High School Physics Waves, Cont.

Hello and welcome back to educator.com. Today we’re going to be continuing with more waves. Last time we talked about our introduction to waves, we got our feet under us.0000

Now we’re going to start talking about some more complex ideas.0007

So we’ve got the basics of waves under our belts and we’re ready for the tough stuff.0011

First thing we’re going to discuss is how waves interact with each other, called interference.0014

It’s an idea that’s going to have a lot of ramifications and it’s really important.0019

Then we’re going to also consider when a wave emitter is moving in relation to the receiver.0022

So when, if it were sound waves, when the listener and the speaker are moving at possible rates.0027

They might be moving away, one might be moving in one direction while the other one is following, all sorts of possibilities.0033

Alright, first we’re going to talk about how waves interact with each other.0039

When two waves are in the same medium at the same time they interact.0042

The result though is remarkably simple, it’s actually really easy.0046

Their individual effects simply add together. Visually we superpose, so if one wave is here and another wave is here, we… and our base line is here.0050

We’re going to add this little bit on top of this and the point there would be that.0059

We do that point wise for every little point and we can draw a new wave that gives us what the total is of the two waves put together.0064

We’ll have a lot more visual examples coming up soon.0072

Algebraically we just add our equations together, if we got one wave equation, y1, that describes the first wave by itself.0075

y2, that describes the second wave on its own, then we put the two together to get the result in wave of what happens in the medium when they’re both going at the same time.0082

We call this phenomenon interference because the two waves are interfering with each other.0092

If the waves interfere in such a way so as they actually create a larger result, then they’re working together it’s called constructive interference because they’re building something.0097

If the result is smaller, it’s called destructive interference because they’re actually lowering, they’re making things smaller, they’re destroying the result.0106

Constructive interference, increasing. Destructive interference, decreasing.0113

Furthermore, while the interfering waves do cause a different result in the medium, it’s important to point out that they have no effect on either’s movement.0118

The waves travel unimpeded as if the other one wasn’t there at all.0124

There is in real life, in real situations, there is some slight changes in the way they move but for the most part this is actually true in real life.0128

You know, just like in some cases we made some slight things where we didn’t account for everything that could have.0134

This is actually closer than neglecting air resistance, which we did a lot. It’s not quite perfect, but for our purposes it’s pretty darn realistic to actually say that the waves travel fully unimpeded by another waves presence.0140

Interference, the idea is much easier to understand with some visual examples so let’s consider two positive pulses.0154

We’ve got the red pulse and the blue pulse. As they come together, they’re going to make a third, a combined pulse that is going to be the purple pulse.0160

The purple pulse will be the resulted…what will happen in the medium. In real life, if we were seeing the two things come together, we wouldn’t see the red and the blue pulse anymore.0168

We would just see the purple pulse. Remember, we’re going to see the added together thing, we’re going to see the final result.0176

If we know what the two pieces are coming together we can know what the third is. But just seeing just the third, we don’t know if it’s a wave on its own or if it’s two waves put together until we study it more closely.0182

We’ll see these start to move together and we’ll be able to talk about it as it goes.0191

First one, we see…it’s a little rough as a sketch. We see the distance from here to here. Winds up getting added here to here.0196

We get that point and so on and so forth throughout the whole thing. Here to here, it’s added.0208

Here to here, and so on and that’s how we get this whole curve.0212

It’s just everything added pointed wise and we go through the whole thing and we can see the purple wave sort of just grows on top and once they separate it just winds up being each wave on its own.0217

As the thing moves, the purple wave. The purple wave that’s the result of the thing, what you’d see if you were actually looking at it is a combination of the two waves that’s making it up.0232

It rides on top of the other ones. Alright, that’s with two positive pulses.0240

What’s going to happen when we have it with the waves, not the pulses?0246

So if red and blue are perfectly opposite, they’re just out of phase with one another, they’re synced so they have opposite oscillations.0248

There out of phase by π. If we look at example 3 in the…actually I’m sorry, we’re going to have that, we’re going to have that in this one, example 3, we’ll talk about that.0257

Sorry, not example 3. 2, example 2. Anyway, with waves instead of pulses, if the red and the blue, if they have…0266

This is length A and this also length A and they have the same frequency and whatnot, then they’re going to cancel each other out perfectly.0275

This height here is canceled out by this height here, because remember it’s a negative height. This height, and really they’re not quite perfect, sorry my drawing is not exactly accurate.0281

But this height should be as tall as this height, this height as tall as this height, and so on and so forth.0292

When you add them always together, one’s positive and one’s negative, and they’re the same size. Boom, they cancel to zero.0296

If we had it with instead, them being synced together we’d actually wind up getting larger positives and larger negatives.0304

The purple would sync together with the two of them and we’d get an even bigger result, we’d get a larger wave that’s the combination of the twos waves putting it together.0312

So we’d get an even bigger thing in the end. Those are both in sync in one way or another.0322

Either they were out of phase, where they opposed each other in the first example.0330

They’re in phase, where they were constructively working. First one being destructive interference, the second one being constructive interference.0333

Now we’re looking in the third example that is not just one or the other. Here the two add together and we get massive constructive interference.0340

But here, we get destructive interference. So we show up, we drop to zero, so this thing actually gets shifted around in more ways than just having things…just think of it as growing taller or getting smaller.0350

