Wave Interference - Video Tutorials & Practice Problems
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Wave Interference & Superposition
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Hey guys. So by now, we've seen all the different characteristics of a single wave that travels through a medium like a wave on a string. But in some problems, we're gonna see two or more waves or pulses that meet along the same medium. So think of two pulses or two waves that meet along the same string. Now, these two waves are going to interact with each other. So that's what I want to talk about in this video. This is called wave interference and superposition. Let's check it out. So in our problem here, we have two pulses that are gonna up approach each other. So imagine that you and your friend have the have two ends of the rope and you're both whipping it up once you're gonna create two wave pulses that travel towards each other. In the second example, it's the same idea. We have two pulses except now one of them is going to be inverted. But the idea is the same, we wanna calculate the amplitude of the resultant wave pulse in both of these problems here. So let's go ahead and take this and take a look at this whenever you have two or more waves or pulses that meet along the same medium, they're going to temporarily disrupt each other. The word that you're gonna see for this is interfere, so they're gonna interfere with each other. So what happens here is that if you and your friend whip a, a string up, once you're gonna create two wave pulses and when they start to overlap, they're gonna disrupt each other. But this disruption is temporary because afterwards, they're both gonna basically just keep moving on along their merry ways as if nothing ever happened. So that's why this is a temporary disruption. So what actually happens here during this disruption? Well, as they pass through each other as these waves or pulses pass through each other, they're gonna combine the word you're gonna see for this is superimposed there. So you're gonna superimpose these waves, which is just a fancy word for combination. And what you're gonna do is you're gonna form a new wave in which the height is basically just the sum of the heights, the two waves that make it up. So if you have these two wave pulses as they fully overlap each other, what happens is you're gonna create an even bigger wave like this? All right. So what happens here is that these two waves can only interfere with each other when they have the same frequency and there's two types of, of interference that I want to talk about. So in this first problem, we're gonna have two wave pulses and they're basically both above the X axis. And this is called constructive interference. Constructive interference happens when you have two waves that meet each other and the displacements are gonna be in are gonna have the same signs. For instance, both of these displacements are gonna be positive. So what happens to the amplitude of this new resulting wave? Well, if you have a wave in which the amplitude is one and you have another wave in which the amplitude is one, when they combine the heights are gonna combine, you're gonna have H one plus H two or Y one plus Y two. And basically, you're just gonna create a new wave in which the new height or the new amplitude is just two. So you just say that the amplitude of this wave pulse is two, that's it, that's really all there is to it. So in the second example, it's the same setup except now what happens is that one pulse is gonna be inverted and they also don't have the same amplitude. So you'll notice that one of them has an amplitude of 0.5 the other one has an amplitude of negative one. So now what happens? Well, it's the same idea when you have these two waves that overlap each other. Basically what happens is that they sort of partially cancel each other out. This is called destructive interference. This is where you have displacements that have opposite signs. So I like to think of constructive and destructive as constructive. You're building, constructive, sounds like construction and you're building a bigger wave. Destructive sounds like destruction. And basically, you're destroying the wave, right? So what happens here? Well, basically what happens is that the new wave is gonna have a height that is H one plus a two Y one plus Y two. And because these things are opposite signs, you're gonna create a new wave in which this thing is gonna be slightly downwards like this. So you're gonna create a new little wave like this. And so the amplitude of our new wave is gonna be negative 0.5. And then what happens afterwards is again, these wave pulses just separate as if nothing ever happened. So that's really all there is to it guys, let me know if you guys have any questions with this.
