cylindrical tube. This could be an organ pipe. Somehow you've got to drive pressure waves in there, so maybe there's a little hole and you can drive some air in and out, or something. And then you'll get a sound wave going one way and a sound wave going the other way.
Now you're thinking, "It's an open tube. That sound wave is just going to cruise on by and you're not going to get any standing wave." It's a curious fact that when you have a sound wave traveling inside of a pipe and all of a sudden there's no more pipe, you get a very strong reflection of that wave. It's not obvious why but the fact of the matter is that when the walls disappear suddenly, you get very strong reflection. It's a boundary and you get a strong reflection. And of course, if you have a wave going this way and then it reflects, you've got waves going both ways and you'll get a standing wave in there. So open pipes on both ends really do generate standing waves inside.
So let me draw you a picture. Watch, this is going to be my pipe. It's open at both ends. And I'm drawing a graph. I'm drawing it inside of the pipe, which is a little bit weird, but the reason I'm doing that is I want the horizontal axis to represent distance along the pipe. Here's x=0 and here's x=L. Once I've explained that, this vertical axis has nothing to do with vertical distance in the pipe, so let me just look at the graph. It's a graph of over-pressure as a function of position. Now, what do I mean by "over-pressure"? That's just the pressure minus atmospheric pressure. Remember, what's a sound wave? There's atmospheric pressure all over the place, and when I make a sound wave I'm increasing the pressure here, and right behind it I'm decreasing the pressure. So it's really over-pressure, which can be positive or negative, and that's what the sound wave is. It's high, low, high, low, high, low, following one another.
So what's going on in this standing wave. This is clearly the lowest possible frequency of a standing wave if you have a node at both ends. Now, why have I drawn a node at the two ends? You're thinking, "It's an open pipe. Can't I have plenty of over-pressure there?" No. The open end of the pipe is open to the atmosphere. It's p atmosphere right at the end of the pipe, and if it's p atmosphere, there can't be any over-pressure there. That means zero over-pressure, and that's what I'm graphing: over-pressure versus position. Open at both ends, so there's a node at both ends.
And the standing wave has this shape. At a particular moment in time it's the solid line. What does that mean? There's atmospheric pressure here, atmospheric pressure here, and high over-pressure in the middle. The air has rushed into the tube and there's high pressure in the middle. And then what's going to happen? The air is going to rush out of the tube, and so you'll have briefly a negative over-pressure, a low pressure inside. And the air is flowing in and out, and that's the standing wave that we're talking about. And this is the graph.
So now that I see the graph, I can immediately write that one-half wavelength--this is a half a wavelength and here is the rest--over 2 is equal to L. It's the same old condition that I had for a standing wave on a string, and in fact all the math now is going to work just the same as all the math that is involved with transverse standing waves on a string. For example, this is the longest possible standing wavelength, called the fundamental. And the fundamental frequency, v over [1] is v over 2L. Same old formula that we had for transverse waves on a string. There's a difference. On a string, the velocity is the velocity of a wave traveling down the string. It's given by the square root of t over . Here, what's the velocity? It's the velocity of an air pressure wave. That's just the speed of sound. The speed of sound is just a number. In air it's about 331 meters per second. It depends a little bit on temperature and pressure. But basically this is just a fixed number. And so a little bit different. You're just stuck. You can't adjust it with some tension knob, and so the fundamental frequency is just whatever it is. Unless you can adjust the length of your tube, you're stuck.
Here's the fundamental, and what would first overtone or the second harmonic look like? Well, this would be the graph. You would just have a node in the middle. The air pressure would be high here and low here, and still you have to have nodes at the end no matter where you are. And so this would be [2]. It would be two halves [2] equals to L, and f[2] would be equal to 2f[1]. So the formulas and the ideas now follow in exact analogy to what we had with waves on a string.
There is another kind of a tube. A lot of musical instruments are not open at both ends. Some musical instruments are closed at one end, and we call this a closed tube. You know, you might think a closed tube should be closed on both ends, but then you don't really have a musical instrument because there is no way to get air in and out and there's no way to get sound in and out. So when you talk about a closed tube, you really mean closed on one end and open at the other. So I'm trying to draw the graph again of over-pressure on the vertical axis and just position on the horizontal axis. And as before, it's open over here so there's got to be a pressure node, an over-pressure node. Here it's atmospheric pressure. On this end you don't have a node. In fact, you have an antinode. You can have maximal pressure when you're pushing up against a wall. You can have maximal pressure; you can have maximum underpressure. You have an antinode over here. So that's the new twist when you've got a tube that's open at one end.