It’s actually been to the side and it gets more complicated. The only way to figure out what this would be is to actually know what the blue one is, what the red one is and add their wave equations to figure out what our purple wave equation would have to be.0363

We can also figure it out point wise and look at things by knowing what this value is and knowing what this value is and then just adding those two values and wind up getting the third value for the purple one.0375

With all of these cases, the result has to do with the phase of the cycle. One wave is compared to the other.0386

In the first example, they were out of phase because they were destructively interfering. They were out of phase by a phase angle of π.0392

Because they are…we’ll get a little bit more of that now, so we’ll not worry about it right now.0399

The phase angle, there is some amount that they’re off. They’re out of phase by some amount, I think that’s pretty clear at this point.0404

When they’re in phase they’re always working constructively together. The second example where we saw it grow larger throughout.0411

In that case, they’re going to be in phase together because they’re always working together.0417

Then, there is also a sort of middle thing where they’re neither fully in phase nor out of phase and they’re only partially...they’re between the two.0422

So, you’ll get sometimes constructive interference and sometimes destructive interference.0429

So a wave, if you have two waves interacting with one another it’s not just going to be frequency and wave length and what time we’re looking and what location we’re looking at.0433

It’s also going to have to be…you could start one wave a little bit after you start another wave and even if they’re the exact same looking wave, they’d be shifted by a certain amount, right?0442

That’s where the idea of phase shifting comes in, phase angle.0452

So we need some algebraic ways to talk about this slight shifting of phase from one wave to another phase.0455

To do this, we introduce a new variable to our wave equation. Phase shift.0461

Once again, this guy right here, yet another new Greek letter, he’s pronounced either fee or fi, fee fi fo fum.0466

But seriously, fi or fi, kind of depends on who you are. I don’t think there is actually a standard pronunciation. Me, I’m probably a fy guy…no I’m a fi guy. I’m a fi guy.0471

If somebody tells you they think it should be pronounce another way, pronounce it that way in front of them but, you know, check a pronunciation dictionary if you really want to know.0485

Anyway, this guy right here is spelled like that. Spelled like that, so that right there, fy or fi.0492

Y…so our wave equation is a, amplitude, times sin of kx. K being what we talked about before, 2 π over λ minus ωt.0501

Where ω is once again where we talked about before 2 π over period plus the new guy fi, fi.0512

Plus the new guy. So the new guy winds up, winds up giving us the ability to shift, right?0519

So kx-ωt, these things are determined here, but by this, we could have the same thing. We could have this wave get described both in x and t.0526

If we wanted to we could shift it forward by some little amount here as opposed to having to say ‘Look at a different time’, it allows us to have a time factor for the whole thing.0537

That allows us to have one that’s the original equation without the fi and another one that’s a new equation with the fi. I guess I’m a fi guy, not a fy guy.0544

Whatever we put up, the beginning one would be without this, it would be zero. Fi would be zero for this one.0553

The second one would have some fi and would could calculate what that is.0560

Now there is no given equation to determine what the value of fi is going to have to be.0564

We have to figure it out by solving for it from known data points.0567

You’re going have to compare the values you know and figure out what it’s got to be from what you know.0570

One equation has to be and what the other equation, what something’s it has to involve is and then solve for.0575

Well we know the resultant wave comes out to be this at x=1, t=2 and it comes out to be this at x=1, t=3 and from that we’re going to be able figure out.0583

It takes some effort, but by basically doing a lot of math and solving for it, you’ll be able to figure it out eventually.0593

Alright, on to standing waves. When a wave hits a rigid surface, it reflects and is inverted.0600

This means if we have a continuously generated wave being bounced off a surface, the incident, original and reflected wave will interfere with each other.0606

You shoot a wave at a hard surface and it comes in and it bounces off and it gets flipped…sorry, it’s not going to just be a direct flip of the other one.0614

It’s going to be a flip like this…and that will require…if that was up, I think this one would actually wind up being up but then down, so it’d be out of phase.0627

Its…sorry, little bit hard to be sure because it has to do with the fact that it’s being reflected. So it’s going backwards but it’s also being inverted at every moment.0636

You have to know what going in at each time is being spat out going the other direction.0645

This idea means that we got in one medium, if we’re bouncing off a hard surface, off a rigid surface that means that we’ve got the first wave that’s going in.0653

But now we’ve also got the reflected wave coming out, so if we got two waves they’re suddenly going to start interacting with each other.0662

Its certain wave lengths, a standing wave is set up. A standing wave is a special phenomenon where certain points on the wave, called nodes, remain stationary.0668

We’re going to explain more why it’s called a standing wave and what that means as we go on.0677

This is due to the interaction between the incident and the reflected waves. The original and the new wave going in and out of phase.0680

Let’s look at a visual example. This one, we’re once again using the purple-pink to refer to the color that comes out when they’re added together.0688

Then red and blue are the two original waves. So if we have two waves going forward…sorry. We have the incident wave, the red wave, and it’s moving to the right like this.0697

And it’s hitting the wall and being bounced off the wall and it’s being reflected like this. The red one slides forward, but then if we slide if forward a bit we’ll wind up getting a point where the reflected wave is going wind up being reflected in the other direction.0705

It’s going to be reflected and it’s going to be the opposite. So the top one, we’ve got a larger wave but in the bottom one the reflection is now going the other way because how they come in and come out, you’re going to wind up canceling at a certain point.0723

