Standing Sound Waves - Video Tutorials & Practice Problems

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concept

Standing Sound Waves

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5m

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Hey guys. So in previous videos, we saw how transverse standing waves worked, which were standing waves on strings. But in some problems, you're gonna need to know how longitudinal standing waves worked. The best example you're gonna see is a sound wave that's traveling through an open or closed pipe. Remember sound waves are longitudinal waves. And they're gonna ask you for things like the fundamental frequency or the different harmonics depending on whether that pipe is open or closed. So I'm gonna show you the differences between longitudinal and standing wave and transverse standing waves in this video. And we're gonna see that they're actually very similar. Just a couple of key differences between what the end points of these waves are doing. So let's check this out real quick when we had transfer standing waves. Remember that we had waves on strings. Both of the end points here basically had to return to a displacement of zero. Both of the end points had to be nodes. Now in a longitudinal wave or longitudinal standing wave, it's actually different because it depends on whether the pipe is open or closed. So for an open pipe, what happens is that both of the end points are going to be open. That's why it's, that's why it's called an open pipe. And the way that this works here is that both of the ends have to be anti nodes here. So the way this works is that the only way you can set up a standing wave inside of an open pipe is if the end points here are actually going to be anti nodes, you're gonna see a wave that looks like this. And then the other sort of half of that wave looks like this. So what's kind of funny about these problems is that a lot of textbooks will actually draw it exactly this way as if it were a transverse standing wave. So the idea behind an open wave is that, you know, both of the ends are gonna be anti nodes. Now for a closed or stopped pipe, what happens is that one of the ends is open and the other end is going to be closed. That's this one right here. Now, what happens here is that the only way you can set up a standing wave is if the open end is an anti no. So you have to have the anti nodes here, but the closed end has to be a node. So the way that this is gonna look is it's gonna look like this. So you're gonna have something like that and this is gonna be a node and then this point right here is gonna be an anti node. This is the only way you can set up a standing wave inside of an o inside of a closed pipe. All right. So that's really just the differences between them. It's really what's happening at the end points between the two pipes. Now, the equations are for, for the, for the frequency and the wavelengths are actually be very similar to what we've seen before. So, in fact, for an open pipe, the equations are gonna be the exact same as they were for transverse standing waves. So it's gonna be NV divided by two L. And then for the wavelength we're gonna have two L over N and the allowed values for this are N equals one and two and three and so on and so forth. Any integer. Now for a closed wave or closed pipe, it's gonna be a little bit different. So here what happens is that the fundamental frequency or the frequencies are gonna be uh NV divided by four L and then your wavelength is going to be four L divided by N. So basically just the two LS now become four LS. Now, what's really important about this is that the allowed values are gonna be only odd integer. They have to be odd. There's no way that you could have N equals two instead of a closed pipe. So these in these uh values here have to be odd values. But that's it. That's the only difference here guys. So let's go and take a look at our problem here, right? So we have a uh pipe which travels through sound and that's that pipe is gonna be 5 m long. So that's gonna be RL. So in the first part, we want to calculate the fundamental frequency. So remember fundamental just means N equals one and it works the exact same way that it does for standing sound waves. So N equals one here, if the pipe is open at both ends. So basically, for part A, we want to figure out F one for an open pipe. All right. So how do you do this? We're just gonna stick to this equation over here. So we have for the N equals one. Basically the, you know, this just becomes one and our fundamental frequency becomes V over two L. That's the equation. So what's actually cool about these standing sound waves is that they're actually a little bit simpler because we're always gonna assume that the speed of sound, right? That's the medium here. Um is the 343 m per second unless we're otherwise explicitly told. Uh otherwise, right? So that means that the velocity that we're gonna use is 343 and we're gonna be using two times the length of five. So that means that your first fundamental frequency is gonna be 33 34.3 Hertz. That's it. That's the uh answer. Now, for part B, we're gonna be taking a look at now what happens if you have a third overtone? If the pipe is open at one end and then closed at the other? So what does that mean? The third overtone? Well, remember what if the allowed values for N, so, so we have N equals one that would be the fundamental frequency for a closed pipe. But remember we have one, N equals 13579, so on and so forth. So what we wanna do the third overtone is gonna be the third tone over the fundamental frequency. So we're gonna go, this is the first one. This is gonna be 12 and three. So we're actually looking for N equals seven. So the third overtone for a closed pipe is actually going to be F seven. It's the third allowed frequency over the fundamental one. So that's gonna be a little bit different, right? So we have F seven for a closed pipe is just gonna be NV, divided by four L. So this is gonna be seven times the V which is 343 divided by four times the length, which is five. If you go ahead and work this out, you're gonna get 100 and 20 Hertz. All right. So that's basically the difference guys. So let me know if you have any questions and I'll see the next one.

