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Current time:0:00Total duration:14:19

Standing waves in open tubes

Video transcript

if you blow air over the top of a soda bottle you get a tone I've got a soda bottle right here I'm going to show you listen to this so the question is why does it make that noise how come you get that loud sound it has to do with something called standing waves or a very closely related idea is resonance and so we're going to talk about this how do these work so let's go into here what I've really got I'm going to model this I'm going to say that I've just got a soda bottle I'm going to model it like it's just a a tube a simple tube and one end is closed this is important this end over here is closed I tried to shade it in to show you that this end is blocked off that's the bottom of the soda bottle this is on its side and this end over here is open and so what happens you've got this closed end you've got this open end you've got air in between what's the air do well when I blow for the top the air starts to move around but this air on the closed end it's pretty much stuck if it tried to it wants to oscillate back and forth that's what these air molecules want to do but every time it tries to oscillate just bumps into this closed end loses its energy and try it again bumps in loses energy so it doesn't really go anywhere whereas on this side the size open and shoot this air can just dance like crazy oscillate back and forth it wouldn't go that far I'm exaggerating here so you can see it but this end will oscillate much more than this other end this closed then the air just stays there in the middle it'll oscillate somewhat somewhere in the middle and so if you wanted to see this I made a little animation so you can see this happen here's what it would look like you see that the closed end the air is not doing anything at the open end the air can also late wildly and in the middle it's varying amount that gets smaller and smaller as you get toward that closed end okay so that's that's this bottle that's how we're modeling this bottle here so you can do this for if you cut the bottom out if you cut the bottom out of the bottle you'd also be able to set up a standing wave it would look like this so let's say you've got an open end on both ends so now we've got an open end on both ends this side is open this side here is open this means the air now on this side isn't stuck anymore this air can oscillate like crazy this air over here can oscillate like crazy and it turns out if you blew over a bottle that was cut open on both ends or if you just had a PVC pipe and you blow it over the top you'd get another resonance you'd get another standing wave and in the middle this air molecule would just stay still these would oscillate like crazy on the ends and this is what that looks like it looks a little bit like this so both ends oscillating like crazy and then right in the middle air not really moving at all so this is a standing wave it's a standing wave I don't I actually don't like that name I like the name dancing wave I mean the air is still moving this air is moving back and forth this air is not but lots of the air is moving back and forth and they call it a standing wave because no longer remember with a wave with a wave you had this compressed region and what it looked like it looked like the compressed region was moving down the line with some velocity so this is a moving wave but when you set up a standing wave I'll show you again this standing wave doesn't really this one here say it's not really the compressed reason doesn't region doesn't look like it's moving down the line everything just kind of bounces back and forth so how do we describe this mathematically that's the hard part this is the part that drives people crazy what we could do we could try to draw lines let's draw some lines that represent where all the air particles are in their equilibrium position equilibrium positions a fancy name for this is if the air was is our open open tube and this is what the air is at when you're just not messing with it the air is just in this position just hang it out and I'm gonna draw these lines here so that we know this is where they want to be and when they get displaced from that position we'll be able to tell how far they've been displaced so if you did this you took a PVC pipe you blew over the top this is what the air looks like before sometime afterward it might look like this now the airs displaced so check it out this this one got dis placed all the way over to there this one went to there this one went to there this one which is the smidgen over this one didn't go anywhere this moment' to the right and this one went to the right and this one went to the right this one went way over to the right this is at the open end so you've got varying amounts of displacement at different points so what we're going to do we're going to graph this I'm going to make a graph of what this thing is doing so let's make a x-axis the horizontal axis this will represent where I am along the tube and then we'll make a vertical axis this will represent how much displacement there actually is so this top end so this will be displacement the amount of displacement of the air molecule and then this is just position along the tube where exactly am i along the tube I'm just going to call it X and so if we graph this what are we going to get well we're going to get is right here this air molecule at this X position has displaced a lot to the left and usually leftwards negative so on this graph I'm going to represent it down here somewhere I'm going to just pick a point down here and I'm going to graph I'll pick a different color so we could see it better I'm going to graph this that's a lot of displacement this one didn't displace at all now it's just right in the middle so that's got to be on the axis because that represents zero displacement this is zero displacement over here and then over here displaced a lot to the right so that would be a lot of displacement to the right and in between it's varying amounts and it would look like this you'd get a graph that looks something like that and what is this this is a standing wave this is what we'd see but it wouldn't stay like this these particles would this one in the middle that keeps on not doing anything but this one over here would then move all the way this way and oscillate back and forth and so what you would see this shape do if you played this in time this would start to move back to equilibrium so this spot would start to move up to here you'd get another point in time where it looked like this everything not nearly displaced as much as it was before and then you wait a little longer it goes flatline everything's back to its equilibrium position then these this one over here starts to move to the right so now it's a little bit further to the right than its equilibrium position and eventually it's flip-flops like this and so you'd get a graph