Hey, 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 going to set up a standing wave for is at x equals 0. So it's basically just that the left support is some kind of wall or structure or something like that. What we want to do in this problem is 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, 3 or 4 loops. It actually doesn't really matter. So for something where it isn't, you know, just like the fundamental frequency, it's it's gonna look something like this. Right? There's gonna be, like, 3 or, let's say, 4 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 are nodes and antinodes of a standing wave. The nodes are basically 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 places that are kind of we're gonna be gonna be almost as if they're stand 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 imagined like a jump rope that's going up and down. Those are the antinodes, the place where the jump rope is the highest. So these right here are the antinodes. Alright. 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 I'm focusing 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, the general sort of format for a standing wave is gonna be a times and there are 2 signs. There's a sine of kx, and then there's also a sine of omega t. Alright? So there are 2 signs in this equation, and remember that, basically, the a sine kx is actually all encased in a bracket, and then what happens is that thing oscillates over time. Okay? So here's what we're gonna do. For nodes, well, 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 0 forever. So, basically, we want the y equals 0 for all values of t. That's the most important thing here. It's kind of like your condition for what these nodes are mathematically. Oh, oh, sorry, y of x and t is just gonna be a times, sine kx sine kx, and that's gonna be in brackets. And then we have another sine 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 0. So 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 interested in is we're trying to figure out what are the values of x that I plug into this equation where no matter what I plug in for t, I'm always just going to get 0. So it's kind of like we've kind of just ignored the second sign, and we're only dealing with the first one. Okay? So let's just write this out. We're basically just dealing with where where well, this where does a sine kx equal 0? So let me actually just start replacing some of these numbers here. So in other words, we have our amplitude, which is 0.0025 times the sine, and then we have 0.75pi. That's our k value, and that equals 0. Alright? 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 0 anyways. Right? So, essentially, what we're asked to find is, where does sine of 0.75pix equals 0? What are values what are what is the value or values of x that will make this equation equal to 0? Now what you might see in your problems in your in your textbooks is you might see sort of, like, this derivation where they're talking about, well, where does sinekx equal 0? And to kind of imagine this, let's say let's 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 0, 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 0? Well, it hits 0 at 0, but then there's also this point over here where the y value hit 0, where the sine graph hits 0. So in the words, sine of x. Right? So this happens at pi. And then if you complete the cycle, then it hits 0 again at 2 pi. Then if you basically just, you know, keep it going again, it's gonna hit 0 again at 3 pi. If you notice that there's a pattern that's going on, what happens is that the sine graph will always be 0 whenever the x equals some kind of integer multiple of pi, so 1 pi or 2 pi or 3 pi and so on and so forth. So what the what the textbooks would do in these sort of, like, in these sort of derivations is they'll say the sine of kx, so sine of k x equals 0 whenever whenever k x 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 0 or 1 or 2 or 3 or 4 and so on and so forth. Right? So in other words, sine equals 0, sine of kx equals 0, whenever kx, whenever the thing inside the parenthesis ends up being either 0 pi or 1 pi or 2 pi or 3 pi and so on and so forth. Okay? So why is this helpful? Because, basically, we can solve for the x values that will make this happen. So in other words, sin x sin of kx equals 0 when x equals n pi divided by k. Okay? So in other words, what happens is when x equals 0 pi over k or 1 pi over k or 2 pi over k or and then so on. You can just repeat that over and over again. So let's just plug in some numbers here and actually solve for what these numbers are. So So this is just gonna be, well, really, 0 divided by anything. It's just 0. But then what about 1 pi over k? This is really just gonna be 1 pi divided by 0.75 pi. So what happens is you can cancel out these 2 pis, and you get 1 over 0.75. And then what happens is you get 2 pi divided by 0.75pi, and you can cancel out the pis again. And then 3 will be the same thing. 3pi 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. Sine of k x will equal 0, whatever x equals 0 or, 1.33 or 2.66 or 3, and then so on and so forth. Right? So, basically, that's what sort of all of these values will be. 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 because if I sort of go back up here, 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. I'm, actually, not sure. So it's just 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 meters, 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, but it's really important, 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.
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Standing Wave Functions
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