When you cancel at that certain point, the resultant wave, since they’re now perfectly out of phase, you’re going to cancel the whole thing and be a flat line.0738

When you’re going like this and you’re in phase, you’re going to make a larger one. So it’s going to be a larger wave than the original but then it’s going to flatten out and it’s going to be a larger wave and flatten out, then a larger wave and flatten out.0746

The points that stay the same, like that one right there, are called nodes. There is a node here and also on this graph, there would be a node here and a node here.0759

So this node here and also one here and one here because they’re staying at zero the whole time they way we’re working this.0769

Might be having the case that we’re having the wave look differently so that wouldn’t be a node, but if we had another surface here then we would have nodes there.0775

Alright, so this produces a wave that stands still even when it’s moving. What does it mean to stand still?0783

Well it means that the nodes stay fixed as the rest of it moves up and down. For this example we’ve got time going from red to green here.0788

The curve starts off red but then it begins to turn to orange then drops to yellow and turns to light green and then a more vibrant green. Just like this.0798

One possibly easy way to see it would be if you were to imagine a jump rope. If you were looking at a jump rope straight on.0811

There’s a person here and there’s a person here holding the two ends of the jump rope, right?0819

The jump rope goes like that but it’s going to also as you look at it, it’s going to wind up going through a number of different phases if they’re spinning it around.0825

As it spins around, it’s going to go up and down, and up and down. That’s a standing wave, with a node here and a node here. There’s no node in the middle for that because it stays the same.0832

So there’s no nodes in the middle because it moves around.0843

In this case though, the standing wave that we’ve got right here, we’ve also got a node in the middle.0848

So there’s a node in the middle here and a node here and a node here.0853

If we got this, which is how we’ve designed it. So we’ve got…a wave could have the front start like this and then go like that in a different version.0856

The way that the wave behaves from where its originating has to do with how it’s created, which is why I’m sort of hand waving so much here.0871

In our case, if we had it between two fixed hard objects then pluck the string, both the end points where it’s held down would be held in place.0880

If on the other hand, if we had a tube that had a hard end on one end, but then we were blowing into the other end to create sound, like say on the flute.0887

The waves would be moving up and down and it wouldn’t have a node at that end, we’d actually have an anti-node because it’s able to do its full size.0895

Standing waves is a whole thing that can be explored much more in depth than we’re going to.0902

We’re going to at least get our feet wet and get an idea of what’s going on here.0907

A node is a stationary point of the wave, so these three points on this, there’d be three nodes on this one, here they’d be two.0911

Anti-nodes, the exact opposite to a node, it’s the point of largest possible amplitude.0917

In this case, where both the end points are fixed, the number of nodes will be one more than the number of anti-nodes.0923

There’s an anti-node here, anti-node here on this line.0928

Those lines would be anti-nodes.0934

Standing waves can’t just occur at every wave length.0938

To determine what wave lengths work to create a standing wave, let’s look a little more closely at how our standing waves are being built.0941

The distance between the starting and the reflecting point is l.0947

If we’ve got two walls set up with an l distance between the two walls, we might have a standing wave that looks like this.0951

We’ve got our node in the middle, a node here, and a node here.0956

In this case we managed to repeat one whole wave, right?0959

One whole oscillation, so we’ve got λ as the distance. So in this case, l is equal to precisely λ.0963

We could also have standing waves like this, where it goes from here, so a node here and a node here.0968

Well it doesn’t manage to make an entire wave, it manages to make only half the wave.0975

This one here would be λ over 2. This one manages to make four nodes, so it manages to do one whole wave length here, 1 λ, but then here it manages only to do half a λ.0980

So it’s λ over 2. The way it’s going to work out, is that we’re going to have to have λ over 2 be the portions that it comes in, otherwise it won’t be able to fit.0996

Its not going to be able to have a fixed node at the end. It’s going to have to have a fixed node at the beginning and at the end.1005

We’re only going to be able to have a whole number of nodes in between, which means it’s going to have to come in λ over 2 chunks, right?1011

The wave length is going to determine whether or not it’s going to fit in as a standing wave.1019

For this one we’d have the length of the distance is λ over 2 or the length is equal to 3 λ over 2.1025

For the same length, we known seen at least three different standing wave patterns.1031

So if the standing wave has its end point fixed as nodes, we can only have a whole number of half wave lengths between the end points.1035

Some whole number, n times half wave lengths, λ over 2 is equal to the distance between them, l.1042

Equivalently, we want to know just the wave length from this, which is probably what we want to know.1049

The wave length, λ equals 2 times the length, over n. It can be any n from one on up to any arbitrary n you choose.1053

One marching on forever to infinity. Whatever n you plug in there, it will work because that’s just going to mean how many times you have to go up and down between the walls. Right?1062

You could have it that you go up and down, not up and down, actually yeah, that’s the number of up and downs you have.1072

I was going to say that the number of nodes is one more than the n we’re choosing, but the number of anti-nodes, in this case, there are other forms of standing waves like the flute example.1081

But for the string that’s held down on two ends and then plucked in the middle, the waves that get set up in it when it pulls off a standing wave are going to have to come from that length and we are going to have anti-nodes that are equal to whatever n we choose here.1090