like that and so this is what happens if you watch this graph this graph would dance up and down this part would move all the way to the top and then all the way to the bottom and it's good to know that does not represent an air molecule moving up and down these air molecules do not move up and down they move left and right and this graph that we're drawing represents the amount they have displaced left or right and so this graph this peak called a standing wave because this peak does not this looks like a peak right on it wave on this graph right here this peak does not move to the right it dances up and down now that's what this thing does I wish we could have called them dancing waves but they're called standing wave these Peaks move up and down and the node this guy just stays right here if this was a regular traveling wave if you'd see this node move to the right you'd see this peak move to the right it doesn't do that anymore so we call this a standing wave and I already said it but this point in the middle given a special name this point right here is called the node this is a node and these points at the end this location here in this location here that oscillate wildly are called antinodes so the anti nodes are points where it oscillates wildly and the nodes or points where it doesn't oscillate at all this particle does not move and this point on the graph just stays at zero so the tricky part is how do we represent this mathematically this is how we represented graphically how do we represent this mathematically let me clean this up a little bit the question is how much of a wavelength is that how much of a wavelength is this well if you remember one whole wavelength I'm going to draw a whole wavelength over here one entire wavelength looks something like this so here's a graph just to represent a wave versus X an entire wavelength is when it gets all the way back to where it started so from some point in the cycle all the way back to that point in the cycle would be one wavelength how much of wavelength is this we'll look this is only it starts at the bottom and then it makes it to the top but that's it it stops there so the question is well is that a whole wavelength not it's only half of a wavelength so if we wanted to know how much of a wavelength is this in terms of the length of this tube say this tube has a length L for this first one we'd realize that okay that's half a wavelength so L 1/2 of a wavelength is fitting into a length L so it's 1/2 of a wavelength equals a certain distance the distance that 1/2 of a wavelength equals for this first standing wave we've set up is just 1/2 lambda what that means is well then lambda equals 2l so this is it the lambda of this wave is 2l and we call that the fundamental frequency or the fundamental wavelength and it's a special name because this is the one you'll hear if you blow over a tube this is the one that you'll hear it's going to sound loud this is the wavelength you'll hear but that's not the only one you can set up the only requirement here is that these ends are going to oscillate like crazy and we know have to be antinodes over in this case we had a node in the middle to antinodes at the end the question is what other standing wave could you set up another one would be okay got to be anti node here got to be anti node on the other end but you might have multiple nodes in the middle instead of just one node say we did something like this say we had a wave like that now anti node on this end anti node on this end it's got to be because the open open tube the open ends have to be the anti nodes for the displacement of the particle and now we've got two nodes in the middle though so we've got two nodes in the middle to anti nodes how much of wavelength is this let's check it out so this was a whole wavelength the whole blue and so this green is all the way up to the top and then all the way to the bottom look that's a whole wavelength so in this case Heol the length of this tube is equaling one whole wavelength for this second this is called the second harmonic this is also set up you don't hear it as much but if you were to analyze the frequencies you can see that there's a little bit of that frequency in there too a little bit of that wavelength so in this case lambda equals L so this is called the second harmonic so I'm going to call this lambda 2 lambda 2 just equals L this is the second harmonic and you can find the third harmonic let's see what else would be possible let's try another one you know it's got to be anti note on this end because it's open anti note on this end because it's open but instead of having just one or two nodes in the more likely to have three so I'm going to come all the way up to the top and I'm going to come all the way back down to the bottom and then I'm gonna go all the way back up to the top again this is anti note on this end and I know it on this end now you got one two three nodes in the middle and so how much of a wavelength is this let's try it out let's reference our one wavelength so it starts at the bottom and then it goes all the way up to the top and then it goes all the way down to the bottom but this one keeps going this is more than a whole wavelength because that's just this part that's one whole wavelength now I'm going to go all the way back up to the top so this wave is actually one wavelength and a half this amount is one extra half of a wavelength so this was one wavelength and a half so in this case L the total distance of the tube that's not changing here the total distance of the tube is L this time a wavelength then there is fitting and one and a half wavelengths fit in there that's three halves of a wavelength that means the lambda equals 2l over three so in this case for lambda three this is going to be called the third harmonic third this is the third possible wavelength that can fit in there this would be 2l over three and so these are the it keeps going you can have the fourth harmonic fifth harmonic every time you add one more node in here it's always got to be anti note on one and anti-node at the other these are the possible wavelengths and if you wanted the possible all of the possible ones you could probably see the pattern here look too well and then just L then 2l over three the next one turns out to be two L over 4 and then 2 over 5 2l over 6 and so the if you wanted to just write them all down shoot lambda N equals this is all the possible wavelengths to L over N where N equals 1 2 3 4 and so on and so if I look at if I had n equals 1 in here I'd have 2 well that's the fundamental you plug in N equals 1 you get the fundamental if I plug in N equals 2 I get to L over 2 that's just L that's my second harmonic because I'm plugging in N equals 2 if I plug in N equals 3 I get to L over 3 that's my third harmonic this is telling me all the possible wavelengths that I'm getting for this standing wave so that's open open in the next video I'm going to show you how to handle open closed tubes