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About this Lesson
- Type: Video Tutorial
- Length: 10:50
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- Posted: 07/01/2009
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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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Anytime you have two waves passing by each other through the same place in space, they're going to interfere with one another and something, which appears to be relatively complicated, is going to happen. What's really happening is not so complicated at all. The principle of superposition says if you've got wave number one, and wave number two, you can just imagine that wave number one is continuing on its way unaffected by the other. Wave number two is continuing on its way unaffected by the other, and at every point in space, and at any time, if you want to know the displacement of the medium, you just add up the displacement that number one would have caused, and the displacement number two would have caused, and add them up like numbers.
This principle of superposition allows you to understand, for instance, what's happening when two pulses go by one another. When it's two sign waves, the graphs start getting a little bit difficult to look at. It's worthwhile looking at them, and it's worthwhile also just looking at the mathematics, because it turns out the mathematics is really fairly straightforward because you're really just adding, and it teaches you something. So let's begin with the situation where you have a string - so we're talking about a transverse wave - and you're sending one sinusoidal wave down that string with a certain frequency. And you send another wave down that same string in the same direction with the same frequency.
So what's going to happen when you send two waves together? So they're going to be interfering all over the place, because it's just two continuous waves that are both on that same rope. So what will happen? Well, let's look at the mathematics first. Here's the formula for a traveling wave. It's got amplitude A, and it's traveling in the positive direction because it's kx - . And, you know, omega is related to the regular old frequency. This is just the angular frequency, 2 divided by the period. k is the wave number, and tells you the velocity or the speed of the wave, in this case, traveling to the right.
Here's another one - y. Let me pick one with the same amplitude and the same frequency, and since they're on the same string, they have to have the same speed, so if they've got the same omega they're guaranteed to have the same k. Let me add a phase to this one. So when you add a phase to a sinusoidal function it's just like shifting it over so that, for instance, when x is zero and t is zero, so when this one is at a max, this one won't be at a max, so the two waves are somehow out of sync. So what do you get when you add them?
Well, the mathematics looks a little bit intimidating here. There's a wonderful trigonometric identity which we'll make lots of use of. It's just a trig identity. You can prove it fairly quickly. If you add cosine of something, alpha plus cosine of something else - beta. This trig identity is just very handy. It's twice the cosine of times . So it's very inobvious, but it's correct. This little trig identity allows us to add this cosine to this cosine. It's of this structure. So we're going to get twice cosine of - we're going to average these two arguments, so we're going to get kx - . It's not divided by 2, because I'm adding it - this plus this, and then dividing by 2. I just get kx - back plus - that's going to be this term.
And here in the difference, the kx's and 's are going to cancel and we'll be just left with . Don't forget that the cosine of plus is equal to the cosine of minus . That's a property of the cosine function. So here's what I just said. The sum, that's the superposition of these two traveling waves, is itself - and take a look at this - that's a number - cosine of . So this is all just a number times cosine of kx - +. So we've got a traveling wave. If you've got one traveling wave and you add another traveling wave and you superpose them, you simply get another traveling wave, and it's got the same old omega and the same old k, same speed, same frequency. It's got a different phase. Not the phase of either of the two that you started with, and it's got a different amplitude. The amplitude is 2A cos . Instead of looking at the math let's just think of some examples. Supposing that was equal to zero. So I've got two waves and they're both completely in sync, the same amplitude, the same phase. It's just two sine waves traveling together - cosine waves. What do they do? Well, if is equal to zero, you just get cos(kx - ) and cos of zero is 1. You just get 2A. In other words, you just double them up. Let's just look at a picture of this.
