Standing waves on strings
- [Instructor] If you've got a medium and you disturb it, you can create a wave. And if you create a wave in a medium that has no boundaries, in other words, a medium that's so big, this wave basically never meets the boundary, then there's nothing really stopping you from making a wave of any wave length or frequency whatsoever. In order words, there's not really any naturally preferred wave lengths, they're all pretty much as good as any other wave length. However, if you confine this wave into a medium that has boundaries, this wave is gonna reflect when it meets the boundary and that means it's gonna overlap with itself. And when this happens, you can create something that are called standing waves. And we'll talk about what these mean in a minute but the reason we care about them, is because when standing waves happen, they select preferred wave lengths and frequencies. Only particular wave lengths and frequencies are gonna set up these standing waves and what ends up happening is that these often become dominant and that's why these standing waves are important to study. So, let's study some standing waves. Let's take a particular example. Let's say you've got a string, whoa, not that many strings. One string here and you nail this string down at both ends. So, you're gonna prevent any motion from happening at the end of this string. This string can wiggle in the middle but it can't wiggle at the end points. And this isn't that crazy. A guitar string is basically a string fixed at both ends. Piano strings are strings fixed at both ends. So, the physics behind standing waves determines the types of notes you're gonna get on all of these instruments. And by the way, this point over here, we're basically making sure that it has no motion. So, by nailing it down, what I really mean is that there's gong to be no motion at this end point and no motion at this end point. And instead of calling those no motion points, physicists came up with a name for that. They call these nodes. So node is really just a fancy word for not moving at that point. So, for this string, there's gonna be nodes at each end. And we'll see, when you set up a standing wave, it's possible that there's nodes in the middle as well but there don't have to be. For this string though, we're making sure that there have to be nodes at each end. So, why does a standing wave happen and how does it happen? Well, let's say, you give the end of the string here a little pluck and you cause a disturbance, that disturbance is gonna move down the line because that's what waves do. It's gonna come over here. Once it meets a boundary, it's gonna reflect back to the left. Now, it turns out, when a string hits a boundary where it's fixed, when it hits a node, in other words, it gets flipped over. So, you might have tried this before with the hose. If you send a pulse down the line and you try to see how it reflects, it gets reflected upside down. It doesn't matter too much for our purposes, but every time it's gonna reflect, it flips its direction and it keeps bouncing. Now, let's say instead of sending in a single pulse, we send in a whole bunch of pulses, right. We send in like a simple harmonic wave. Now when this thing reflects, it's gonna reflect back on top of itself because this leading edge will get reflected upside down this way and it's gonna meet all the rest of the wave behind it and overlap with it, creating some total wave that would be composed of the wave traveling to the right, plus the wave traveling to the left. And you can use super position and interference to figure out what that is, which, for most wave lengths, is just gonna be a mess. So, if you just send in whatever wave length you want and let it reflect back in on itself, the total wave you get might not really be anything special. It might just be sort of a mess in here. Nothing really all that interesting. However, there'll be particular wave lengths that set up a standing wave, and I'll show you what that looks like in a minute. How do you find these special wave lengths? You simply ask yourself, what wave lengths could I draw on this string so that there was a node at each end? What wave length would fit inside of this region and have a node at both ends. So, instead of trying to add up a complicated super position of waves, we can figure out the special wave lengths simply by drawing them and seeing which ones fit. So, let's try it out. Let's get rid of all this. Alright, what wave lengths would fit in here? Well, we know what a simple harmonic wave looks like. It looks something like this. So, the question we need to ask is if we start at the zero point, because I wanna make sure I have a node at the left end, what might the shape of this graph look like so that I reach a node at the other end as well? Well, the first possibility, look at it, I start at a node, when do I get to a node again? I get to a node when it takes this shape right here. There's another node. So, the first possibility, which is gonna be the longest, largest possibility, would be a wave that just kind of looked like this. Looks kinda like a jump rope. This would be the first possible standing wave you can set up on this string. And that means it's special, it's called the fundamental wave length. This is the big daddy. This guy dominates all the other wave lengths we're gonna meet. Yeah, there's other standing waves you can set up on here but this one's the big alpha dog and if you let this string vibrate however it wants, it's gonna pick the fundamental wave length. And so, what would we see happen? The string's not just gonna be suspended in air like this, it's gonna be moving around, but these are called standing waves because this peak no longer looks like it's moving right or left. This peak is just gonna move up and down, so a lot of times when we draw these standing waves, we draw a dash line underneath here that mirrors the bold line because all this peak's gonna do is go from the top to the bottom, then back to the top, it's just gonna oscillate. It's gonna look like a jump rope but it's not revolving, it's just moving up, then down, then up, then down, it takes this shape, then it would be flat, then it might look something like this and then it comes down to here and then it goes back up, and it keeps going up and down. We call it standing, it's more like dancing. It's kinda like a dancing wave but we call them standing waves because these peaks don't move right or left. So, that's the fundamental wave length. What would the next possibility look like? Let's see, we gotta go from a node. We know we have to go from node all the way to another node. That was the first one. So, let's just keep going and go to the next one. And that would be the next possible standing wave because it'd have to fit within here. What would that look like? It would come up, it would go down and then it would come back up, so that it meets this node on the other end. That would be the next wave length. Sometimes this is called the second harmonic. Second 'cause it's the second possibility, harmonic because these are resonances and this term is used a lot when you talk about resonances with musical instruments. What would the third harmonic look like? Well, we gotta start at a node, we go to this one, that was the fundamental, this is the second harmonic, so that's gonna be the third harmonic. So, this one's gonna come up, it's gonna go down, it's gonna go back up and then it's gonna come back down and that would be the third harmonic. And you can see you could keep going here. You