Hi, I'm Mr. Norris, and this lesson is all about how we can measure waves on a string.

This is part of the measuring waves topic.

This lesson is an example of how in physics, sometimes at just the right conditions, something really like interesting and unexpected can happen.

So let's have a look.

The outcome of this lesson is that by the end of this lesson, you should be able to find the speeds of waves on a string by measuring things called standing wave patterns.

Here are keywords that we're gonna be focusing on in this lesson.

Standing wave, tension, inverse, proportion, and control variable.

Now, each of these terms can be explained when they come up in the lesson, but on the next slide, there's gonna be an example sentence of each word being used.

Once I've gone through each sentence, you might want to pause the video just to reread and make sure you've kind of taken in these keywords so you're as prepared as possible for the lesson.

So a standing wave is a stable pattern of oscillation that does not appear to travel across a wave medium.

Tension is the name of the force applied to a stretched string or rope.

So without any tension, a string or rope would be slack.

Division is the mathematical inverse, opposite, of multiplication.

So the inverse of times two is divided by two, and divided by two can also be written as times 1/2 or times one over two.

So inverse kind of means opposite in maths, mathematical relationships.

A proportion is an exact mathematical comparison, such as double, 1/2, triple, 1/3, quadruple, 1/4.

They're all proportions.

And finally, control variables are quantities that must be kept the same during experiments so they don't affect results.

You might want to pause the video now just to review those keywords.

So we'll move on when you're ready.

This lesson is divided into three sections.

In the first section, we'll look at what these standing waves are and how they can be created on a string.

The second section, we'll look at how we can collect some data on standing waves and how we can calculate the speed of the waves on the string.

And then finally the third section, we'll look at what conclusions we can draw from our data and how the experiment could be extended.

Let's get going with the first section on how standing waves are created.

So every wave you'll have studied so far in this topic I think will have been a progressive wave, and that's a wave that travels, a wave that travels through or along a medium.

So here's an example of a progressive wave travelling along a string.

Here it comes and it's travelling from left to right and I've paused the animation there 'cause something would happen when the wave hits the clamped fixed point and it's just playing again now.

But you can clearly see that wave is travelling along the string.

However, it is possible to create waves that appear not to travel, and that's what a standing wave is.

So here's some examples of a standing wave oscillation.

So this is a stable pattern of oscillation that does not appear to travel.

So it still kind of looks like a wave.

It's a oscillation, but it doesn't appear to travel.

Here's the first one.

So look at that pattern of oscillation up and down.

That wave is not appearing to travel like the progressive wave did on the last slide.

Here's another standing wave.

So look, it's oscillating, but no part of that wave actually appears to travel down the string.

And here's a third one.

This is also a standing wave pattern because no part of that wave appears to travel across that wave medium across that string.

So here's how standing waves are created, how they occur.

I'm gonna go through the explanation first and then show the animation so you know what to look for before it appears.

So the first step is that a wave has to travel down the string.

Then the wave reflects from the fixed point at the end of the string.

So the wave now travels back up the string, but then that reflected wave is gonna meet and combine with the next wave travelling down the string.

So I'm gonna show you an animation of that now now you know what you are looking for.

So here's the wave being created, travelling down the string, it reflects back, but then it's meeting with the next wave being created.

And as a result, the overall pattern is this standing wave pattern.

Here it is again, wave travels down.

As it reflects back from that fixed end, it's meeting with the next wave, creating that standing wave pattern.

And here's a really important point.

You only get a standing wave if the waves have exactly the right frequency for that string.

So there are specific frequencies that produce standing waves on a string where the wavelength matches exact multiples and fractions of the string length.

So not every frequency can produce a standing wave, only certain frequencies where the wavelength matches say double the string length or half the string length, for example.

If the waves have any other frequency, you won't get a standing wave.

So here's an animation where we're not getting a standing wave because the frequency doesn't match an exact multiple or fraction of the string length.

We're still gonna see wave pulses travelling.

So it's not a standing wave 'cause we can see wave pulses travelling left or right down the string.

So this is the wrong frequency, it's not producing a stable pattern of oscillation.

We can still see hints of waves travelling left and right up and down the string.