It’s not like all wave lengths necessarily work at the same time. It will change depending on it, but this is the things that could work with it.1106

This is what standing waves would be allowed with in that, because it will be bouncing off of both walls simultaneously right?1113

One bounces this way and then sets up with another standing wave but then it bounces off the other wall.1119

We’ve got them all stacked on top of one another and the only way that they’re not going to wind up canceling each other out is if they’re in this precise thing that we can have that were it goes in phase and out of phase in such a way so as to sync up properly and cause it to just keep flipping between the two extremes.1125

Back to this precisely, we know that the length has to be equal to 2l/n for any n.1142

That’s just going to determine what number of anti-nodes, what number of nodes, how many bumps and valleys there are for our wave.1149

Furthermore, since the speed of waves in a medium is almost always going to be fixed, a given wave length implies a certain frequency.1159

If we know what the wave length is and we know what the medium speed is, then we know that we’ve got some frequency that’s going to be setting up that standing wave.1167

Any frequency that causes a standing wave is called a resonant frequency because it cause the medium to resonate.1175

Resonate, it picks up the motion and it amplifies it itself, so resonance is a really important property.1181

Huge amounts of study, if you’re heard about catastrophic failure of bridges in some cases, that’s possibly because of resonance.1188

In many cases it’s because some motion gets picked up and amplified and amplified and amplified till eventually it breaks.1196

If you’ve ever seen or heard of an opera singer being able to sing at a high enough pitch to break a glass.1201

That’s because the opera singer is able to hit the resonance frequency in the glass, so it’s able to set up vibrations in the glass, able to put energy into it and so the glass ultimately breaks because glass doesn’t like going through a lot of motion.1207

Its very brittle. So frequency that causes a standing wave is called a resonance frequency, this is a huge field of study but we’ve at least dipped our toes in it and got some idea of what it means.1220

It means something that’s able to set up a standing wave, that’s able to have something that just lives inside of it bouncing around.1230

Onto the Doppler Effect. So we finished with our idea of interference and we’re ready to talk about something new.1238

Interference, adding them together, and standing waves exist because the way interference allows things to interact.1244

What happens in the Doppler Effect, what happens when the emitter or the receiver is moving?1250

To figure that out, let’s first consider the much more simpler case, when they’re both still, right?1256

For this picture, we’re going to have the lines means the tops of waves, wave peaks.1259

The number of wave peaks you receive is the frequency right?1268

If you receive three wave peaks every second then that means you’ve got a frequency of 3 hertz right?1272

Because taking in five wave peaks every second is the same thing as taking in five whole waves, right?1278

There’s only one wave peak per wave. So, I mean, there’s of course one wave valley, a peak in the negative where we consider the peaking being just the top.1287

The number of wave peaks is the counts and that’s why it much more easy to visually represent it.1293

We can have a thing emitting the whole way and each one of those lines mean a wave peak coming out.1297

If you’re standing over here and its emitting these waves at regular intervals. These are the same distance for each one.1302

It’s putting out some frequency and its spitting out it that it has some regular frequency, so regular intervals one the waves because we have the mediums…the mediums speed is going to be constant.1312

Regular frequency means regular distance between the wave peaks.1327

The receiver here, is going to wind up hearing however many wave peaks hit him.1332

In this case, we’ve got these many coming towards, we don’t’ know how many are going to hit him in one second because we don’t know what the speed of this medium is.1338

We’re able to at least get a visual representation as it goes off in both sides.1345

What happens when its starts to move?1347

If the emitter is still producing the same frequency but now it’s moving towards you.1352

We’ve got those same distances, same circles but now as it moves along, it emits it, but then it moves inside of the circle.1356

It pops out the circle that’s growing around it but then before it makes it, before it would have been pop out a circle, stay in the same place, right?1365

But now we’ve switched instead to pop out a circle and before the next time you’re about to pop out a circle. Instead of staying in the same place in the middle, you’re now over on the right side, right?1372

You pop out a new circle and now you’re going to wind up having two circles around you.1382

You’re going to be building them out closer to one side right? Because as you move, you’re putting out these pulses out of you.1389

Your frequency, we can consider the wave peaks as just being pulses, right, it makes it a little bit easier to think about.1397

You put out these pulses around you but you’re moving, so while they’re spreading out evenly, you’re getting ahead of the point that they’d have to be to all be equal distance from that same point.1402

You’re moving forward, they’re wind up bunching up in front of you. If it’s moving you, I switched to being the emitter.1414

If the receiver is standing still and the emitter is moving towards the receiver, it’s going to bunch up those peaks in front of it, it’s going to make basically this wall of peaks in front of it that’s going to hit the receiver with really quickly.1420

On this side, the receiver is going suddenly receive a lot more peaks coming in at once because the emitter’s motion has effetely caused those peaks to get bunched up.1435

When they come towards the receiver, it’s going to get a whole bunch more of them at once because they’ve been bunched up by the emitter’s motion.1446

Similarly, if the emitter’s moving away from you, it’s going to look like this. You experience a lower frequency because there extra space between the wave peaks.1454

It’s bunched up one side, but in exchange it’s caused more space, it’s caused a widening of gaps between the wave peaks.1461

Its going to sound like the frequency has gone down, sound…I haven’t necessarily made this just about sound, it’s about any wave in fact.1469