And look at this picture. If you're going to have a node here and an antinode there, that's a quarter of a wavelength, not a half of a wavelength. So the formula now looks a little bit different than what we had. Instead of a half a wavelength, it's a quarter equaling to L and therefore the fundamental frequency, which is v over [1], is now v over 4L, not v over 2L. And so with a closed tube like this, you have a lower fundamental frequency than when you have an open end on both sides.
And now you can look at the higher harmonic. What's the next standing wave that you can fit into a closed tube? The next one--remember, you've got to have a node on the open end. This is over-pressure. And you've got to have an antinode over here. So this is the next shorter wavelength, next higher frequency picture that you can draw. And again, you've got to stare at the picture and think about the counting. This is one quarter wavelength, two quarters, three quarters wavelength. Three-quarters is fitting into L. So that's the second harmonic, so f[2], the second harmonic or the first overtone is v over [2] is equal to not 2f[1]; it's 3f[1]. It's an interesting feature of closed pipes that you don't get the even harmonic. So the character of the sound in this kind of a musical instrument is a little bit different. You can detect in some way, just with your hearing, whether you're getting music with even and odd overtones or just the odd overtone. So it just kind of changes the character, and so some people find this music more pleasing and some people find doubly-open more pleasing. The physics is simply the physics of standing waves with this new feature that you can have node on one end and antinode on the other, which you couldn't have with strings.
You know, there is another language that some people like to use when they're thinking about standing waves of longitudinal waves, like sound waves in a pipe. I've been thinking exclusively in terms of over-pressure, but here's a completely different way of thinking about the physics. It's going to turn out to be equivalent; it's just another way of thinking about it. When you have some longitudinal wave, I can think about, where is the pressure high and where is the pressure low? I can also just think about, what is the medium doing? What is the metal doing? As this sinusoidal pressure wave is going by, this little chunk of Slinky is just moving back and forth. Its longitudinal motion is kind of analogous to the little piece of a rope going transverse up and down, but this is moving longitudinally. So instead of graphing and thinking about over-pressure, I can instead graph and think about displacement from where that little chunk of medium really wants to live in equilibrium. Here it's a piece of Slinky, and in a musical instrument it's a little chunk of air which is moving back and forth as the pressure wave is moving by.
So let's go back to that last situation. This is the fundamental mode of a closed pipe. I've drawn here pressure versus position. And instead of drawing pressure versus position, let me draw--and this is over-pressure--let me the draw the sideways displacement from equilibrium position. So think about that for a second. When you're over here at the closed end, air molecules cannot displace so they're in a node. They're at a displacement node. They were at an over-pressure antinode. Those things always go hand in hand. Where the pressure is the highest, that's where the air has been squeezed in from both sides, and the spot in the middle--the largest over-pressure--is the spot where the sideways displacement is zero from where it wants to be. So node of this graph matches with antinode on this graph. And vice-versa. Think about the open end where the pressure is obligated to be atmospheric. That's where the air is rushing in and out the most. That's where the sideways displacement of the air molecules is maximum. It's the antinode.
So it's another graph. It's another way of thinking about it. But the counting and the physics come out the same. This was one-quarter wavelength fitting in L. This is one-quarter wavelength fitting in L. So it's worthwhile going back and looking at the overtones of this closed pipe. And also look at the double open, fully opened pipe. You can draw the corresponding graphs. And just remember that they're just always out of sync with one another. Whenever this one is at a node, this one is at an antinode. And you'll see that the counting is the same; the frequencies that you get for the standing waves are the same. It's just another way of thinking about it.
So whether you think about sideways displacement or over-pressure, the physics of these wind instruments is really just the physics of standing waves. They're longitudinal instead of transverse, and that allows us to have this nifty feature of antinodes at the end. But otherwise the math and the ideas are the same. And the one key thing if you want to solve these problems and really understand them is that when you have an open end, you've got to have a pressure node that's a displacement antinode. And when you've got a closed end, you've got to have a displacement node, and that means an over-pressure antinode.
Waves
Standing Waves
Longitutinal Standing Waves Page [3 of 3]