So it's not a standing wave.

So let's do a check of what you've understood so far about what a standing wave is.

Which one of these does show a standing wave? Is it A, B, or C? Which one? Decide now.

Okay, you should have made a decision.

Well done if you said that answer B shows a standing wave.

Answer A clearly shows a progressive wave 'cause the arrow indicates that that wave is travelling down the string and also, the string continues out that window.

Answer C can't show a standing wave because it's showing kind of two different wave amplitudes, which might be travelling into each other.

So it's not a stable pattern of oscillation like a standing wave has to be.

Next question.

Which of these does not show a standing wave on a string? Is it A, B, C, or D? So three of them are standing waves and one of them is not.

Which one is not a standing wave? Make a decision now.

Well done if you said an answer of D.

D is not a standing wave.

All of the others are standing waves.

In all of the others, you could just see each part of the string would just oscillate vertically with no apparent motion up or down the string.

Sono apparent motion left or right of any wave pulses.

Whereas in D, those wave pulses look like they might travel left and right, and there's different amplitudes as well.

So D is not a standing wave.

Final quick check for this one.

You only get a standing wave if you get the what of the waves exactly right for your system? What word fills in that blank? Have a think now.

Right, make sure you've decided an answer for what word goes in that blank.

And well done if the answer you came up with is frequency.

You only get a standing wave if you get the frequency of the waves exactly right for your setup or your system or your exact string that you've set up.

Okay, so here's the equipment that you could use to create and observe standing waves on a string in the lab.

You've got a string.

That's where the standing waves are going to appear.

This is an oscillator which oscillates up and down and it's set oscillating by a signal generator and you can set it oscillating at whatever frequency you choose on the signal generator.

So it controls the frequency of the oscillations, controls the frequency of the waves you're creating on the string.

And then they have these hanging masses which set the tension in the string.

That pulley is used just to allow the string to go off the end of the table and you can hang the masses on it to put tension in the string.

Okay, we need to say a quick word about how we represent standing waves.

So diagrams of standing waves actually show the vibrating string in two positions at once.

So for the first kind of standing wave, the string would be oscillating from that position to that position.

So oscillating up and then oscillating down.

But we can represent that standing wave by showing both those positions at once.

And if those oscillations are happening very quickly, we just see a blur between those two positions.

But that's how we can draw that standing wave.

What about the next kind of standing wave at the next exact frequency that standing wave is produced? That string would oscillate between that position and that position.

But again, we represent that standing wave by drawing the string in both positions at once like that.

So that represents that oscillation.

And for the next standing wave that happens at the next frequency that produces a standing wave, the string will be oscillating between that position and that position.

And we draw the string in both positions at once.

So we represent that standing wave with a diagram that looks like this.

And finally, a fourth standing wave at the next frequency that produces a standing wave would look like this, the string oscillating between that position and then that position.

And we draw both at once.

So that's how we represent that standing wave pattern.

We draw the string in two positions at once.

We also need to say how we can then work out the wavelength, which remember, wavelength is represented by the Greek letter lambda.

How can we work out the wavelength of a standing wave pattern? Well, what we do is we just measure the length of the string with no waves present and we could call that capital L, which is shown on the diagram.

Measure that length of the string.

And then the wavelength of the waves can be worked out from that measured string length.

So we're gonna look at the second standing wave pattern first 'cause that's kind of the simplest to see.

So let's look at the second column of the table.

How can we get wavelength from the string length for that second standing wave pattern in the second row of the table? Well, you might be able to see that on that second standing wave pattern, the length of the string, well, one wavelength fits on it exactly.

Look, there's one wavelength and it fits on the pattern exactly.

So the wavelength is just gonna be the length of the string.

What about the next standing wave pattern down? Well, there's one wavelength and that's obviously gonna be 2/3 of the length of the string.

So the wavelength is 2/3 of the length of the string.

What about the next standing wave pattern? That's one wavelength.

One peak, one trough.

So that is obviously half the length of the string that you've measured.

So that's how we can get the wavelength from the length of the string.

What about that first standing wave pattern? Well, we've only got one peak, but then not the trough on the string.

So the full wavelength would actually look like this.

That peak would have to be extended with a trough.

