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- [Narrator] So, let's say you had a string here, and you give that string a little yoink up and down. You'd see a little wave pulse travel down that string. But here's a question, if someone else on the other end of this string also sent a wave pulse down the line toward the first wave, what would happen when they overlap? So, let's try to figure this out. Let's say you had a wave coming in this way, and yes, this is square. Kinda weird. It'd be hard. You have to be pretty talented to do this on a string, but this doesn't have to be a string. Let's say it could be a sound wave, an electromagnetic wave, any wave at all. The fact that it's a square is just gonna make it easier for us to analyze. So, you got this wave coming in this way and then another wave coming in this way. So, to be clear, this is the same string. There aren't like, two strings here. There is one string, two waves coming at each other. So, to be real, I mean, honestly, there's only one string, so this should be string coming here, and then there's a pulse up this way, so this string shouldn't be here. This string moved up to that point. It got disturbed, then it comes back down to zero, and then there shouldn't be two strings here. You don't have two strings in the same spot, so this wouldn't be, the string up there, and then it comes back down, but in these examples I don't wanna have to erase these all the time. It'd make the video really long, so let's just any time there's a string underneath a pulse we're just gonna pretend like there's no string in there. So, what would happen? What would happen when these pulses overlap? Well, let's just find out if I take one, and I move this here, and then another wave's gonna move over the top of that one. I'm gonna get wave interference. This is the term, wave interference, for when two or more waves overlap in the same region. So, what's gonna happen? Well, the string can't be in two places at once. There can only be one string and one shape of that string. And the way you find out what the total wave is gonna look like is simply by adding up the contributions of the two waves that are overlapping. So, in other words, if I wanna know the height of the total wave, I'm gonna call that height YT, T for total. That's just gonna equal the height of the first wave, I'll call that Y1, plus the height of the second wave, and I'll call that Y2. So, if you're familiar with the wave equations you could just plug in those two wave equations here, add 'em up, and you get a total wave equation. But a lotta times you don't have to resort to the full blown mathematics of the wave equation. You can kinda just look at the picture and figure out what the total wave would look like. So, let's do that. Let's put a little backdrop here, so we can add these up. So, we'll call this one unit high, and this is gonna be two units high, and this is gonna be three units high. It could be meters or centimeters, but it doesn't matter. We'll just say one unit, two unit, three unit. And now to figure out what the total wave's gonna look like I just add up the contributions from each individual wave. So, both waves are zero over here, so that's easy. Zero plus zero is zero. And then it gets to here. The blue wave, we'll call that wave one, has a value of one unit high. The pink wave, we'll call that wave two, has a value of two units high. They're going in different directions. Doesn't matter. Right now they're overlapping, so one unit high plus two units high is gonna be equal to three units high. My total wave would look something like this. So, if I were to ask what would the wave actually look like? The string, if this were a string, would actually look like this. It would just be one big three unit high wave, while those two waves are overlapping. So, that one was kinda easy. How does this get harder? Well, let's say we ask the question what do these two waves look like when they're only partially overlapping? So, maybe when they get to this point, where they're only halfway overlapped, what's that total wave gonna look like? We're still gonna use this rule. We're gonna add up both contributions to get the total. So, over here we have zero, and zero plus zero is zero. Until you get to here. Now the blue wave, wave one, has a value of one unit high. The height of this wave is one unit. The height of the pink wave, wave two, is zero units. One plus zero is one, so my total wave would look like this in that region, and now in this region the blue wave is one unit high, the pink wave is two units high, one plus two is three, so it would look like this in this region. Now over here, since the blue wave dropped down we have to figure out a new value, so the blue wave has a value of zero. The pink wave has a value of two. Two plus zero is two, and so my total wave's gonna look like this. So, the total string when those are overlapping halfway would look something like this, which if it was a string would be really hard to do, 'cause it's hard to get an exactly square wave, but electronic signals can have square waves, and this is what they would look like if they were overlapping. Now I wanna warn you about one thing. This idea of wave interference is a cool idea, but you gotta be careful. The term interference is a little misleading. Yes, while these waves are overlapping they create a different wave. They get distorted because the total wave will be the sum of the two waves. But these waves pass right through each other, which is great, 'cause when our phones send out a text message or a call to someone else, everyone else's phone is also sending out a message in that same air. Those electromagnetic waves are traveling right through each other. If they bounced off of each other, right, if these waves like bounced or got corrupted, and the information got changed so that the shape isn't the same after as it was before, it'd be really hard to make phone calls and send text messages, but the wave interference is only happening while they're overlapping. The waves make it through unaffected. So, the interference is only happening while they're overlapping, otherwise they pass right through each other unaffected, which is good. So, let's look at one more example that's a little more challenging. So, I'm gonna get rid of these, and let's say you had these two wave pulses, the same square pulse, and then you got this weird triangular pulse coming in, and they're gonna overlap. So, this wave pulse makes to here. This triangular pulse makes it to here. Oh, and you get the state of Nevada. So, the string is not gonna take the shape of Nevada. The string can't be in two places at once, so what's our total wave gonna look like? We're gonna use the same rules that we did before for wave interference. We're gonna add up the values of each wave at a particular point to get the value of the total wave at that point. So, what do we get? We've got zero plus zero over here, so that's easy, and now at this moment the value of the blue wave is one. The value of the pink wave is zero, so zero plus one is one. And then here it's a little weird, like you've got this pink wave changing, but over here it's easy, 'cause the blue wave has a value of one. The pink wave let's assume this drops down one as well, so this is negative one unit. Blue wave is one, pink wave is negative one. That's gonna be zero. One plus negative one is zero. And after there it's gonna be zero, but what is it in between? Well, the simplest answer is actually the correct answer here. It just drops down like this. Why does it do that? Well, let's consider a point in the middle. This point in the middle, this blue wave has a value of one. The pink wave has a value of negative 1/2, so one plus negative 1/2 is positive 1/2. Or, consider a point over here. The value of the wave right here for the blue wave is one. The value of the pink wave is like, negative 3/4, so the value of the total wave would be positive 1/4. That's why this drops down linearly if this is linear. This pink wave just keeps taking a bigger and bigger bite out of this blue wave, and the total wave would look like this. So, if we get rid of these this would be what the total wave looks like when these two waves overlap. So, I should say that this technique of just adding up the values of each wave at that point is called the superposition principle. It's a very lofty, intimidating name for something that's actually pretty simple. To find the total wave, you just add up the values of the individual waves. So, recapping, wave interference is the term we use to refer to the situation where two or more waves are overlapping in the same region. And, to find the value of the total wave while they're overlapping you can use the superposition principle, which just says to add up the values of the individual waves at a given point to find the value of the total wave at that point.