2

Problem

Problem

The fundamental frequency of your closed organ pipe is 200 Hz. The second overtone of this pipe has the same frequency as the 3^{rd} harmonic of an open pipe. What is the length of this open pipe?

A

0.85 m

B

0.51 m

C

0.69 m

D

0.43

3

example

Example 1

Video duration:

3m

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Hey, everyone. Welcome back. Let's take a look at this problem. So we have the length of a closed pipe that's shown over here, which is 2.75 m long. So I know that's basically my L and this is 2.75. All right, I want to calculate the frequency of the standing wave that's shown here. Remember standing waves visually kind of just look exactly like transverse waves. So in the first part, I wanna figure out what's F. All right. So what is F and the second part, I want to figure out what's the fundamental frequency of this pipe or the fundamental frequency is just gonna be F one specifically. All right. So let's take a look here. So in order to figure out f what I really actually have to figure out first is which end I'm dealing with. Because my equation here for F is I have to know what end to plug in. Now, by the way, we're also just gonna be using our closed pipe equations because we're not dealing with an open pipe. So you can kind of just cross those out already. All right. So we're gonna be really be dealing with this equation over here for FN. All right. So let's do that. So this FN is gonna be N times F one, but we actually don't know what that is. We're gonna calculate that in part B. So we're not going to use this equation. We're actually gonna use the other equation. Now, remember for closed pipes, we don't use two L in the bottom. We used four L, we use four L over here uh for the bottom of this equation. Now, if you look at this, we already know what V is. We're just gonna use 343 we know what L is. We have to actually figure out in this problem is we have to figure out what N is. And the way we do this, this is by actually looking at the number of nodes and anti nodes in our closed pipe. Now, remember what happens is that you always need a node here at the uh closed end of a, of a pipe. And then you'd always need an anti node over here at the open parts. So really, we just have to sort of count out how many loops we have. Remember that a closed pipe like this where it just kind of opens out, this is gonna be N equals one. But if it loops over itself, then that's gonna be the first heart or that's gonna be the first overtone. So really what we're shown here is that this is N equals three. All right. So this is not the simplest standing wave, you're gonna have one anti node or sorry, actually 22 nodes and an anti node here. So this is gonna be N equals three. So now that we figure out that N equals three, now we can go ahead and solve for what this, what this frequency is. So this F three is just gonna be three times the speed of sound, which is 343 divided by four L which is four times 2.75. Now, if you work this out, well, you should get, as you should get. Uh Let's see, this is going to be uh 93.5 Hertz. All right. So check my notes, that's gonna be 93.5 Hertz. That's the first part of the problem. Now, the second part is actually way more straightforward because now that we know what the NTH frequency is, we could always just work backwards to figure out the fundamental frequency just by using this relationship over here. It's N times F one. All right. So we want to figure out what F one is, we would just go from F three. This is just gonna be N times F one. So this is gonna be three times F one. Uh I'm sorry, this is gonna be Yeah. Yeah. OK. So that means that in order to figure out F one, I could basically just take the F three that I just calculated and divide by three. All right. So F one will just be F three divided by three. So this is gonna be 93.5 Hertz divided by three, which is just gonna give me 31.17 Hertz. All right. So that is the answers to the frequency and fundamental frequency of this closed pipe. Let me know if you have any questions. Thanks for watching. I'll see you in the next one.

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