You’re going to wind up experiencing fewer of them getting to you. You’re going to receive fewer wave peaks per second because it’s moving away from you.1477

We can also make these same ideas for the person if the emitter was still and the person was moving.1486

It’s going to wind up being the case that this person is going to be able to sort of out run these peaks to some extent and it’s going to be harder for this peak to catch up with the person.1495

Its going to increase the space effectively. Similarly, with the receiver running towards the emitter, it is it’s going to run through the wave fronts.1506

You could think of it as a very slowly moving bunch of bushes, I not quite sure what the best thing to do here is, like bubbles.1515

If you had a constantly strung together stream of bubbles right? Between every bubble there was one meter, right?1521

They’re moving along, they’re floating through the space, you could either run away from the bubbles and they could pop on you much more slowly because it takes that much longer for each one to get up to you.1530

Because not only does it need to cross the distance between the bubble in front of it and itself, it has to cross that plus you running away from it for a while.1538

If you ran towards the bubbles you’d wind up popping them on you much faster because it doesn’t have to just cross that distance, it now only has to cross to you, right?1546

The motion of the receiver is going to effect the experienced frequency.1553

Finally we’re ready for a formula, but I want you to keep in mind the reasons why this…what the expectation for how the frequency should shift, right?1559

If the emitter is moving towards the receiver, frequency should go up, because we’ve got them bunched together.1569

If the emitter is moving away from the receiver, frequency should go down because there’s now space between the wave peaks.1573

Same thing with the receiver moving towards…well similar ideas is the receiver moving towards the emitter is going to mean bunched up effectively is going to mean that they pop sooner.1579

It’s going to mean that the frequency goes up if the receiver is moving towards the emitter. If the receiver is moving away from the emitter, the frequency is going to go down.1589

You have to be able to imagine this because the formula is going to require you to understand what to expect before you’re able to use it and we’ll see why in just a moment.1597

We’re going to need a bunch of factors. V, speed of waves in the medium, so if we don’t know how fast the medium is moving, if those bubbles are moving at 1 meter per second, it’s a big difference if those bubbles are flying towards you at a 100,000 meters per second.1606

If they’re zipping along you don’t have any chance at out running them right?1620

If they’re moving along at only 1 meter per second, you’ve got a really good chance at being able to out run any bubbles.1623

You might be able to lower your rate to zero frequency.1627

The E, the speed of the emitter. How fast it’s moving is going to have an effect.1632

Vr, the speed of the receiver. How the fast the receiver is moving is going to have an effect.1635

FE, the emitted frequency, what everyone would hear if everyone was sitting still. What the person on the emitter would hear if they were doing it.1638

Hear is probably the wrong thing, I keep using sound here because we’re used thinking the Doppler Effect in sound, but it actually works with any wave.1646

Once again, works with light waves, that’s how we’re able to tell how far planets…not necessarily how far planets, what the motion of planets is.1654

It is either red shifting or blue shifting in the electromagnetic spectrum. It’s a little beyond us right now to talk about to in depth, but it is one of the uses for the Doppler Effect.1663

Anytime that you know there is going to be motion, you can compare what the emitted frequency is, what the expected emitted frequency is to the actually received and to be able to figure out some things about the motion going on.1673

If we have all these different ideas, we’re finally ready to have a formula.1683

Put together, we have the receive frequency, f. So the receive frequency f is equal to the speed of the waves in the medium, plus or minus the speed of the receiver, divided by the speed of waves in the medium, plus or minus the speed of the emitter, times the frequency of the emitter.1689

Now this isn’t like the quadratic formula where it’s plus or minus and it means both of those at once and you’re supposed to get two different answers.1705

For the Doppler Effect, you have to choose which one you’re going to use and the choice is important and it’s up to you.1712

You have to know what the expected is. You assign plus’ or minus’ by thinking through those above examples about the emitter and the motion.1718

If the wave fronts are going to get bunched up or bounced…or gets contacted by the receiver faster, you’re going to want to use the appropriate sign.1725

For example if the receiver was moving towards the emitter, we’d want a plus sign because we’d want a bigger number on top.1733

If the emitter was away from the receiver, we’d want a plus sign on the bottom, because we’d want a bigger number on the bottom.1745

You want to think in terms of bigger number on the bottom means lower frequency. Smaller number on the bottom means larger frequency.1754

Bigger number on top means larger frequency. Smaller number on top means smaller frequency.1760

You have to think about which you’re going to want here, so there’s some flipping, it’s really takes real thought here, so don’t just rush into it, don’t just throw things into this formula.1765

You have to understand what you’re expected output is and be able to think about that and that’s one of the really important to learn in physics.1774

Is being able to know what you’re sort of expecting and being able to guide your use of the formulas. Guide your use of information based on how you’re doing it.1781

Math is an important set of machinery, but you have to build the scaffold of understanding to know how to use that machinery.1788

This is just one more example of when it’s up to you to pay attention to what you’re doing.1796

Ready for some examples. If two waves have the equations y1 equals this and y2 equals this.1800

Now keep in mind, there are no units with these so this is really just something totally in a vacuum.1809

This is not actually applicable to real life things, because we don’t know if this in meters, we don’t know if this is in centimeters, we don’t know if this is in like the king’s feet.1814

It’s something, but we can talk about it mathematically and we can get a good idea.1820

