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Standing Waves

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Standing waves, also known as stationary waves, are waves formed by the combination of two waves with equal amplitudes and frequencies. Because of the interference between these waves, standing waves don't travel at all. There are points along the wave, called nodes, which don't seem to move at all but vibrate in place.

One of the most commonly seen types of waves in introductory Physics classes is standing waves. Standing waves are very, very important and they actually represent one of the first types of waves that was ever studied because they represent the types of waves that are associated with harps and violins and guitars and all kinds of stringed instruments. Standing waves occur whenever you have a medium that is constrained at the end points alright. Now we're going to talk about one type boundary condition there's a couple other ones but this is the most commonly seen in introductory Physics. The rigid boundary, at a rigid boundary we pin the boundary down the medium down at the boundary so that it can't be disturbed. So the medium can be disturbed in between the 2 boundaries but the boundaries themselves got to remain where they are. So this is a rigid boundary, can't move. So let's consider a string like a violin or guitar string that has rigid boundaries at both ends and he's got length l.

Alright now here's the idea the wave has to fit in between these 2 boundaries, because if I were to just draw a random wave, look at it here we go. It doesn't work it has to fit so that means that only certain wave lengths are going to be allowed. Alright so let's see some situations in which this occurs, alright we've got this one right here where the wave does fit. These 2 points right here represent equilibrium points on the way, they're called nodes of the standing wave. The opposite is this guy right here, this is a place where the wave moves the most, it's called an anti-node. Now if I begin with something like that, the string is then going to vibrate back and forth like this. So the standing wave will go between this and that back and forth, back and forth.

The nodes stay where they are and the anti-nodes go the maximum displacement, people in between do less of the anti-nodes and more of the node right? So it just goes like that, now this isn't the only one, we also could imagine having this guy. Now as this one vibrates, the part that's down is going to go up and the part that's up is going to go down at the same time and these guys are just going to vibrate like that. So the other wave form that we'll see is that one, now obviously this is a node I mean same thing as it was before and we picked up another node in the middle. When we jumped up to the next level we picked up another node and look what else we picked up another anti-node. So now I got 2 anti-nodes and 3 nodes. Alright what about this guy? Well now we bumped up again, so we're going to pick up another anti-node and another node and let's just go ahead and look at it, there it is right? So in general when we do this we're going to count and we usually count by just counting the number of anti-nodes. So this is going to be 1, this one is going to be 2, this one is going to be 3 and now let's try to determine the wave length.

Well in this case this is only half the wave length, because the whole wave would go like that. So that means that l is one half of the wave length. So the wave length will be 2l, alright what about in this case? Well, this is a wave length so we'll just say lambda equals l and be done with it alright? What about here? Well here the length l is a wave length and a half, so it'll be three halves wave length and that means the wave length will be two thirds l. In general you should be able to convince yourself that the wave length of the nth harmonic these are called harmonics will be given by 2l over n alright? Well what about the frequency? Well frequency is always equal to speed divided by wave length, so when I take the speed and I divide by this wave length I end up getting n times v over 2l. Now v over 2l is the lowest frequency that we get, it's the frequency associated with this dude right here which only has half a wave length showing. That's called the fundamental frequency and we'll call it f1 so this is f1. So that means that fn is n times f1 alright. This makes what we call a harmonic series. It's harmonic because the frequencies are all integer multiples of the same frequency. What that means is, if we wait a certain amount of time, all of the waves that fit in between these 2 nodes will repeat the same period over and over and over again and that's what makes them sound good together it makes what's called a harmony.

Alright now what about if I want to change this fundamental frequency? Well here's a fundamental frequency how will I change it? Well it depends on 2 things, it depends on the speed and it depends and it depends on the length. So what happens if I increase the length of this string? Well it's going to make the fundamental frequency go down, so long equals low frequency, equals deep alright. What about if I make the strings shorter? Well now it's going to make this fundamental frequency bigger, so that means short equals high pitched and we've seen that before right? If you look at a bass it has really long strings and it's really low sounding. Whereas if you look at a violin it's got much shorter strings and it has a much higher pitch. Same thing happens with the wood wind instruments but we'll talk about that later. Alright what about the speed? How can I change the speed? Well we know that the speed of a wave depends only on the medium. Well the medium in this case is the string, alright the string has 2 important properties that are associated with the speed. The tension that its held under and how much mass density, how heavy is that string?

Alright now if I increase the tension, and of course we'll do this on a stringed instrument by twisting that little knob at the top then what that's going to do is it's going to increase the speed. Increasing the speed, increases the fundamental frequency and makes it sound more high pitched alright. What happens if I increase the mass density? Now this isn't something that I can do, if I've already got the guitar the mass density is fixed, but if you look at the strings on the guitar, you'll find that the ones that make the lower tones are thicker. They weigh more and that's because a larger mass density gives rise to a smaller fundamental frequency a lower note. And so that's why we have the lower notes with these thicker strings, so that it's easier to get the lower notes and I don't have to turn down the tension you know so that it won't work anymore. So anyway that is standing waves with rigid boundary conditions. Now there's another boundary condition that I haven't talked about in this video and that's associated with the boundary condition that is an anti-node instead of a node. Those are called open boundary conditions or free boundary conditions.

Now those are a little bit more difficult to visualize because a free boundary condition means that instead of pinning it down you're got to let it move as much as it wants to. Now that's a, in order to visualize that, what I usually do in class is I take like a rod and I put a ring on it and I attach the string to the ring and I just kind of oscillate it up and down like that. And then the ring can go up and down. But that's not really something that we often see with strings. Usually with strings it's the rigid boundary conditions but we will see these free boundary conditions when we're looking at sound waves that are standing waves. But anyway for right now these are standing waves.