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.