Everyone. Welcome back here. So let's take a look at this example problem here. It's a little tricky because we're given a wave function of a standing wave. It's this big long equation over here. Now, we're told that the left end of a wire that we're gonna set up a standing wave for is that X equals zero. So it's basically just that the left supports of some kind of a wall or structure or something like that. What we want to do in this problem is we want to derive an expression for the distances of the nodes. What does that mean? Well, if you set up a standing wave, let's sort of draw this out what this might look like. Right? Let's set up a standing wave and let's just say it has, I don't know the three or four loops, it actually doesn't really matter. So for something where it isn't, you know, just like the fundamental frequency, it's gonna look something like this, right? There's gonna be like three or let's say four loops that's gonna hit the support and then it's basically just gonna double back over itself like this. Let me draw this a little bit nicer. So I can sort of visualize what's going on here. Now, remember what happens is there's nodes and anti nodes of a standing wave. The nodes are basically the places where the loops cross and these are gonna be places where if you imagine sort of like a jump rope that's doing something like this or a wave that's sort of doing something like this. It's basically just the place that are kind of we gonna be almost as if they're sort of standing still. So these are the nodes, right? So these are the nodes like this and these are the anti nodes, the places where it actually hits the maximum. So it's kind of imagine like a jump rope that's going up and down. Those are the anti nodes, the place where the jump rope is the highest. So these right here are the anti nodes. All right, we want to do is we want to figure out an expression for the distances of the nodes, right? So imagine that the standing wave basically just continues on like this. What we're gonna see here is that there's an X value for this node, there's an X value for this node and this node and we can actually sort of set up a pattern for this because they're actually gonna be sort of multiples of each other. So that's basically what we want to do here. We want to figure out what are the values of these sort of X's over here. So that's really what, what I'm focused on focusing on here. And I'm looking for, what are the distances of these points? How do I do that? Well, let's take a look at the equation here because remember that the general sort of format for a standing wave is gonna be a times and there's two signs, there's a sign of KX and then there's also a sign of Omega T. All right. So there's two signs in this equation and remember that basically the A S sign KX is actually all encased in a bracket. Um And then what happens is that thing oscillates over time. OK. So here's what we're gonna do for nodes where the key thing to remember about these nodes is what happens to the Y value when you actually sort of evaluate the wave function over here. The key thing that you have to remember is because these nodes are basically points where no matter what happens with the, the jump rope. And what happens with time, the Y values of these things, we always want to be zero forever. So basically, we want the Y equals zero for all values of T. That's the most important thing here. That's kind of like your condition for what these nodes are mathematically. So Y equals zero for all values of T. Now, why is this important? Well, let's take a look at the general form of our equation. And we see that Y equals oh sorry, Y of X and T it's just gonna be a times uh Sine kxskx and that's gonna be in brackets. And then we have another sign of Omega T. We want this wave function, whatever we plug in values of X and T, we want this thing to basically just equal zero. So what we do in this case is because we don't care what the values of T are, we basically kind of just ignore it. What we're really looking interested in is we're trying to figure out what the values of X that I plug into this equation where no matter what I plug in for T, I'm always just gonna get zero. So it's kind of like we kind of just ignore the second sign and we're only dealing with the first one. OK. So let's just write this out. We're basically just dealing with where, where, what, where does a SKX equal zero? So let me actually just start replacing some of these numbers here. So in other words, we have our amplitude which is 0.00 25 times the sign of and we have 0.75 pi uh that's our K value. Um And that equals zero. All right. So basically what we're trying to find here is we can simplify this a little bit because we can actually just divide both sides by the amplitude 0025 that will just cancel 0025. And if we just divide by something, it doesn't matter because it's still just gonna be zero anyways, right? So essentially what we're asked to find is where does sine of 0.75 pi X equals zero? What are values or what are, what is the value or values of X that will make this equation equal to zero? Now, what you might see in your problems in, in, in your textbooks is you might see sort of like this derivation where they're talking about, well, where to s sign KX equal zero. And to kind of imagine this, let's sort of sort of look at what a sine graph looks like. So if you look at, if you sort of visualize what a sine graph looks like, which is not a standing wave, right? We're just looking at what a sine graph looks like it's gonna look like this. So it starts at zero, remember? And then it goes up and then down and then it goes up again and then it goes down again. Now, what are the values where it hits zero? Well, it hits zero at zero. But then there's also this point over here where the Y value hits zero where the sine graph hits zero. So in the words sin of X, right? So this happens at pi and then if you complete the cycle and it hits zero again at two pi, then if you basically just, you know, keep it going again, it's gonna hit zero again at three pi if you notice that there's a pattern that's going on. What happens is that the sine graph will always be zero whenever the X equals some kind of integer, multiple of pi, so one pi or two pi or three pi and so on and so forth. So what the, what the textbooks would do in these sort of like in these sort of uh derivations is they'll say the sign of KX. So sign of KX equals zero whenever, whenever KX equals N times pi. Now, the most important thing to remember about this is that this N is not for the number of loops. So this does not mean that N is like the number of loops that we have on our standing wave. Really? This N is kind of just an integer. It's what happens when N equals zero or one or two or three or four and so on and so forth, right? So in other words, Sine equals zero. S of KX equals zero whenever KX, whenever the thing inside the parentheses ends up being either zero pi or one pi or two pi or three pi and so on and so forth. OK. So why is this helpful? Because basically we can solve or the X values that will make this happen. So in other words, SXS of KX equals zero when X equals and pi divided by K. OK. So in other words, what happens is when X equals zero pi over K or one pi over K or two pi over K or, and then so on, you can just sort of repeat that over and over again. So let's just plug in some numbers here and actually solve what these numbers are. So this is just gonna be, well, really zero divided by anything is just zero. But then what about one pi over K? This is really just gonna be one pi divided by 0.75 pi. So what happens is you can cancel out these two pies and you get one over 0.75. And then what happens is you get two pi divided by 0.75 pi and you can cancel out the pies again and then three will be the same thing. Three pi divided by 0.75 pi you'll cancel that stuff out. So what does this actually work out to? Well, for basically, for the grand finale X equals or sorry sign of KX will equal zero, whatever X equals zero or uh 1.33 or 2.66 or three and then so on and so forth, right? So basically that's what sort of uh all of these values will be. Um And so if I go back up to the equation or if I go back up to my graph, I can sort of visualize where this is gonna happen. Uh Because if I sort of go back up here, uh what this is gonna be is I'm gonna have a node here at 1.33 that's gonna be in, I think this is gonna be in millimeters. Um I'm actually not sure. So this is gonna be 1.33. This is gonna be 2.66. That's gonna be my second node. The third node is gonna be at 3 m and then so on and so forth. So these are just gonna be the values and you can continue this pattern on forever. So hopefully, this makes sense. This is a little bit of like a derivation. Um But it's really important uh because you might be asked for that on a test. So that's how you derive the distances of the nodes given a standing wave function. Thanks for watching.