Hey, guys. So, up until now, we've seen some basic diagrams and equations for standing waves, but in some problems, you're going to be given the wave function for a standing wave and you're going to have to calculate some of its properties. So, the example they're going to work out down here actually has a standing wave function like this. We're going to calculate things like the length of the string or the period of oscillation. So, I'm going to show you how to do that in this video, how to calculate the properties of standing waves by using its wave function. So we're going to see a wave function that looks sort of similar to the ones that we've seen so far, but it is going to be a little bit different. So, I'm going to go ahead and walk you through it.

Remember that the wave function for a standing wave is really just a combination of the original and the reflected waves that sort of get added together. So the wave function is going to look like this. You're going to have that y of x and t, right, as we've seen so far, is going to be \( ASW \sin(kx) \sin(\omega t) \). So this is the wave function for standing waves only. Notice how it looks a little bit different than the wave functions that we've seen so far. The ones that we've seen so far have either been a sine or a cosine plus or minus \(\omega t\), and in this sort of wave function, we have two sines. We have a sine for \(kx\) and a sine for \(\omega t\). The other difference here is that we have an amplitude, \(ASW\). So this is the amplitude of the standing wave and basically, it's just going to be twice the amplitude of the original and reflected waves that make it up. So, what happens is these two things sort of interfere with each other and their amplitudes actually double. So that's the amplitude of the standing wave. Other than that, that's really just the differences. So all the relationships between k and \(\omega\), all of these relationships here still hold. So we're going to be able to use all of our equations here.

Let's go ahead and check this out. We have this wave function here for a thin string that's under tension tied at both ends, and it's vibrating in its third harmonic. So remember, third harmonic means that \(n = 3\). It's actually giving you what the n number is. So we have this wave function here, and we're told that the amplitude here is 5 centimeters. So this is our \(ASW\). Now, the first sine is going to be \(kx\). Right? So this is actually going to be \(k\). And then the second sine is going to be \(\omega t\). So that actually this is going to be our \(\omega\). So it's actually giving us the values for \(k\) and \(\omega\) here and also the amplitude. Let's take a look at the first one; we're going to draw a sketch of the standing wave. So how do we do that? Well, we need to know the amplitude, right, and we also need to know how many loops there are. So, you know, the amplitude here is 5 centimeters. It's going to basically oscillate between 5 and negative 5. Now, how many loops are there? Remember the number of loops is always going to be related to the number \(n\). Right? So the value of \(n\) here. If \(n = 3\), there are 3 loops. So really what this is going to look like is this. We have one loop like this, a second loop like this, and then a third one like this. And then the reflected wave is going to be basically like that, right, so it's going to look like this. So there are 3 loops, that's \(n = 3\). Other than that, we know that there are 4 nodes. Right? There's 1, 2, 3, and 4, and there are going to be 3 antinodes, 1, 2, and 3.

Alright. So that's really the sketch of what the standing wave is going to look like. Let's move on to the second part. We want to calculate out the amplitude of the waves that make up this standing wave. So what does that mean? Well, remember that if \(ASW\) is equal to 5 centimeters, what we said is that the amplitude of the standing wave is actually twice the amplitude of the waves that make it up. So that just means that the \(a\) is just really going to be 5 centimeters divided by 2, which is going to be 2.5 centimeters here. So these sort of you could kind of imagine that the one wave is going to look like this. Right? It's going to be 2 and a half, and it's going to look like this, and the other wave is going to be the same but backwards. And when they sort of add up together, they're actually going to multiply by 2 to create that big standing wave that has twice the amplitude.

Alright. So let's move on here. We're going to calculate the length of the string. So what does that mean? Well, we have \(l\), right, that's what we want to calculate. We actually didn't know what the length of the string was. So how do we do that? Well, remember that there's a relationship between the length of the string, which is \(l\), and the wavelength, depending on which standing wave we're at or which value of \(n\) that we have. So remember that \(\lambda_n\) is going to be \(2l/n\). So we can actually go ahead and write an expression for \(l\). So \(l\) is just going to be, if we rearrange this, we're going to have \(n \times \lambda_n / 2\). So if we plug in some numbers, we know that \(n = 3\), so we have \(3 \times \lambda_3 / 2\). So, we're almost ready to plug in. The problem is I actually don't know what the \(\lambda_3\) is. What's the wavelength for a standing wave or \(n = 3\)? So how do I figure that out? Well, remember that we have the wave function of this equation here and we're told that this that the \(k\) value, the value of \(k\) here is equal to 0.34. So what happens here is that we can write a relationship between \(k\) and our \(\lambda\). So we have that \(k = 2\pi/\lambda\). So that means if we flip the values like this, then that means that \(\lambda\) is just going to equal \(2\pi/k\). So it's going to be \(2\pi\) divided by the \(k\) value which is 0.034, and you're going to get the wavelength of 185 centimeters. So now all we have to do is just plug that value in for here and then solve. So we have that \(3 \times 185 / 2\). Remember, we're going to keep everything in centimeters. You're going to get 278 centimeters. So it's 278.

Alright. Now for the last problem, we're going to calculate the period of oscillation. So what does that mean? Remember that the period of oscillation is the value \(T\). And how do we figure that out? Remember, we only have one relationship for \(T\). It's actually just going to be \(\omega = 2\pi/T\). So we want to calculate \(T\), we're going to actually have to relate this to the angular frequency. So \(\omega = 2\pi/T\). So therefore, \(T = 2\pi/\omega\). Remember these things are just going to flip around like this. So how do we figure this out? Well, remember we actually have the value for \(\omega\) because it's actually given to us in our wave function. \(\omega\) is just going to be the 50. So that means that we have \(2\pi / 50\) here, and that's in radians per second, so we don't have to do any conversions, and you're just going to get a period of 1.3 seconds. Alright? So that's it for this one, guys. Let me know if you have any questions.