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About this Lesson
- Type: Video Tutorial
- Length: 12:27
- Media: Video/mp4
- Use: Watch Online & Download
- Access Period: Unrestricted
- Download: MP4 (iPod compatible)
- Size: 133 MB
- Posted: 07/02/2009
This lesson is part of the following series:
This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.
Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.
Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.
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It's important to understand how a wave travels through a medium, but what's really interesting, when the physics starts getting really interesting, is when this wave interacts with something, when a wave hits a boundary and it reflects or it goes through. If you want to understand about how loud a stereo sounds and it's in your room, you just need to know about the simple properties of the wave traveling through the air in the room. But ultimately you want to know what happens when it reaches your ear and interacts somehow with your body. If your next door neighbor is playing their stereo you want to know what happens when the sound waves hit a wall and do they make it through or not.
Even waves interacting with other waves. What happens if there's sound going one way in a room, and sound going another way in a room. What happens in the region in the middle? The interaction of waves with waves is also really interesting and important in an awful lot of physics situations.
So let's start off with just a qualitative discussion. The first of these things that I mentioned was reflection off of a wall, like if you are speaking. You're yelling in the Grand Canyon. There's this big, rigid wall. There's a sound wave propagating over. It's going to echo. You're going to hear a reflection. I'd like to just sort of think through what that looks like. Imagine a wave on a string, a pulse, which is sort of the simplest picture that I could draw. So here's this pulse heading towards the wall, and there's a boundary. There's a rigid, fixed end of the rope and it can't budge. So what's going to happen? You could imagine that nothing at all would happen; the wave would just kind of die. But there's got to be something happening because energy is being transmitted by this wave, and energy is conserved. So either it's going to go into friction and heating, but if there's no friction here, then energy is conserved. That wave is going to have to reflect.
The way it reflects is pretty straightforward. When the wave reaches the boundary, think about what's happening in a wave. There's tension in this uplifted piece of rope, and that tension, normally speaking, is pulling the next chunk of rope up. That's how the wave propagates itself. This piece here is pulling up the next, which pulls up the next, and now, all of a sudden, it's pulling on the wall, but it can't budge. So by Newton's Third Law, if it's pulling up on the wall, the wall pulls down on it. So all of a sudden there's this external force pulling this wave pulse down, if you like, and for a brief moment in time, if this wave is nice and symmetrical, you could get this scenario where you've got just a flat stretch of rope.
Now, does that mean the wave is gone? No, this is a snapshot in time. The wave is still there. It's hidden because this rope is on its way down. There was this external force pulling it down. It's on its way down. It's passing through zero, and a moment later it's going to pop up on the down side. The conservation of energy - it can't be going to the left. You've got this energy traveling. The wave is going to be heading now off to the right. And this is what happens when a wave hits a boundary with a rigid edge. You get reflection of the wave and the reflection refers to the fact that it's heading backwards, and in this case you also get an up pulse turning into a down pulse.
Now, what happens in the similar-looking scenario of a wave pulse approaching, not a fixed wall, but sort of a floppy end of the string? I've indicated it here with just a little loose frictionless end of the string. So once again, this piece of the rope is tugging up the next piece, and so the pulse moves over, and now what happens? You're tugging on the end, but the end doesn't have any more rope to tug on. So you've got extra pull here, which is going to start moving this thing up extra high. For a brief moment in time this thing is going to be way up high. If this is a nice symmetrical pulse it'll be twice as high as the amplitude of that pulse, and then what happens? We haven't lost any energy to friction, the wave is going to reflect, and this time you know exactly what's going to happen. This thing is going to fall back down and the pulse is just going to continue reflected, but without being upside down. It's now an up pulse going out.
So the physics is qualitatively reasonably simple to understand, and the tricky business is just to remember is the end fixed, then the wave inverts itself, it goes upside down when it reflects, and if the end is free, it just reverses direction.
So that's what happens when you reflect off of an end, a boundary. What happens if you're transmitting through a boundary, like the person in the next room and their sound waves are hitting the wall, how about transmission? So let me draw you a picture. There are a couple of possible scenarios. Let's imagine transmission from a skinny rope into a thick one. So here's the boundary, and you've got this pulse cruising along, and what's going to happen?
If this was a really, really thick rope, it would look like a big wall, and I would expect to just get an upside down wave heading backwards. And if this one is just thicker but not infinitely thick, two things will happen. You will get an upside down reflected wave, but also this is a rope, it can have a wave in it, and it will. This is pulling it up so you just get an up wave, not as big as the original one, a down wave, not as big as the original one. This is the reflected wave and this is the transmitted wave. Interesting thing - this shape is different from this shape because the speed is different here. Remember, wavelength times frequency is equal to wave speed. And wave speed in a rope goes like squared of tension divided by mass per length. Big mass per length here means smaller speed. So if this pulse is wiggling the end at a certain frequency, then the wavelength over here, since the velocity or speed is going to be smaller, the wavelength is going to be smaller, so this pulse will be squished a little bit in shape, and traveling along more slowly than this one was.