So the peak is half the wavelength, the trough is the other half.

So what we've got is that the wavelength must be double the length of the string 'cause that wavelength in purple is gonna be double the string length.

So that's how we get the wavelength of the waves for each standing wave pattern from one single measurement of the string length.

We don't need to make any other measurements, just measure the string length once and get the wavelength from that string length by calculation each time.

Right, you're ready to do a task on your own setup of equipment now.

So use a metre rule to measure the length of your string and then work out the wavelength that each standing wave pattern is gonna have for your string length.

Now, if you're watching this video, you might not have the equipment in front of you, so we're going to imagine an example length of 1.

2 metres and what you can do now is work out the wavelength, what the wavelength of each standing wave pattern would be if the string length is capital L is 1.

2 metres.

So you can use column two to work out column three.

Have a go at doing that now.

You'll need to pause the video and use your calculator.

Okay, let's see how you got on.

If you used the example data of a string length of 1.

2 metres, well, then the wavelength of the first standing wave pattern would be 2.

4 metres 'cause it's double.

The wavelength of the second standing wave pattern would be 1.

2 metres because it is the length of the string.

The third pattern would be 2/3 of the length of the string.

So 2/3 of 1.

2 is 0.

8.

And then half the length of the string is 0.

6 if you've got a 1.

2 metre length string.

So very well done if you've got all of those.

So that completes the section on how we can create and analyse standing waves on a string.

So now we're ready to actually collect data on waves on a string and calculate the speed of the waves that produce the standing wave pattern.

So we can recap.

The wave speed can be calculated using the wave equation.

Wave speed is the frequency of the wave times the wavelength.

V stands for wave speed, f for frequency, and the Greek letter lambda stands for wavelength.

And of course, the units for wave speed can depend on the units used for wavelength.

If wavelength is measured in metres, frequency and hertz that's gonna give speed in metres per second.

Whereas if the wavelength is measured in centimetres, then speed is gonna be in centimetres per second.

The standard unit would be metres for wavelength.

We normally want speed in metres per second.

So let's just do a practise calculation of that but relating to standing waves.

So have a look at the string on the left.

It has a length of 1.

2 metres, like our example data.

That standing wave pattern, let's say it occurred at a frequency of 14 hertz.

So what is the speed of the waves on that string that create that standing wave pattern? Well, how do you do this? Get all the data from the question.

There's the string length, there's the string frequency.

We're gonna need to use wave speed.

It's frequency times wavelength, but what is the wavelength of those waves? Well, that is one wavelength, it's just appeared in green.

One wavelength, in this case, is the length of the string.

So we can say that one wavelength actually is that 1.

2 metre string length.

So the wave speed is then frequency times wavelength.

The frequency is 14, the wavelength is 1.

20 metres, 1.

20 metres.

That gives a wave speed of 16.

8 metres per second.

Right, you're gonna do one now but for a different standing wave pattern.

So this string also has a length of 1.

2 metres, but this standing wave pattern occurred at 21 hertz.

So what is the speed of waves on this string? So I'd like you to have a go at doing that calculation now, but following through exactly the same thought process as we did before.

We're gonna need to use wave speed, it's frequency times wavelength, but you'll need to figure out what the wavelength of that standing wave pattern is first.

Off you go, pause the video.

Have a go now.

Right, let's see how you got on.

That is the length of the string, that's the frequency of the standing wave pattern occurred.

We're gonna need to use wave speed is frequency times wavelength, but what's the wavelength of these waves? Well, that's one wavelength.

That's 2/3 of the length of the string 'cause the string is split into three by the three pulses on the standing wave pattern.

So wavelength is 2/3 of the length of the string.

That's 0.

80 metres.

And then the wave speed is the frequency times the wavelength.

The frequency is 21, the wavelength we've just said is 0.

80.

Multiply that on your calculator, you get 16.

8 metres per second.

Hmm, interesting.

That's the same as the other standing wave pattern on the left-hand side of the screen.

So maybe have that in your heads going forward.

Right, so we're now ready to do the main part of this lesson, which is to actually collect some data from standing waves and find the frequencies at which the different standing wave patterns occur, so in real life.

