Standing Waves - Video Tutorials & Practice Problems
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1
concept
Intro to Transverse Standing Waves
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Hey guys. So in earlier videos, we saw the waves can interfere with each other in this video, I'm going to show you that in some cases, waves can interfere in a special way to produce a special type of wave called the standing wave. So I'm gonna introduce you to transverse standing waves on the strings. Now, there's a lot of conceptual information to understand here. But the hardest thing about standing waves is understanding what exactly is going on. So I'm going to break it down for you guys and show you exactly what a standing wave is. And to do that, I'm basically going to show you three different scenarios to kind of understand what a standing wave is all about. Let's go ahead and take a look here. So what I want you to do is imagine that you have a string in your hand that whipping up and down and that the string is tied to a fixed end like a doorknob. So your oscillator, the thing that's moving up and down is your hand and it's tied to the doorknob. So let's take a look at the first scenario here. You're going to take the string and you're gonna whip it up just once you're going to create a single pulse that moves to the right. So what happens when that pulse actually reaches the doorknob when you have this pulse that reaches the doorknob, this and that disturbance has to go somewhere. Basically, what happens is that the doorknob is actually going to flip this pulse upside down and now this pulse is going to be going backwards. So, in general, what happens here is that when a traveling wave reaches a fixed end point, like a doorknob, it's going to create a reflected wave or pulse that travels backwards, but it's going to be upside down, it's going to be inverted. All right, that's the first scenario. Let's take a look at the second one and the second one here, now you have the string and instead of whipping it up twice or once, you're actually just gonna whip it up and down once to create a single wave, you're gonna wait a few seconds and you're gonna create another up and down to create these two waves here. So you have two waves that are moving to the right. So what happens later on? Well, basically what happens is is that the right wave is gonna hit the door handle first and it's going to invert and now it's going to travel to the left. So basically, it's going to invert upside down. And now this wave is going to be traveling to the left. So that's what this wave becomes. However, the left wave is still moving to the right. So the left wave is going to be over here and basically, it's going to have travel to the right like this and you're going to end up with a wave pattern that looks like this. So basically, what happens is that you have these two waves now that are sort of moving towards each other and they're going to interfere. So what ends up happening is you have to end up with a superposition of these waves and it's gonna look something kind of like this, it's gonna look like sort of this weird pattern. So this is going to be the next wave if you will. Now in general, what happens is you can change the width of the wave by changing your frequency. You can basically wiggle the string up and down faster or slower. If you uh wiggle it super fast, you're gonna create these, you know, really, really sort of compact wave pulses like this. And if you create a sort of, if you wiggle it super slow, you're going to create these sort of larger wave pulses. So what happens here in the second scenario is that for most frequencies that you whip the string up and down the waves are going to interfere somewhat randomly, you're going to create these sort of random interference patterns which are going to have a jumbled up wave that looks like this, it's not going to create any sort of like pattern. Now, let's take a look. Now, at the third scenario, now let's change your frequency. Let's go a little bit slower or faster. And what happens is that for very, very special frequencies, you're gonna create some interesting patterns. If you slow down your wiggle and you basically create a single arc that goes from left to right from your hand all the way to the door handle. When it reaches the door handle, it's gonna flip upside down, it's gonna go backwards like this. And if you keep this frequency, you're basically going to create a wave that looks as if it's just kind of just going up and down, but it's not actually moving left to right. It kind of looks like a jump rope. If you were to look at it from the side, it kind of just looks like this here now. So for very special frequencies, you're gonna create an interference pattern that's very special. And you're going to create a wave that looks as if it's stationary, it looks as if it's not moving. So because this wave looks as if it's standing still, we call it a standing wave. So that's what a standing wave is. It's just a special wave pattern that you get for a very special frequency. This doesn't happen for all frequencies as we saw on the second example here, for any sort of general frequency, you're gonna create this sort of jumbled massive waves that interfere randomly. All right. So let's talk about these frequencies. Now, we said that this is a very special frequency, but there's actually an infinite number of these special frequencies which create standing waves if I start off from this pattern here and I take the strain, I whip it up and down even