My picture is going to be a little bit different than what I just had a formula for, because what I had a formula for is so trivial. Here's one wave. Here's another wave. Both of them are traveling, let's say, to the right, the blue wave and the purple wave. So this is a snapshot in time. And what I've done, just to make the problem a little bit more interesting is I've made the two amplitudes different. My formula doesn't work anymore, but you can just look at this graphically and see that when you add these two waves you just get a sinusoidal traveling wave, which has an amplitude, in this case, of just this amplitude plus that amplitude. It's the red wave, and it's also traveling to the right with the same speed and the same direction and everything.
If the two waves had been equal in amplitude, then the formula that I just derived would have worked just fine and we would have gotten that the result was exactly twice as big. So that case is really intuitive and makes a lot of physical sense to me.
Let's go back to the formula and take a look at what would happen if was 180 degrees or pi radians. I should really be using radians everywhere here. Pi radians 180 degrees means the two waves are entirely out of sync. When you add 180 degrees to a wave, or pi radians, that's going to take you exactly a half a wavelength or a half a period out of sync with where you started, so when one's up, the other is going to be down. It's like comparing the cosine of zero with the cosine of 180 degrees. One is 1, and the other is -1.
So what's going to happen when you add them? Well, the formula says I've got a cosine of pi over 2 radians, which is zero. The two waves have canceled, and that's exactly what happens. This one is a little bit more counter-intuitive, and the fact that this really does happen with waves in nature provides us with all sorts of really interesting and counter-intuitive wave phenomena. Here's my blue wave, which is traveling to the right, and here's my other lighter blue wave, the one that's 180 degrees out of sync with it, so when one's up, the other's down. When one of them is taking the cosine of 90 degrees, the other one is taking the cosine of 270 degrees, and both of those are zero and you're getting zero. Just stare at this and convince yourself that I'm really adding two traveling waves, which are out of sync, and the sum is shown. It's zero. And this is right. What I'm telling you is that if I'm broadcasting a beautiful, pure tone from a speaker, and then right on top of that I broadcast that same tone - remember, same wavelength, same frequency, same speed, but exactly out of phase. That's the trick, which is hard to do, but you can do this. What you get is nothing. They cancel. That's what waves do. It's really very cool and it's just the principle of superposition.
Let me do one more example where I have these two waves, and let me pick now a random phase, just something that's not this dramatic zero or 180 degrees. Let me make the phase 60 degrees, convert that into radians, plug into this formula, and now it's just not so obvious, but when you look at it, it's actually pretty easy. You still get a traveling wave and it's slightly out of phase with both of the original ones, and then your amplitude is not double, because there's this cosine of phase divided by 2.
In this case, the mathematics tells you the answer and it's kind of nice just to look at the picture. Here is one of my original waves traveling to the right. And here is another wave, the blue one, traveling to the right, and they are out of phase with one another. The peak of one is occurring at a different place in time as the peak of the other. Just add them. At every point, you can simply add - purple plus blue. Here, they are opposite, so they're basically canceling, and you get read, which is near zero. Here, they're both positive. They're adding up. If you do that - if you didn't know the formula you might think you got some really complicated curve, but in fact, you get a beautiful sinusoidal curve, and its amplitude is not the sum of the two amplitudes that's given by the formula, but it's traveling to the right.
Adding two sine waves with the same wavelengths and traveling in the same direction with the same amplitude, just out of phase, is not the most general problem in the world that we've solved. But that trig identity is really going to allow us to think about much more physically interesting examples, like for instance, what happens if you have two waves going in opposite directions? We'll get some really interesting physics effects then. What happens if you have two waves going in the same direction and they've got the same amplitude, but their wavelengths or their frequency is just a little bit different? So they start off in phase, but then after a while they get out of sync. Again, we'll get some really interesting physical situations.
In fact, it turns out that if you could just answer this problem in complete generality of what happens when you add one traveling wave to any other traveling wave, you can really construct any physical situation and we're really almost all the way there. We've got the mathematics that we need, and it's really just a question of working out the details and thinking about all the interesting physical situations and applications that arise from this principle of superposition of traveling waves.
Waves on Top of Waves
Interference Page [1 of 3]
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