We can use it as a practice problem but it’s not good physics, it’s just a reasonable practice problem for in a class, which is what we’re doing.1827

Its okay here, but you want to make sure that you know what units you’re using. You want to mark units down for the real thing, when you’re actually doing a lab for example.1834

If the waves are superimposed on one another, we place the waves together. What’s the value for y?1845

X equals 2. T equals 22.1849

The new value, the y that we’re going to make is just y1 + y2. We just add the two together.1852

It’s as simple as that. 3 sin of 1.2x + 600t + 4sin of 0.9x + 625t + 0.3.1858

That’s what the general formula is. If we want to plug in specific values to be able to get what the location is going to be like.1880

What the height is going to be at x=2. At 2 far down and 22 time later in the future. We have to plug them in.1888

X equals 2. T equals 22. That means y is going to be equal to 3 sin 1.2 x 2 + 600 x 22 + 4 sin 0.9 x 2 + 625 x 22 + 0.3.1897

We simplify these things out a little and we get y is equal to 3 sin of this big number, 13,202.4 + 4 sin of 13,751.4.1930

Now before we actually figure out what this is, I have some very important thing to talk about.1950

When you’re working with waves, you work are working with radians. Let me repeat that again, when you are working with waves, you are working with radians. Right?1956

Remember radians from trig, when things that go in terms of 2 π. 2 π is your one whole way around the circle.1970

Whenever we’re working with waves, we work with radians. Those phase angles we talked about, they aren’t phase angles in terms of 45 degrees or 127 degrees, or a 147 degrees.1977

They’re in radians, so they’re going to be some number.1987

Now remember you can convert from degrees to radians but we’re working in radians.1992

That’s why in terms of 2 π divided by λ, or 2 π divided by the period.1995

Because we’ve got 2 π is really fundamental to the way we’re using waves. It’s really fundamental.2000

Radians is the currency of physics. Now keep in mind, we do still occasionally use when we’re surveying things.2006

We use degrees, right? We’ve had plenty of problems where we say something goes off at a 50 degree angle or something is below something else by 15 degrees.2016

In this case, we’re now switching over to radians wholeheartedly. Radians and waves go together like, I don’t know, chocolate and peanut butter.2025

I’m actually not a big fan of chocolate and peanut butter, sorry if you are. Anyway, bad example, don’t need to tell you about my personal life.2035

W equals 3 sin, so we’ve got this, right? So we have to plug in it into radians.2042

If you are using a calculator and you probably are, you need to make sure your calculator is in radians when you’re working with waves.2047

In some way there is some mode function in your calculator, some way to get from degrees to radians.2054

Make sure you’re going to switch it over before you’re going to do stuff with this.2059

That’s where you’re going to have to start thinking in terms of. Now what value might we expect this to output when we plug this into a calculator?2061

We probably expect something pretty big, right? We’ve got sin acting on some giant number, 13,000 and sin acting on some other giant number, even more 13,000, so we’d expect a big number.2069

Remember, sin just repeats itself every 2 π. If sin repeats itself every 2 π, sin really only gives a value between -1 and +1.2078

The biggest and the smallest. The lowest and the tallest value that the unit circles going to give.2086

In reality, this number can never get very big. It’s going to repeat and do a lot of bouncing around.2092

It’s going to be kind of complicated because these don’t really interact that well as two equations.2098

They’re not clearly in phase, they’re clearly out of phase.2101

It is never going to actually turn out to be a very large thing. It’s just determined by those amplitudes.2106

Remember, it’s not like when you get to a really far number and your wave equation, it just off the amplitude. The amplitude is always maximum height it achieves.2110

We plug in these numbers and we wind up getting the kind of lack luster, 0.552.2119

You just plug in x and plug in t. If you’ve got two equations working together, you just put them together and you plug in separately.2127

Then you just add everything together and remember even more important when you’re working with waves, you’re working with radians.2134

We haven’t talked about that before but it’s a really important thing to notice, is that everything is in terms of radians.2139

If you switch to using degrees, it can screw up a whole mess of things. So double check on your calculator, make sure you’re switching in radians.2146

Try taking the sin of 90 and see what comes out. If the sin of 90 gives you 1, you’re not in radians.2152

Example two. We’ve got a wave that’s a wave equation of y1 equals a, amplitude, times sin of kx minus ωt.2158

Another wave, y2 is added to it. Superposed and they completely cancel each other out.2166

Fully destructive interference. If we’ve got that…so let’s say this is the baseline, alright.2172

We want to now figure out what the equation of y2 is going to be.2180

Well graphically, we know that y1 is going to look something like this, right?2182

I don’t know what’s it’s precisely going to be, but it’s going to look something like this.2190

Now, for y2 to come along and cancel out, it’s going to have to be the perfect mirror opposite of this.2193

It’s able to come up with a resultant thing here that just turns it to zero.2203

Y2 has to be its negative at all points, so one thing we can do is say, ‘Y1, well –y1 equals y2, right? Because y2 is you’re negative”.2208

True, but we can’t have a negative in front because amplitude doesn’t come negative.2217

Amplitude always positive so we need to be able to shift this around by a phase shift.2221

We do know, clearly they must have the same frequency.2226

Y1 frequency is equal to the y2 frequency.2230

Which also means that y1 t equals y2 t.2235

I’m not saying y1 x t, I’m just saying the guy that belongs to it.2240

Similarly we also know that y1 amplitude equals y2 amplitude because they’ve got to be the same height for them to cancel each other out.2244