faster. Eventually I'm gonna create another pattern in which I have a single wavelength along the string. When it reaches the door handle, it's going to invert and it's going to travel backwards and you're going to create this other standing wave pattern. That's kind of going to look like this, right? It's going to have this weird sort of loopy pattern like that. And if you vibrate this thing even faster, you're going to create another standing wave pattern and where you have three loops and then four loops and then so on and so forth here. All right. So basically here we have the, the letter N is the number of loops in our standing wave pattern. So for the most basic one that we created for this one here, we have one loop. And for this one, we have two pairs of loops and this is N equals two and then you again have N equals three and so on and so forth. All right. So the most important frequency that you need to know is called the fundamental frequency which is given by the letter F one. It's basically the, um it's basically just the frequency uh where N equals one. So anytime you see the fundamental frequency, it just means that N equals one, it's the frequency that you need to whip the string up and down so that you can create this pattern right here. The most basic type of standing wave. Now, the reason this fundamental frequency is important is because once you figure that out, you can actually figure out all of the other sort of general what are called harmonic frequencies, harmonic frequencies are basically just all the other frequencies for all the other standing wave patterns that you can possibly get where you have more than one loop. And they're basically just multiples of this frequency, this fundamental one. So the general equation you're going to see is that FN right, for any number of loops is just going to be N times one. That's just a basic introduction to standing waves. Let's go ahead and take a look at an example here. So we have a string between two supports and it vibrates in a standing wave pattern. So we have three loops right here. In the first part, we want to draw a sketch of the wave. So for this first part here, we have N equals three. So what does that tell us? Well, remember N is just the number of loops. So we have three loops where we have three if we have N equals three, we're going to have three loops. So a sketch, this wave would look like this, we're going to have one loop. This is going to be the second loop and this is going to be the third one. So what this wave pattern would look like is, is it would kind of look something like this here. So you have these three pairs of loops and that's N equals three. So now the second part here, uh sorry, actually, in the first part where they're telling us here is that the frequency of the standing wave pattern is 15 Hertz because N equals three. What they're saying is that F three is equal to 15 Hertz like this. So let's take a look at the second part. Now, the part we asked us to find out what the fundamental frequency is. Now, remember anytime you see the word fundamental, what that means is that N equals one. So if you have N equals one, what they're asking this is what is F one right here? And to figure this out, we're just going to use this equation right here. What we know is that F three is equal to 15 Hertz. So the relationship here is that F three is just three times F one and that's going to be um and we can actually solve this F one right here. So F one is just going to be this 15 divided by three and that's going to be five hertz. All right. So, the idea here is that, to create a standing wave pattern with three loops, you'd have to vibrate the string up and down at 15 Hertz. But to create a standing wave pattern with just one loop, you'd have to slow down that frequency and you'd only have to vibrate, they get five hertz to create that more sort of simple wave pattern. All right. So now let's take a look at the last part here. What's the frequency for a standing wave with five loops? So they're actually saying that here, N equals five. So if N equals five here, what is F five? We just, again, you're gonna use this equation right here. N times F one. So the idea here is that you would have five times F one. So this would just be five times the fundamental frequency of five hertz. And we just figured out over here and you're going to get 25 Hertz. So the idea here is that every time you add five Hertz, you vibrate this thing five Hertz faster, you're gonna end up with one more loop in your standing wave pattern. All right. So that's it. For this one guys. Let me know if you have any questions.
2
Problem
Problem
By whipping a string up and down, you determine the fundamental frequency to be 4 Hz. If you attached the string to a motorized oscillator and increased the frequency to 28 Hz, how many loops would this standing wave have?
A
14
B
4
C
28
D
7
3
Problem
Problem
One of the harmonic frequencies for a particular string under tension is 325 Hz. The next higher harmonic frequency is 390 Hz. What harmonic frequency is next higher after the harmonic frequency 195 Hz?
A
260 Hz
B
130 Hz
C
196 Hz
D
4 Hz
4
concept
Equations for Transverse Standing Waves
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6m
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Video transcript
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.
5
Problem
Problem
The figure below shows a standing wave on a 2.0-m-long string that has been fixed at both ends and tightened until the wave speed is 40 m/s. What is the frequency of this wave?