can create infinitely many of these. But let's analyze what's going on up here. What's actually happening in these standing waves? Note that there's gonna be points, like right here on this third harmonic, and if I draw its mirror so that I can get this. So, this is what it would look like maybe a little after, actually exactly one half of a period after this bold line. So, you wait, this peak moves down to here, this valley moves up to this peak, this peak moves down to here, they're oscillating back and forth. But note, this point right here just stays put. That's not even gonna move. That's a node and so is this point right here. These points are happening 'cause when those waves line back up, remember, the wave travels to the right, bounces back to the left, and at this point right here and this point right here, you're getting destructive interference between those two waves. Similarly, at these points, where you're getting the maximum displacement, the two waves are lining up in such a way that they're interfering constructively. So, the nodes are the destructive points where the wave cancels. That makes sense 'cause there's nothing happening there, there's no motion. And these maximum displacement points are the constructive points. We should give those a name. What do you think we call those? If you guessed anti-node, then you're right. These are called anti-nodes because that's where there's the most motion. Now, you often have to figure out mathematically, in terms of the length of the string, what are the actual wave lengths you can get. So, drawing the picture allows you to find those. But how do you actually get them mathematically? Well, I've kind of created a horrible mess here. So, let me clean this up. Sorry about that. Get rid of all that. Let me just add some strings in here. And so this doesn't get too abstract, let's just say the length of this string, it's pretty big, let's just say it's 10 meters. A really long string, you secure it at both ends. So, the first standing wave looked like this. Jump rope mode, looked like that. Now, if the string has a length of 10 meters, what would be the wave length of this wave? You might say 10 meters but no. This is not one whole wave length, remember, If we looked at this wave pattern we had over here that we were using, this is one entire wave length. You had to go all the way to here to get through a whole wave length, this was only half of a wave length. So, this jump rope is only half of a wave length. What would a whole wave length look like? This would extend all the way out here, all the way back up, I can't really get there so I don't go off screen. That would be a whole wave length. So, this would be 10 meters and then another 10 meters. That means that the wave length of this wave, even though a whole wave length isn't fitting in here, if there was a whole wave length on this string extended, this wave length would be 20 meters. What was the next wave? Remember it looked like this. It had one node in the middle, whereas this first fundamental wave length had no nodes in the middle. And again, if this string is 10 meters, what's this wave length equal? Well, that's easy. This is one whole wave length. So, that would just be 10 meters. So, the wave length here would be 10 meters, 'cause one whole wave length fit exactly within the string's length of 10 meters. And the third harmonic looks something like this. It has two nodes in the middle. Note, you keep picking up another node in the middle. Fundamental has no nodes in the middle. Second harmonic has one node in the middle. Third harmonic'll have two. The fourth will have three and so on. So, what is this wave length? This one's a lot harder for people to figure out. So, let's look at this. One wave length is all the way to here. So, this is a wave length. But our string is this long. So, what fraction of this length is this wave length? Well, look at, this wave length is two thirds of the entire length of the string. So that means we could just say that this wave length is two thirds of 10 meters which is, we could write it as 20 meters over three, and we'll keep going here. I'll draw the rest. This is the fourth harmonic. How big is this wave length? Well, this wave length covers half of the string. So, this wave length is gonna be half the length of the string and that's gonna be half of 10 which is five meters. And we can keep going. I can draw the fifth harmonic down here. And it would look like this and you could ask yourself, how big is this wave length? Well, this wave length, let's see. One, two, three, four, five, we got five of these humps in here. So, this wave length's gonna be two fifths of this entire length. So, I'm just gonna write lambda is two times 10 would be 20 meters, so two fifths of 10 would be 20 meters over five, oh, which we could simplify as four meters. But what if they asked you for like the 43rd harmonic? If they're like, hey, what's the wave length of the 43rd harmonic? I don't wanna sit down and draw like 43 of these things and try to figure out what fraction it is. And you don't have to. There's a pattern here. So, let me show you the pattern. So, this is gonna look kinda weird but I'm gonna write this first fundamental wave length as two times 10 meters over one. And then I'm gonna write the second harmonic as two times 10 meters over two. And I'll write this third harmonic as two times 10 meters over three. This fourth harmonic is gonna be two times 10 meters over four. This fifth harmonic could be written equivalently as two times 10 meters over five, since that's 25ths. And now, hopefully, you see the pattern. You realize, okay, I see what's going on here. If I want the wave length of the nth harmonic, n could be like the first, the second, the third, so n is really just an integer one, two, three and so on. I could figure out what that wave length would be simply by taking two times the length of the string, I'm gonna write it as L, so this applies to any string of any length as long as it's got nodes at the end points. So, take two times L and then just divide by n. So, in other words, if I want the wave length of the 84th harmonic, I'll just take two times the length of my string and divide by 84. If I wanted the 33rd harmonic, I'd take two times the length of the string over 33 and that would give me the wave length of that harmonic. Now you should remember when we derived this equation, we drew these pictures and these pictures all assume that the end points are nodes. So, this equation assumes you have a node. Node, standing wave on a string, which honestly, is almost always the case, since on all instruments with a string both ends are fixed. So recapping, when you confine a wave into a given region, the wave will reflect off the boundaries and overlap with itself causing constructive and destructive interference. For particular wave lengths, you can set up a standing wave, which means the wave just oscillates up and down instead of left to right. In these standing waves, the points where there's no motion are called nodes. And the points of maximum displacement are called anti-nodes. You can find the possible wave lengths of a standing wave on a string fixed at both ends by ensuring that the standing wave takes the shape of a simple harmonic wave and has nodes at both ends, which if you do, gives you a formula for the possible wave lengths for a node node standing wave as being two times the length of the string divided by the number for the harmonic you're concerned with.
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