At this point, we know that the equation for y2 is going to have to have the same amplitude, going to be using that same function sin.2251

Its going to have to have the same k, right? Because it’s got the same frequency, same wave length, all that sort of thing…frequency.2259

Since we’re assuming that velocity of the medium is the same, speed of the medium is the same.2268

Frequency, the fact they’ve got the same frequency implies. Same that they’ve got the same period implies. The must have the same wave length.2272

Kx minus ωt, but that’s still just the exact same thing.2278

So one last thing we have to do, add in φ. We add in φ and that gives us our phase angle change.2282

So we’ve got some phase shift. How do we figure out what angle φ has to be?2292

Well let’s look at the units circle. The units circle is a perfect circle.2295

If we start here at 0 and go up, we’ve got positive right?2302

This guy effectively has +0. He’s got an angle φ of 0, so he’s really easy going.2307

If we want to make the exact opposite, what’s the perfect opposite to the circle at this point?2312

The perfect opposite of the circle is to start here and go this way, you’re negative.2318

You’re just the negative mirror of what’s going on if you go positive.2322

If you’re the negative mirror, what’s going on if you go positive?2325

That means over here you start at 0, here you start at π.2328

If we add in π, we get a negative, we’re able to flip the whole thing.2331

Normally we’re looking starting from the positive side of the circle, now we’re going to start from the negative side of the circle.2336

That means φ equals π for this. So we’ve got y2 equals a times sin kx minus ωt plus π.2340

We add that in and we’re set. It’s as simple as that.2354

We are able to figure out where it has to be located based on the fact that we know it’s going to be completely in opposition the whole time.2361

So it has to have a phase shift that will put it out of phase all the time and the thing that’s going to make it out of phase, full destructive is if its phase shifted π over.2368

That’s where you get the opposite part of the sin wave.2377

We know that everything else has to be the same in order to figure this out and this is an easy example to some extent because we know that we’ve got this perfect phase shifting.2382

Now if it were going to have to be a phase shifting of three quarters π, it might be harder to see if coming when we probably have to have a couple of values given to us and more things filled out.2390

But it’s the same basic idea, we’ll plug in y1 and we’ll plug in: a sin k – ωt equals that.2400

So we know that y = y1 + y2. You plug in all your values and you’ll eventually be able to get enough information to solve for what φ has to be.2406

You’ll have to use of the information you know from trig, but at that point, it’s just solving for what φ has to be from that.2416

That’s how you can do it. In this case we can do it graphically because we can see what has to happen.2422

This will hopefully give you a slightly deeper intuitive understanding of how this stuff is working, what it means to have a phase angle.2427

Where are you starting from on the circle effectively? How are you different? What is your different way of looking at the same wave?2432

The same wave but you’re starting in a slightly different place so you just a different vantage point from the beginning.2440

String length of 2 meters is put under tension so that waves travel along it of v equals 450 meters per second.2446

We’ve got some string, it’s tensioned between two things. Two meters and we know that the velocity of this medium is 450 meters per second.2452

What frequencies would produce standing waves on this string? What frequencies would produce standing waves?2466

Well, we know that λ is equal to 2l over n, for any n contained in the natural numbers; 1, 2, 3, and up.2473

We can figure out what λ is. We know what the speed is, if we can know what the λ is, we can figure out what the frequencies have to be.2488

We know that for it to have a standing wave it’s got to have a wave length that’s equal to 2 times the length divided by n.2497

That’s going to give a whole infinite array of possibilities, but it will give us what the possibilities are.2504

We know that λ has to equal 2 times 2 meters over n. Anything that’s 4 divided by n and we’ll have the wave length.2510

Anything that’s 4 over n. If we want to know its frequency is, we know that velocity equals frequency times λ.2523

That’s going to give us that frequency is equal to the velocity over λ. Which is equal to 450 meters per second divided by 4 over n.2531

Which gives us n times 450 over 4. Or 112.5n.2543

You plug in any n and multiply it by 112.5 and that gives you a possible frequency, a possible resonate frequency.2553

If gives you one frequency that would produce a standing wave.2562

Some examples frequencies would be f = 112.5, the next one up; 225, the next one up; 337.5 and so on and so forth.2565

These are all in hertz. Both this and this are in hertz.2578

If you’re curious, the λ was in meters, right? Because we’ve got some length, 2 meters divided by n, and n doesn’t have any units, it’s just a number.2585

So 4 over n, we’ve still got 4 meters. Which makes sense because wave length has to be in length.2593

If we want to produce standing waves we just have to determine what wave length possibilities are and use that to figure out the frequency.2599

Example four. If you’re standing just off the side of the road and a police car is flying down the road at a very fast 40 meters per second, with the siren going at a frequency of 800 hertz.2607

If the speed of sound is 340 meters per second, what frequency will you hear as the car is approaching?2623

Once it passes you, what are the sounds you’re going to hear?2629

Clearly we’re going to need to use the Doppler Effect because the formula for that gives us the change in frequency due to the motion of the emitter, motion of the receiver.2632

In general, it was f = v, speed of the medium plus or minus v, speed of the receiver divided by v, speed of medium plus or minus v, emitter, times the frequency emitted.2641

