Standing Sound Waves - Online Tutor, Practice Problems & Exam Prep

Longitudinal standing waves, such as sound waves in pipes, differ from transverse standing waves primarily at their endpoints. In an open pipe, both ends are antinodes, while in a closed pipe, one end is a node and the other an antinode. The fundamental frequency for an open pipe is given by , and for a closed pipe, , with allowed values of n being odd integers for closed pipes. Understanding these principles is crucial for solving problems related to acoustics and wave behavior.

Standing Sound Waves

Hey, guys. So in previous videos, we saw how transverse standing waves worked, which are standing waves on strings, but in some problems, you're going to need to know how longitudinal standing waves work. The best example you're going to see is a sound wave that's traveling through an open or closed pipe. Remember, sound waves are longitudinal waves. And they're going to ask you for things like the fundamental frequency or the different harmonics depending on whether that pipe is open or closed. So I'm going to show you the differences between longitudinal and standing waves or and transverse standing waves in this video, and we're going to see that they're actually very similar. Just a couple of key differences between what the endpoints of these waves are doing. So let's check this out real quick.

When we had transverse standing waves, remember that we had waves on strings, both of the endpoints here basically had to return to a displacement of 0. Both of the endpoints had to be nodes. Now in a longitudinal wave, a 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 endpoints are going to be open. 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 antinodes 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 endpoints here are actually going to be antinodes. You're going to 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 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 going to be antinodes. 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 antinode, so you have to have the antinodes here, but the closed end has to be a node. So the way that this is going to look is it's going to look like this. So you're going to have something like that and this is going to be a node and then this point right here is going to be an antinode. This is the only way you can set up a standing wave inside of a closed pipe. Alright. So that's really just the differences between them. It's really what's happening at the endpoints between the two pipes.

Now the equations for for the frequency and the wavelengths are actually very similar to what we've seen before. So in fact, for an open pipe, the equations are going to be the exact same as they were for transverse standing waves. So it's going to be nv/2l, and then for the wavelength, we're going to have 2l/n. The allowed values for this are n=1, 2, 3, and so on and so forth, any integer. Now for a closed wave or closed pipe, it's going to be a little bit different. So here what happens is that the fundamental frequency or the frequencies are going to be, nv/4l. And then your wavelength is going to be 4l/n. Basically, just the 2l's now become 4l. Now what's really important about this is that the allowed values are going to be only odd integers. They have to be odd. There's no way that you could have n=2 inside of a closed pipe. So these 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 pipe which travels through sound and that pipe is going to be 5 meters long. So that's going to be our l. So in the first part, we want to calculate the fundamental frequency. So remember, fundamental just means n=1, and it works the exact same way that it does for standing sound waves. So n=1 here if the pipe is open at both ends. So basically, for part a, we want to figure out f1 for an open pipe. Alright? So how do you do this? Well, we're just going to stick to this equation over here: nv/2l. So we have for n=1, basically, you know, this just becomes 1 and our fundamental frequency becomes v/2l. That's the equation. So what's actually cool about these standing sound waves is they're actually a little bit simpler because we're always going to assume that the speed of sound, right? That's the medium here, is, 343 meters per second unless we're otherwise explicitly told otherwise. Right? So that means that the velocity that we're going to use is 343 and we're going to be using 2 times the length of 5. So that means that your first fundamental frequency is going to be 34.3 Hertz. That's it. That's the answer. Now for part b, we're going to be taking a look at now what happens if you have a 3rd overtone if the pipe is open at one end and then closed at the other. So what does that mean, the 3rd overtone? Well, remember, what are the allowed values for n? So we have n=1, that would be the fundamental frequency for a closed pipe. But remember we have 1 n=1, 3, 5, 7, 9, so on and so forth. So we want to do the 3rd overtone is going to be the 3rd tone over the fundamental frequency. So we're going to go, this is the first one, this is going to be 1, 2, and 3. So we're actually looking for n=7. So the 3rd overtone for a closed pipe is actually going to be f7. It's the 3rd allowed frequency over the fundamental one. So that's going to be a little bit different. Right? So we have f7 for a closed pipe is just going to be nv/4l. So this is going to be 7 times the v which is 343 divided by 4 times the length which is 5. If you go and work this out, you're going to get a 120 Hertz. Alright? So that's basically the difference guys. So let me know if you have any questions and I'll see you in the next one.

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

Example 1

Hey, everyone. Welcome back. Let's take a look at this problem. So we have a length of a closed pipe that's shown over here, which is 2.75 meters long. So I know that's basically my \( l \), and this is 2.75. Alright. 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 want to figure out what's \( f \). Alright? So what is \( f \)? In the second part, I want to figure out what's the fundamental frequency of this pipe. Remember, the fundamental frequency is just going to be \( f_1 \), specifically. Alright? So let's take a look here.