A
20 Hz
B
160 Hz
C
40 Hz
D
10 Hz
6
example
Example 1
Video duration:
5m
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Video transcript
What's up everyone. So let's see if we could work this example problem out together. We're gonna put together a lot of information that we've seen from waves and standing waves in this problem. Let's break it down. So we have a string that's fixed at both ends. It's 8 m long and has a length or sorry, a mass of 0.0 or sorry, 0.2 kg. So in other words, I'm told that the length is eight and M is 0.2. But I'm also is that this is a string and it's subjected to a tension of 100 newtons. Now, we've seen tension before in these types of problems with waves and strings and that's the variable ft. So in other words, the force of tension on the string is 100 newtons. So the first thing we want to figure out in part A is the speed of waves on the string. That's just the wave speed. So let's put these things together and see if we can look at part A over here. Now, remember we have a couple of equations for velocity but strings particularly we had one equation which is the square root of tension over mu. That's what we're going to take a look at here because it applies to strings only in this case and transverse waves. So let's take a look at this V equation. Um Notice how also we don't know anything about the standing waves. So we can't use fundamental frequencies or any of these other harmonic frequencies, anything like that. So we're gonna have to use this equation here. So this equation says that V is equal to the square root of the tension divided by mu. Now remember the mu is the linear mass density. And just so for your recollection that the linear mass density is really just the mass divided by the length. That's what that MU value becomes. All right. So if I just go ahead and plug all this stuff into my calculator, this is really just going to be 100 divided by the mass which is 0.2 divided by the length, which is if you plug all that stuff into your calculator, what you should get is a velocity of 63.2 m per second. All right. So that's the first part. So figuring out just the speed of waves that uh sort of behave as they move along on this, on the string here. All right, let's move on. Now to the second part. The second part asks us to find what is the longest possible wavelength for a standing wave So what does that mean? What's the longest possible wavelength? Well, let's take a look at it and actually just draw out what our standing waves will look like for the first few loops, right? Remember that the first loop, the N equals one is actually basically like one, it's only just one bump up like this, right? Whereas the N equals two, that's where you actually have two loops. And remember that's gonna look like a standing wave like like this, it's gonna look something like this and then it's gonna basically just fold back over itself. So you're gonna have two loops like this. The N equals three is gonna look like three loops. So it's gonna look a little bit sort of messier. I know it's gonna be a little bit messy to see here, but it's basically just gonna be like you have three loops, 12 and then three. All right. So this is basically gonna look something like this. All right now. So that's N equals one equals two and equals three. So this N equals one, N equals two and then N equals three over here. All right. So which one is the longest possible wavelength? Well, let's take a look at what happens, right? As the ends get bigger and bigger, right? You have more and more loops, which means that the wavelengths actually get smaller. Notice how it happens here is that the blue wave, the wavelength would just be one would just be one up and down like this, whereas for the green wave, a wavelength would be something like that, so that wavelength would be shorter. So the longest possible wavelength actually corresponds to, let's see, it would be the lowest possible NN equals one. So if you think about what happens here, right, this is actually only one half of the wavelength. So this is N equals one. And really, it's because what happens is that the other wavelength, the other part of the wavelength is actually kind of sort of imaginary, but it kind of continue on continues on down here. And remember that the wavelength is defined as basically just one complete cycle, you have to go up and down and then back up again. So this actually be the complete wavelength for N equals one. So really what happens here is that to calculate your wavelength for one, you're actually not going to use an equation, you won't use the equation. You'll actually just see that it actually has to be double the length of the string that's actually gonna be the longest possible wavelength that you can have. So that lambda one is gonna be equal to 16 m. So this in fact will always be true. The longest possible wavelength of a standing wave will always correspond to N equals one, always, so always N equals one. In fact, I actually want to write to that um so N equals one, always All right. So that means that lambda one is equal to 16 m. All right. So now what we're gonna do is we're gonna calculate the frequency of that wave. Now, there's actually a couple of different ways to do this. Um You basically just could use um one of the easiest ways to do this is actually just to kind of use um this equation over here. This just could be your uh frequency equation. Um So let's go ahead and do that, right? So you could just use um f of one equals um This is just gonna be V over two L and we can just use our wave speed, which we've already seen before, right? We actually know what that wave speed is. This is just 63.2 we calculated earlier in the problem divided by two times the length of the string, which is gonna be eight. So in other words, the fundamental frequency here that you'll calculate should be 3.95 Hertz. All right. So that will be the fundamental frequency of that wave. All right. So those are the three parts of this problem, figuring out the wave speed, then figuring out the longest possible wavelength and then what the corresponding frequency will be. And that actually happens to be the fundamental frequency. So those are all three parts of the problem. Let me know if that made sense. Thanks for watching and I'll see you in the next video.
7
Problem
Problem
A 3m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 60 m/s. What are the wavelength and frequency of the second overtone?