Let's do the practical.

I want you to calculate the wave speed for each standing wave pattern.

So what is the speed of the waves that when they travel on the string and reflect on the other end create the standing wave pattern? Now, if you're watching this and you don't have the equipment at hand, then you can watch this example video of data being collected and fill in your results table based on the data being collected in this video.

<v Instructor>I'm now adjusting the frequency to try</v> to get that first standing wave.

It's quite fiddly (string whirring) and it's just starting there.

And if we pause the video, we can see that we've got a nice standing wave at a frequency of 10.

3 hertz.

And looking closer, you can see that we've got just half of one wavelength across that one metre 20 distance.

So half a wavelength is one metre 20 at a frequency of 10.

3 hertz.

(string whirring) Okay, so I'll adjust the frequency again and we'll see if you can get the next standing wave (string whirring) starting to form.

And there we go.

A nice standing wave at about 20.

7 hertz.

So again, I've freeze the video.

We can see now we've got one full wavelength over one metre 20 and its frequency is 20.

7 hertz.

(string whirring) Adjusting the frequency up again, we can go for the next standing wave.

This one should be three half wavelengths.

Just coming there (string whirring) and freezing the video again, we can see we've got three half wavelengths over one metre 20 at a frequency of 30.

9 hertz.

(string whirring) And finally, we just adjust it one more time.

This time we're aiming to get four half wavelengths or two waves.

(string whirring) Still a bit smaller this time, just a bit more fiddly.

And there we've got the standing wave.

Pause the video.

We can see we've got two full wavelengths just about at a frequency of 41.

2 hertz.

So we can now use those measurements to calculate the wave speed for each standing wave that we've seen.

<v ->Okay, well done for doing that task.

</v> Hopefully your data should look like this if you collected data from that sample video.

Okay, the first standing wave pattern occurred at 10.

3 hertz on the video.

And if you do then the frequency times the wavelength for that standing wave pattern, you get 24.

7 metres per second for the wave speed of the waves on the string.

Then for the second standing wave pattern, that occurred at a frequency of 20.

7 hertz in the video.

And if you do the frequency times the wavelength, again you get wave speed of well, 24.

8 but very close to the previous.

For the third standing wave pattern, that occurred at frequency of 30.

9 in the video.

And if you do the frequency times the wavelength, you get 24.

7 metres per second.

Very, very similar.

And then for the final standing wave pattern in the video, that occurred at 41.

2 hertz.

And if you do the frequency times the wavelength, that gives a wave speed of 24.

7 metres per second.

Again, so very, very similar.

So this takes us to the final section of the lesson where we can now draw some conclusions from that data and think about how the experiment could potentially be extended.

So here's the final three columns of our example data and here's a conclusion which you probably made as soon as the data was collected 'cause it turns out the speed of waves on a string was not affected by the wavelength or the frequency of the waves.

So the speed of waves on a string is not affected by properties of the waves.

That's because the speed of waves on a string is a constant for that string.

It's only set by the properties of the string itself, the properties of the wave medium.

So it's the properties of the medium that set the wave speed, not the properties of the waves, like frequency or wavelength.

Let's talk a bit now about proportions and then you'll see in a moment how this relates to our results.

A proportion is an exact mathematical comparison, like double, 1/2, or 50%, triple, 1/3, quadruple, 1/4, things like that, proportions.

And then the inverse proportion is the mathematically opposite proportion, which is always one over the original proportion.

This probably shows better what that means.

So if a proportion is double, then the inverse proportion is 1/2 because that's basically the opposite of double is 1/2.

If the proportion is triple, then the inverse proportion would be 1/3.

So instead of triple times three, then 1/3 is times one over three times 1/3.

Quadruple times four.

The inverse proportion is a quarter or times one over four.

So if the proportion is multiplied by n where n is any number, then the inverse proportion is times one over n 'cause that creates a fraction which is the inverse proportion of that original proportion.

So how does that relate to our experiment? Well, in our experiment, it turns out the frequency of waves on a string was inversely proportional to the wavelength.

So what that means is when the frequency changed by a certain proportion, the wavelength changed, but not by the same proportion, by the inverse proportion, by the opposite proportion.

