Hey guys. So we talked a lot about the graphs and the diagrams for standing waves. And I've given you sort of a brief introduction, but now we're actually going to get into the equations for transverse standing waves on strings. All right. So let's get to it here. Do you remember that? The idea is that standing waves can only exist for special harmonic frequencies. These are special frequencies allow you to set up a standing wave. So for instance, here we have N equals 11 loop. There's one frequency that's associated with that. Here we have N equals two and there's another frequency that's associated with that standing wave. And then here we have N equals three and then so on and so forth. So there's special values for the wavelength and also the frequency I'm just gonna give you these equations here. Remember that the fundamental frequency F one here is the lowest frequency you can have to set up a standing wave. Now, if you're not given that information, you can calculate it by using the properties of the wave, it's the equation is gonna be V over two L. Now any harmonic frequencies, remember these harmonic frequencies are basically gonna depend on this fundamental frequency. We saw that in the last couple of videos where we just multiplied this number here of these, these F ones by multiples of N. So what happens if we just stick an N in front of this equation? And our FN equation just becomes NV divided by two L. Now last but not least our wavelength equation is going to be two L divided by N. So it's these three equations right here. These bottom two actually work for any value of N, any number of loops that you have in your standing wave pattern. That's really all there is to it guys. So let's go ahead and take a look at our example here. So we have a one point meter, 1.5 m long string. So I mean the length of this string here is 1.5 that's tied between the two supports the speed of transverse waves that we're told is V equals 48 m per second. Now, in the first part of the problem, we're gonna calculate the wavelength and the frequency. So we're gonna calculate lambda and F four, the fundamental tone. Remember that whenever you see fundamental, that's just gonna be N equals one. So we have N equals one here, we're gonna calculate lambda one and F one. And to do that because we're not given any of the other information like any of the other frequencies we're gonna have to stick to these equations over here. So our F one just gonna be V over two L. Uh So we're gonna have, sorry, I'm gonna go with lambda first. So we're gonna have two L divided by uh N equals one. So this is, we're gonna have two times 1.5 and that's gonna give us 3 m. So let me go ahead and just draw this out. Remember the fundamental tone when N equals one is just gonna be a loop that has or, or a pattern that just has one loop like this. So if you'll notice what happens is that our wavelength is 3 m, but the length of our string is only 1.5. How can that possibly be? Well, remember that the total wavelength is a, is a complete up and down cycle. So a complete wavelength would look something like this. So our string is actually only half of that wavelength. So the fact that we got a wavelength of double of the length of our string actually totally makes sense. All right. So our F one here is just gonna be V divided by two L. So this is gonna be 48 divided by two times 1.5 and we're gonna get 32 Hertz. All right. So those are the two answers. We've got 3 m and 32 Hertz. So let's take a look at the second one. Now, part B now we want to calculate the same exact thing, we wanna calculate lambda and frequency except we wanna do it for something called the first overtone. So, what does that mean? Well, it turns out there's actually two different words that will tell you the value of N and those two words are harmonic and overtone. So both these words basically tell you what the value of N is. So we've actually seen harmonic. The first harmonic is just when N equals one second harmonic is when it equals two, third harmonic is when it equals three and so on and so forth, right? So whatever number this is when it says harmonic, that's just the number of the number of loops that you have. Now where things get a little tricky is there's another word called an overtone and basically what an overtone is, is it's a tone that is over your fundamental frequency. So what happens is that N equals two is your second or harmonic, but it's the first overtone, it's the first tone over F one. So the first to overtone is N equals two, the N equals three is the third harmonic, but it is the second overtone and then so on and so forth. So in our problem here, when they say calculate the wavelength and frequency of the first overtone, they're actually saying that N equals two here, it's the first tone over F one. So we're gonna calculate lambda two and F two. So our lambda two is just gonna be two L divided by two, which is our N and we're gonna use this equation over here. And what that happens is we just cancel out the twos and then our wavelength just becomes L which is 1.5 m. So if we draw this out, this should make some sense because N equals two is just a pattern where that has two loops like this. So here we have one complete wavelength inside of our L equals 1.5. So those things are the same number. Now, for F two, there's actually two different ways we can calculate this. Well, first thing we could do is we could use two times F one, right? That's just using, you know, the harmonic frequency, the multiple of F one. So we're gonna have two times 16 and that's gonna be 32 Hertz. That's one way to calculate this or another way is we can sort of do it the long way. Just in case we actually work F two, we could calculate this by using um NV divided by two L. So we're gonna use two times 48 divided by two times 1.5. And you should get also 32 Hertz when you're done with that. All right. So these are the two equations. Um Basically, you know, it's pretty straightforward. Let me know if you guys have any questions and I'll see you in the next one.