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example
Example 1
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4m
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Everyone. So let's see if we can get some practice here. So I can kind of visualize this whole phenomena with wave interference. So I have these two wave pulses A and B that are traveling towards each other. Imagine you and your friend grab the end of a rope and you both flick it upwards and the wave pulses travel towards each other. Now, they each travel with a speed of two centimeters per second and basically everything on the X and Y axis is in centimeters. So we don't have to do anything, any conversion. So in other words, both of these things are moving towards each other with a speed of two centimeters per second. What we want to do is we want to sort of sketch out, not really calculate anything what the wave is gonna look like at different points throughout uh the sort of interaction here. All right, we have, we have actually got a couple of graphs to kind of help us along with that. We're gonna measure this or we're gonna draw this out when T equals one seconds and then two and then three and so forth and so on. And so forth. So let's get started here. Basically, what we're gonna do here is we're gonna take this sort of wave pulse and we're actually sort of given, uh, you know, we're sort of like where the wave pulse starts and stops in terms of centimeters. So let's take a look at what happens at T equals one second. All right. So from T equals 0 to 1, there's one second that passes. And if these things move at two centimeters per second, then that means that they both are gonna move towards each other by two centimeters. So this is what's gonna happen. I'm initially starting here at six, which means that the blue curve, that blue pulse is gonna travel to the right by two centimeters. So you basically are just gonna line up the tip of this pulse and now you're just gonna shift it over by two centimeters after one second. So basically, what's gonna happen is that the top of this little peak is actually gonna happen here now at eight and that means that the bottoms of these two triangles aren't gonna be four and eight, they're all gonna shift over by two centimeters. So it's gonna be six and 10. So that's basically what the wave pulse is gonna look like over here. Then everything else is gonna be flat. What does the red curve look like? Well, the red, the red pulse is gonna do the same, but except it's actually gonna move to the left. So what happens is this peak uh lines up with 14, but then it's gonna move to the left by two. So it's gonna line up with 12 and notice at the top of this peak actually isn't four, it's just two. So basically, it's gonna end up looking like this, right? And then the bottoms of these is gonna look like something like that and then the rest of the wave pulse is flat. OK? So now let's keep going with this. So basically, this is what t equals one looks like. But now what happens is that this pulse continues moving to the right. So in other words, the top is gonna be over at 10 on the X axis. So the top is gonna be at 10 and the bottom is gonna be at eight and this bottom over here is gonna be at 12. So this is gonna be eight and 12. That's where the sort of these uh things sort of line up. And then there are, again, the rest is gonna be flat. What happens to the red one, the red one will move the same exam again. So this will move actually to the left. So in other words, the tippy top will actually be over at 10 m or 10 centimeters. And then what happens is uh the po the bottoms are gonna go from 10 to 8 and then 14 to 12. So it's gonna look something like this. Basically what happens is at T equals two seconds. These waves are actually gonna be, the pulses are gonna be on top of each other. Now, it's not actually gonna look like this because remember what happens is the waves are gonna interfere with each other. And remember what happens is when you have waves that are stacked on top of each other, the resultant wave is actually just gonna be be basically just adding the two triangles together. So what ends up happening here is this actually kind of looks like the wave pulses stack up and they actually add on top of each other and you'll end up with a really, really, really strong wave pulse that goes all the way up to actually six centimeters, right? Because it's gonna add the two and the four, you're just gonna add all those Y valley. So the, the real wave pulse is actually gonna look something like this. All right. So these two things will come together and, and as they pass each other, they'll add up to a very, very, very high wave or high of pulse and then they basically just keep going past each other. All right. So let's look at what happens now after they pass each other. So after they pass each other, uh what happens is again, the blue moves to the right and it goes from 10 to 12 the peak. So it's gonna look something like this. I'm gonna just do this a little bit faster. So it's gonna look something like that. It's gonna be flat everywhere else. And then what about the red one? The red is gonna move two centimeters to the left. So in other words, it's gonna line up and the top is gonna be here at eight and it's gonna go like this now because these things aren't a sort of on top of each other. We don't have to do any addition anymore. They basically have already passed each other and the superposition has already happened. So these things aren't interfering anymore. So this is basically what each of these sort of stages look like throughout the motion of these two wave pulses, which is very cool. You can actually even try it at home if you have a string. Um And you can create two little wave pulses. You're gonna see this weird thing that happens when the pulses hit each other and they'll sort of add up like this. Anyway, folks. That's it for this one. Thanks for, thanks for watching.
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