So for example, when frequency doubled from about 10 to about 20, the wavelength halved, that's the inverse proportion.

Doubling the frequency caused the wavelength to halve.

And that works for this doubling of frequency as well.

When the frequency was about 20 and it doubled to about 40, the wavelength halved from 1.

2 to 0.

6 metres.

So doubling the frequency causes the wavelength to reduce by half, but it works for all proportions.

So for example, tripling the frequency, when the frequency went from a frequency of about 10 to a frequency of about 30, tripling the frequency meant the wavelength reduced to 1/3 of what it was.

So 2.

4, 0.

8 is about 1/3 of the 2.

4 when the wavelength changes that.

So the frequency of waves on a string is inversely proportional to the wavelength because when one changes by a certain proportion, the other changes but by the inverse proportion.

That's what inversely proportional means.

And that happens for these waves because for wavelength and frequency, when one quantity changes, the other changes by the inverse proportion because the wave speed stays the same because wavelength times frequency must give the same wave speed.

So if one has doubled, say wavelength doubles, then frequency must halve to give the same wave speed.

'cause wave speed is the same.

So that's the reason why wavelength and frequency are inversely proportional.

The wave speed is set independently of wavelength and frequency.

So if frequency doubles, then wavelength must halve to give the same wave speed or the wave speed must remain the same because we haven't changed any of the factors that actually affect wave speed, such as the tension in the string or the density of the string or something like that.

Here's some simplified data to illustrate this.

So imagine the wave speed was 24, exactly 24 metres per second exactly.

Well, how many ways can you multiply two numbers to make 24? You could do one times 24 if the wavelength is one and the frequency is 24, you could do two times 12, you could do three times eight, you could do four times six, and they would represent possible wavelengths and frequencies that could give a wave speed of 24.

So you can see if you double the wavelength from one to two, then the frequency must halve to give the same wave speed.

If you triple the wavelength from one metre to three metres, then the frequency must drop to 1/3, eight is 1/3 of 24, to give the same wave speed.

So that's what inversely proportional means relating to wavelength and frequency of waves where the wave speed is a constant set by the properties of the system.

So let's do a check that we've got our heads around this idea.

What is the inverse proportion to double? Well, this is a worked example, so I'll just talk through this one.

The inverse proportion to double is a half, it's the opposite proportion to double is 1/2.

If the wavelength of a wave on a string doubles, what happens to the wave speed? Well, we know that nothing happens to the wave speed because the wave speed does not depend on the properties of the waves, like wavelength and frequency.

So wave speed is unchanged if wavelength doubles, but if wavelength doubles, what happens to the wave frequency? Well, we said that wavelength and frequency are inversely proportional.

So if wavelength doubles, wave frequency halves.

So that really spells out what we've just gone through.

So I'm gonna put a very similar question up, but I want you to have a go at answering all the parts of this question.

Here it comes.

You should pause the video now to have a go at answering all of the parts of this question.

Right, let's see how you got on.

What's the inverse proportion to triple? Well, triple is times three.

So the inverse proportion is divide by three or if something drops to 1/3.

So dropping to 1/3.

1/3 is the inverse proportion to triple.

It's the opposite.

Then if the wavelength of a wave on a string triples, if wavelength triples, what happens to wave speed? Nothing, it's unchanged because wavelength doesn't affect wave speed.

Other factors linked to the properties of the system affect the wave speed.

But if the wavelength of a wave on a string triples, what happens to frequency? Frequency drops to 1/3 or reduces to 1/3 because wavelength and frequency are inversely proportional.

So however one changes, the other changes by the inverse proportions with wavelength multiplies by three, the wave frequency must drop to 1/3 to give the same wave speed.

Wavelength and frequency are inversely proportional for waves on a string.

So what factors could affect the speed of waves on a string? Well, I've mentioned a few of these already.

The thickness of the string, the material of the string, which really means the density of the string, the tension that you've applied to the string.

So how many masses have you put on the end of the string to make sure that string is under tension? So waves can travel along it, otherwise the string would be slack.

So in our experiment, when we looked at different frequencies and saw how the wavelength changed, all three of those were control variables.

So wave speed wasn't affected.