What about the other way? What if you have a pulse coming in on a thick rope and the boundary is with a thinner one? Well, in the limit that this thin rope was really, really thin, it's just not there at all, this is like that reflection problem where you had a loose, floppy end, so I expect that we should get a reflected pulse, which is not upside down, and that's correct. That's what will happen, but of course, you're lifting up this little loose string, and so there will also be a transmitted wave heading to the right.
So reflection and transmission are relatively straightforward, at least qualitatively. The next really important physics that we have to think about is not what happens when you hit a boundary, but what happens when you have waves interacting with waves, rather than waves interacting with a boundary? What happens if I've got a rope and I'm sending one pulse in this way and another pulse in that way? Lots of really interesting physics can happen with all sorts of applications in the real world. Supposing that this pulse is an up pulse coming in. I could draw you a picture like this. This black line is not a wall. It's just to guide your eye. So there's a pulse heading in, and another pulse heading in. Let's just make them symmetrical. You've got two pulses coming in. What are they going to do? The rope is going to do something really complicated here. It's going to have some weirdo shape, but fundamentally, if you think about what's happening, it's really simple.
This is a pulse. It's traveling along to your left, and you can imagine it just continuing completely unperturbed forever. And here's a pulse moving to your right, and you can imagine it continuing to travel along completely unperturbed forever. So when they're far apart you just see these two pulses coming towards each other. What happens when they get together? Just draw a picture. Well, you draw wave number one, and you draw wave number two as though there was nobody else there, and once you've drawn them, just add the curves. In other words, the final displacement of the real rope is just the simple sum of the displacement caused by wave number one, plus the displacement caused by wave number two. It's as simple as that - just adding displacements.
We call this super position principle. It's the super position of waves. You just add the two waves. So when these two waves are coming together, when they begin to overlap, mostly the wave that you get is just the good old fashioned waves that were heading in. In this little region where they're both non-zero, the rope will begin to lift up a little. You'll add the small amount from this wave and the small amount from that wave. So you get a funny-looking shape. When they are totally overlapped for that brief instant in time, you've got a curve plus that same curve. You get basically twice that curve, at least if they're symmetrical. And then, after a while, they just pass on through one another. So here are these two waves and when they interact, this green curve, that's the rope itself, has all sorts of weird shapes, but at the end, these two waves just continue on. This one has just walked on its way through and this one has gone its way through, and after the interaction, it's as though nothing had happened. Very different than what particles do. If a particle comes in and hits another particle, gosh, almost anything can happen. They can break apart, they can go off at different angles. There would be conservation principles of energy and momentum, but very different physics - much simpler with waves than with particles.
This effect here, where you've got two up pulses adding up to a big up pulse is called constructive interference. Interference is the word for when two waves are interfering with one another, and constructive means they're building up. You can have the opposite scenario. You can have destructive interference. That would be when you've got an up pulse coming in and a down pulse coming in. So you've got up and down, and it's the same principle of super position. You just imagine them each traveling on their own way, and when they overlap, think about what happens when they're right together. There's up plus down. You're adding amplitudes. You're adding the displacement, not the amplitude. If this displacement is up and that displacement is negative, you add a positive and a negative and you get zero. For a brief moment in time when two symmetrical pulses pass by, the rope is flat. And it's really true. If you took a snapshot - here I had two unsymmetrical pulses, so the green rope is merely flat, and you know, it can happen. For that brief moment in time, if you take a snapshot, you can't see the waves, but they're there. The rope is in motion, and after a while, this down pulse, which was heading this way is still heading that way. This up pulse, which was heading the other way is still heading the other way.
So this principle of super position is fantastic. It's very easy mathematically. Any wave you like, a sinusoidal wave, any arbitrary wave, if you've got two of them and you interfere them - that means you run them past each other, you simply add the displacements. Lots of practical applications.
You know, some of the times in life when you have waves interacting it seems really weird. Like if you're tuning a guitar and you've got two strings that are supposed to be the same pitch and you play them together, you'll hear this effect that's called "beating." We'll talk about it in a future tutorial. What you hear is it gets louder and softer, louder and softer, and you think, "This is really unusual physics. How can I understand it?" And we'll see it's nothing more than the principle of super position. You've got two sound waves and you add them, and what you get sometimes can be surprising, like when up waves cancel with down waves and you get quiet - nothing for a while. Interesting physics with interference.
Waves on Top of Waves
Reflection, Transmission, and Superposition Page [3 of 3]
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