Standing Wave Functions - Online Tutor, Practice Problems & Exam Prep

Standing waves result from the superposition of two waves traveling in opposite directions, characterized by a wave function of the form y(x,t)=Asin(kx)sin(ωt). The amplitude of the standing wave is double that of the individual waves. Key properties include the relationship between string length l and wavelength λ, given by l = n · λ/2, and the period of oscillation T = 2π/ω.

Properties of Standing Waves from Wave Functions

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.

Example 1

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.

Here’s what students ask on this topic:

What is the wave function for a standing wave?

The wave function for a standing wave is given by:

$y(x,t)=Asin\left(\mathrm{kx}\right)sin\left(\mathrm{\omega t}\right)$

Here, $A$ is the amplitude of the standing wave, $k$ is the wave number, and $\omega$ is the angular frequency. This function represents the superposition of two waves traveling in opposite directions, resulting in a pattern of nodes and antinodes.

How do you calculate the length of a string with a standing wave?

The length of a string with a standing wave can be calculated using the relationship between the string length $l$ and the wavelength $\lambda$:

$l=n·\frac{\mathrm{\lambda }}{2}$

Here, $n$ is the harmonic number. For example, if the string is vibrating in its 3rd harmonic (n=3), the length of the string is:

$l=3·\frac{\mathrm{\lambda }}{2}$

What is the relationship between the amplitude of a standing wave and the original waves?

The amplitude of a standing wave, denoted as $A$, is twice the amplitude of the individual waves that form it. If the amplitude of the original and reflected waves is $a$, then:

$A=2a$

This doubling occurs because the two waves interfere constructively at certain points, leading to an increase in amplitude.

How do you calculate the period of oscillation for a standing wave?

The period of oscillation $T$ for a standing wave can be calculated using the angular frequency $\omega$:

$\omega =\frac{2}{\pi }/T$

Rearranging this equation to solve for $T$ gives:

$T=\frac{2}{\pi }/\omega$

By substituting the value of $\omega$ from the wave function, you can determine the period of oscillation.

What are nodes and antinodes in a standing wave?

Nodes and antinodes are key features of standing waves. Nodes are points where the displacement of the wave is always zero, meaning there is no movement at these points. Antinodes are points where the displacement reaches its maximum value, meaning these points experience the greatest movement. In a standing wave, nodes and antinodes are evenly spaced along the medium. The number of nodes and antinodes depends on the harmonic number $n$. For example, in the 3rd harmonic, there are 4 nodes and 3 antinodes.