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# Standing waves in closed tubes

## Video transcript

okay so last time we saw that for an open open tube or an open open pipe a pipe where both ends were open there were only particular wavelengths that were allowed because you had to have anti nodes at both ends so you had that one we had this one we have this one and we found the wavelength of all of these and then we realized wait a minute we can write down a formula for any possible wavelength in an open open tube and it depends only on the length of the tube L and n and n is the which harmonic we're talking about one is the fundamental - is the second harmonic three is the third harmonic four and so on and this gave you every possible wavelength the question is can we do the same thing if we had an open closed end what happens if one ends closed what if we closed this end off so that it's not open anymore well let's do that right now here we go here's a closed end right here this end is closed this is more like a soda bottle because this end would be open the top that you drink out of and the bottom is closed off we've got air inside if you blow over the top what possible frequencies what possible wavelengths could you set up well what do we know here we know that this end this air is open or this side is open and so this air molecule can oscillate wildly when it what type of note is that going to be that's gonna be an anti note since it could oscillate a lot but this end over here these air molecules keep bumping into the side it's got to be a note over on this end because there can't be any displacement so what do we do here okay so this ends a node we know that I'm going to put this on the axes here because I know this end has no displacement we're graphing displacement again versus X and this is in this case horizontal displacement this end is going to oscillate wildly so I know this is going to be an anti note so I'm going to put it right here and I'm going to draw the simplest possible wave that can go from this anti node to this node is going to look like this anti node and it goes curve right down to the node that's it what wavelength is this well in terms of the length of this tube so if this is L what wavelength is this in terms of L we got to figure out how much of a wavelength this is that's always the trick this is what freaked students out they don't like figuring out how many wavelengths is this but it's not too hard first thing I like to do is draw what I know one wavelength looks like so what is one wavelength look like one wavelength looks like this okay if we start up here a wavelength is going all the way and then back to the same point in the process so there we go that's one entire wavelength at least if this is a versus X graph it is so how much of a wavelength is this pink line for this first fundamental frequency this first fundamental wavelength starts at the top so let's start at the top and it goes and that just goes till the first crossing of the axis and that's the node and so this thing crosses the X is here that's it that's all we got we got a big long what is this this is one-fourth of a wavelength so that's 1/4 and it gets to here that's another fourth here's another fourth and here's another fourth so this is only one-fourth of a wavelength that's hard for a lot of people to see the whole thing's one wavelength half of it is a half so if I cut in half that's a half and I cut this half in half again I get 1/4 so what I find out is that ok the length L equals one-fourth of a wavelength in this case our fundamental wavelength divided by four equals the length of this tube so if I want to know the wavelength is that means my wavelength for the fundamental open closed case is 4l some write that over here lambda equals 4l and this would be the fundamental wavelength for this open closed tube and this is bigger for open open it was 2l so it's 4l how about the next one all right we got to start at an anti-node and we got to end of the node the next simplest case in well we had no nodes in between we only had one at the end so now I need a node in the middle so I'm going to create a node in the middle and then I have to get all the way back to the anti intro to the note on this end so I've got one note here I've got a note at the end and I've got this anti note here and an anti node up here how much weight how many wavelengths is this let's figure it out start at the top alright this goes all the way down past the node and back up so all the way down past the node back up and ends at a node that's how much of a wavelength I've got here so how much is this well it's 1/4 2/4 3/4 so this time L is the length of my tube the amount of wavelengths fitting into L in this case is 3/4 of a wavelength and again if I solve that for wavelength I get so I'm going to get 4l over 3 so the next possible wavelength is 4 L over 3 and if we solve for the next one let's draw it started an anti-node ended a node the next possible one well the last one had one node this time is going to have two nodes and we come down here one node to node and then it gets back to the node at the end so a node at this end it has to be a node at the Sun because it's closed this time have two nodes in the middle and have an anti node on this end and two anti nodes in the middle so how many how many wavelengths is this alright let's figure it out start at the top alright comes down let's just trace it out down his that first ant hits that first node down to the anti node back up to the node now I've gotten to here but I keep going I go up to the top I'm not done yet now I'm back up to here I have to keep going one more fourth of a wavelength so this is more than a wavelength this time L equals well this is one with whole wavelength just to this point right here and I have to add one more fourth of a wavelength to that so this is one wavelength and 1/4 or another way to say that is that this is 5/4 civil wavelength 5/4 of a wavelength because just two here is one whole wavelength and I have to add one more fourth to that so if I solve for lambda I get lambda is 4 L we see a pattern patterns are great for L over five this next wavelength possible is 4l over five and shoot we can do this now now I see enough if I want to write down any possible wavelength it's going to be 4 L loop 4 L 4 L over 3 4 over 5 the next one be 4 L over 7 I get that the possible wavelengths for an open closed tube are 4 L over N except instead of being any possible integer the only allowed integers are the odd ones so I'm not allowed to put in 2 or 4 it's got to be 1 3 5 and so on and so here's the formula this is it if I wanted to know what are the possible wavelengths allowed in an open closed tube this is it it depends on the length it depends on n looks just like the case for open open except it was 2l for that case this case is 4 L on top 4 times the length of the tube and on the bottom it's only the odd integers that means I only get the odd harmonics this 4l over 3 I wouldn't call this the second harmonic I'd call this third harmonic and this 5 down here I wouldn't call this the third call this the fifth you're missing we're missing all the even harmonics on this case for an open closed tube so it's a little bit strange but that's what happens when you have an anti-node at this end and a node required at that end and so you could try this look this is a function of L if you want to test this out next time you're drinking a soda look if the L is large if the length of your tube is large that means the wave length should be large and if the wave length is large well let's see V equals lambda F so if you've got a big wave length that'll mean a small frequency because the speed won't change the speeds are determined by the medium and you're probably not changing the temperature of the medium all that much maybe you've got ice in your drink or something but if it melts it will change a little bit it's not going to change the speed by that much so if you increase the length you'll increase the wave length will decrease the frequency low frequency means a low note you'll hear a lower base e note and so try this I want you to try this out next time you're drinking a soda or or whatever maybe is a healthy beverage whatever it is you've got some soda here it's up to some level this part acts this wherever the water is at acts as the bottom of the tube so right now I only have a tube this length because the top is open and the bottom is basically wherever this water level is because the air can't get past that water level so this would only be a length of that but as you keep drinking keep blowing over the top and you keep blowing over the top listen for what note you hear and you should keep hearing a lower note the lower this gets so once it's down to here now your length the length of your tube is bigger to open at the top close at the bottom you hear an even lower note once it gets down to here when you finish now it's even longer you'll hear a much lower note so the notes you get lower and lower not louder and louder but lower and lower the frequency should sound lower and lower the lower your drink gets because the higher the length bigger the wavelength smaller the frequency