In order to figure out \( f \), what I really actually have to figure out first is which \( n \) I'm dealing with because my equation here for \( f \) is I have to know what \( n \) to plug in. Now, by the way, we're also just going to 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. Alright? So we're going to really be dealing with this equation over here for \( f_n \). Alright? So let's do that. So this \( f_n \) is going to be \( n \times f_1 \), but we actually don't know what that is. We're going to calculate that in part b. So we're not going to use this equation. We're actually going to use the other equation. Now remember, for closed pipes, we don't use 2 \( l \) in the bottom. We used 4 \( l \). We used 4 \( l \) over here, for the bottom of this equation.

Now if you look at this, we already know what \( v \) is. We're just going to use 343. We know what \( l \) is. What we have to actually figure out in this problem is we have to figure out what \( n \) is. And the way we do this is by actually looking at the number of nodes and antinodes in our closed pipe. Now remember what happens is that you always need a node here at the closed end of a pipe, and then you'd always need an antinode 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 going to be \( n \) equals 1. But if it loops over itself, then that's going to be the first heart that's going to be the first overtone. So really, what we're showing here is that this is \( n \) equals 3. Alright? So this is not the simplest standing wave. You're going to have 1 antinode or sorry. Actually, you have 2 and 2 nodes and an antinode here. So this is going to be \( n \) equals 3. So now that we figured out that \( n \) equals 3, now we can go ahead and solve for what this frequency is. So this \( f_3 \) is just going to be 3 times the speed of sound, which is 343, divided by 4 \( l \), which is 4 times 2.75. Now if you work this out, what you should get is you should get, let's see. This is going to be, 93.5 Hz. Alright? So just check my notes. That's going to be 93.5 Hz. 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 \( n \times f_1 \). Alright? So we want to figure out what \( f_1 \) is. We just go from \( f_3 \). This is just going to be \( n \times f_1 \), so this is going to be 3 times \( f_1 \). I'm sorry. This is going to be, yeah. Yeah. Okay. So that means that in order to figure out \( f_1 \), I can basically just take the \( f_3 \) that I just calculated and divide by 3. Alright? So \( f_1 \) will just be \( f_3 \) divided by 3. So this is going to be 93.5 Hz divided by 3, which is just going to give me 31.17 Hz. Alright? 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.

Here’s what students ask on this topic:

What is the fundamental frequency of a standing sound wave in an open pipe?

The fundamental frequency of a standing sound wave in an open pipe is given by the equation:

$f=\frac{nv}{2l}$

where $n$ is the harmonic number (1 for the fundamental frequency), $v$ is the speed of sound in the medium (typically 343 m/s in air), and $l$ is the length of the pipe. For the fundamental frequency, $n$ is 1, so the equation simplifies to:

$f=\frac{v}{2l}$

How do standing sound waves differ in open and closed pipes?

Standing sound waves in open and closed pipes differ primarily at their endpoints. In an open pipe, both ends are antinodes, meaning the air particles at the ends oscillate with maximum amplitude. In a closed pipe, one end is a node (where air particles do not move) and the other end is an antinode. This difference affects the harmonic frequencies and wavelengths. For open pipes, the fundamental frequency is given by:

$f=\frac{nv}{2l}$

For closed pipes, the fundamental frequency is:

$f=\frac{nv}{4l}$

where $n$ is an odd integer.

What are the allowed harmonics in a closed pipe?

In a closed pipe, the allowed harmonics are only the odd integers. This is because one end of the pipe is a node and the other end is an antinode. The fundamental frequency (first harmonic) is given by $n=1$. The next allowed harmonics are $n=3$, $n=5$, and so on. The frequency for these harmonics is given by:

$f=\frac{nv}{4l}$

where $v$ is the speed of sound and $l$ is the length of the pipe.

How do you calculate the third overtone in a closed pipe?

To calculate the third overtone in a closed pipe, you need to understand that overtones are the harmonics above the fundamental frequency. In a closed pipe, only odd harmonics are allowed. The fundamental frequency is the first harmonic ($n=1$), the first overtone is the third harmonic ($n=3$), the second overtone is the fifth harmonic ($n=5$), and the third overtone is the seventh harmonic ($n=7$). The frequency is calculated using:

$f=\frac{nv}{4l}$

where $v$ is the speed of sound and $l$ is the length of the pipe.

What is the difference between longitudinal and transverse standing waves?

Longitudinal and transverse standing waves differ in the direction of particle displacement relative to wave propagation. In longitudinal waves, such as sound waves, particles oscillate parallel to the direction of wave travel. In transverse waves, like those on a string, particles oscillate perpendicular to the direction of wave travel. For standing waves, the key difference lies in the boundary conditions. In an open pipe (longitudinal), both ends are antinodes, while in a closed pipe, one end is a node and the other an antinode. In transverse standing waves on a string, both ends are typically nodes. The equations for frequency and wavelength also differ based on these boundary conditions.