We got the same wave speed every time.

But you could do a different experiment.

You could extend our experiment to actually investigate how one of those three factors in the bullet points, how one of those could affect the speed of waves on a string.

And that would mean the other two factors would have to be control variables so they didn't affect the speed of the waves on the string 'cause you're only investigating how one of them affects the speed of waves on a string because, of course, control variables are the things you keep the same so they don't affect the result.

So let's do a check of that idea.

A student, Izzy, makes a hypothesis.

Increasing the tension in the string will increase the wave speed.

So Izzy doesn't know that for certain, it's a hypothesis.

It's a statement that she's then gonna go and test by doing an experiment.

So which of these would need to be Izzy's control variables, the things she keeps the same so they don't affect the result while she was investigating that question? There might be more than one.

So select from A, B and C which of those would need to be Izzy's control variables if she was investigating how tension or whether tension affects the wave speed.

What would have to be her control variables? What would be the things she needs to keep the same? Pause the video now to decide.

Okay, well done if you said material of string and thickness of string.

She'd have to keep them the same in every test 'cause the only thing she would want to change is the number of masses on the hanger to change the tension.

'cause she's investigating well, does tension increase the wave speed? So that's the only thing she should allow us to change.

And the other two things, A and B, material string thickness of string, she'd have to keep those the same.

Which of these would be Izzy's independent variable that she would change every time? That should be fairly obvious now.

Make a decision.

The thing that Izzy should change every time should be the number of masses on the hanger because that's how she would change the tension.

So that's the one thing she would change and she would keep everything else the same.

And then that would lead to valid results, results that would validly test her hypothesis.

Right, so a final task for this lesson then is I would like you to pull together everything we've learned in this lesson about the method for investigating standing waves on a string.

I would like you to write a brief method for how Izzy could test her hypothesis.

How could Izzy test whether changing the tension increases the wave speed or how could Izzy test how increasing the tension increases the wave speed? I want you to write a written method aiming for five or six short steps.

You could use bullet points or numbered points that just briefly outlines all the steps Izzy would need to do to do her experiment.

You don't need to describe the equipment or the setup.

So the diagram is given there for you.

Okay, so have a go at that now.

Okay, well done for your effort in writing a method for Izzy's experiment.

Here is an example method.

You can check that you have included as many of these points as possible.

Add anything you're missing you think to your answer to improve yours and check that your answer follows roughly this sequence as well so it's a well-sequenced steps in the method with things to do at the start first and the things to do at the end last.

So step one, Izzy should set up the equipment with let's say 100 grammes of mass on the hanger.

It doesn't matter if you said anything around that value.

It doesn't have to be exactly 100 grammes.

Step two, measure the length of string that will oscillate with the metre rule.

That's the length of string between the oscillator and the pulley.

Then what should Izzy do? This is what we did in our experiment as well.

We had to slowly increase the frequency of the oscillator until a standing wave is seen.

Step four, we then have to record that frequency from the signal generator and work out the wavelength of those waves from the length of the string and the standing wave pattern.

And we talked about how to do that earlier in the lesson.

Step five, you then calculate the wave speed for this tension using speed is frequency times wavelength.

But then here's the most important thing because Izzy is, of course, she wants to find out how does tension affect the wave speed? We found the wave speed for this first tension with 100 grammes of mass on the hanger, for example.

But what she needs to do now is repeat steps three to five, but use different amounts of mass on the hanger, say 200 grammes, 300 grammes, 400 grammes.

And that will vary the tension in the string.

The string, that's the string's material, the string's density, the string's thickness, it must be kept the same every time 'cause they're the control variables that she has to keep the same so they don't affect her result.

So that concludes this lesson on measuring waves on a string.

Very well done for completing this lesson.

Here is the summary.

To find the standing wave frequencies for waves on a string, you need to slowly increase the frequency of the oscillator until a standing wave pattern is seen.

Then you calculate the wave speed by doing frequency times wavelength.

And what we found out is the speed of waves on a string, it only depends on the properties of the wave medium, so the properties of the string, like tension or density or thickness.

And the speed of waves on a string does not depend on the properties of the waves, like frequency or wavelength because we've got the same speed every time.