Now remember it’s up to us to figure out plus’ and minus’ and all those sorts of things.2655

Just to mark everything off from the beginning. Velocity here, speed of our medium, 340 meters per second.2660

That’s the speed sounds here. Car is flying down the road at 40 meters per second, so the emitter is moving at 40 meters per second.2665

The receiver, is the receiver moving? No, you’re just standing there, so you’ve got 0 meters per second.2677

Finally, frequency emitted is equal to 800 hertz. In the case where the cop car is going by you, you’re going have your frequency that you’re going to hear is going to be equal to v.2683

Now what do we want to use? Do we want to use plus or do we want to use minus.2697

Well if its moving toward you, if you were moving towards it, you could look at it from either point of view right?2700

The two things are moving towards one another then you’ve got…you’re going to get more, right?2705

It’s going to go up because the two objects are moving towards one another.2713

It’s going to be v plus the receiver because you’re going to be increasing it as you move towards those wave fronts.2717

These two objects moving towards one another divided by v, plus or minus, once again, the emitter is coming towards you so you’ve got more wave fronts coming at you so you’re frequency is going to go up.2727

Because it’s packing those wave fronts at its front. So are you going to use plus or are you going to use minus.2738

Remember, smaller denominator means bigger overall number so you actually use minus v e times the frequency emitted.2743

We’ve got v, so 340, you’re speed is 0 divided by 340. Its speed is 40 times 800. 340 over 300 times 800.2752

We’ve got 906.7 hertz. So that what’s you’ll hear as it move towards you.2767

Now just after it passes you, once it passes you, it’s going to be going the opposite right? It’s going to be going very differently than it just did.2777

Change up the color. So, you’re standing here but now the gulf between you two is widening.2785

You can effectively think of you’re moving away from it, it’s moving away from you. Now we’re going to have to actually use a slightly different version of this formula.2795

All of our constants just stayed the same, but it’s the way that formula changes with those plus’ and minus’ that will give us a different experience.2800

When it’s coming towards you, you’re going to hear a higher frequency than it naturally emits.2809

When it’s going away from you, you’re going to hear a lower frequency than it naturally emits.2812

If frequency is equal to v, plus or minus vr. Which does it become?2817

Well if you’re moving away from an object, if you’re moving away from an object, you’re going to cause yourself to experience less wave peaks because you’re going to be running away from the wave peaks.2821

If you’re running away from the wave peaks, we need a smaller number up top, so v minus vr divided by, now the emitter.2831

Is the emitter going to be bunching up or is it spreading out then from the point of view of the receiver?2841

We’ll it’s spreading them out because it goes farther away before dropping the next peak.2847

That means that we’re going to have to have a smaller frequency. It’s going to have to contribute to making a smaller frequency, so it’s v plus ve.2852

Because it’s making a larger denominator times the frequency emitted.2861

We get 340 once again, minus 0 divided 340 plus 40 times 800 equals 340 over 380 times 800.2866

That gives us the value 715.8 hertz. When it’s moving towards you, you get 906.7 hertz of frequency.2878

When it’s moving away from you, you get 715.8 hertz. And if you were just standing still while it was sitting still next to you, if you were both still next to one another, you’d hear 800 hertz.2890

Alternately, if you were the cop in the car because you’re moving with it, you’re moving at the same speed so to you it’s as if it’s effectively still, it’s going to be 800 hertz for the cop in the car the whole time.2900

If you’re curious what these sounds like because frequency actually what is going into our ears.2912

If you wanted to know what this was, keep in mind, a siren isn’t going to sound like a pure tone.2917

We made this a little bit easier but staying it was just giving out a single hertz and we’ll talk a little bit more about how sound is actually working out after this example.2921

Let’s actually listen to it. I’ve got an example sound here, so here’s an example sound. I’ll be playing it in just a few moments.2927

The first sound you’re going to hear is the 800 hertz sound. Just to give you a sense of what 800 hertz sounds like tonally.2935

Two, you’re going to hear the 906.7 hertz, what it would be as it approaches you.2942

Then three, you’re going to hear 715.8 hertz. Then finally four, you’re going to hear simulation of what it might sound like to have it drive past you.2949

Remember it’s going to actually at some point, swap between these two and if it were driving directly at you, it wouldn’t’ ever swap until it went through you.2958

But because you don’t get hit by the car hopefully, you’re actually going to have this period of time where it’s going to slide between the two.2967

It’s going to slide between the high frequency to the low frequency. Or from the low frequency to the high frequency as it passes you or you pass it depending on the situation.2975

It’s going to slide frequencies because you were actually not going to be passing directly through its point.2985

This works reasonable well when you’re close to the path that it’s taking, but since you can’t ever actually on the path and not get run over.2991

You’re going to actually hear a change when its passing…say when it gets here, the wave changes, is going to about on the distance.2999

In the extreme, we can effectively treat it as a straight line, but as it gets close to you there’s going to be a change in the way the velocities are working.3008

We aren’t dealing with that because that’s much more complicated but that’s why it slides when you hear it in real life.3015

I’m sure you’ve heard a cop’s or car’s siren or something go off that has moved by you quickly while it’s been emitting noise and you’ve heard this consistent slide sound and that’s what we’re going to hear here.3021

Finally, we’ll hear a slide from 906.7. It will start at 906.7 hertz, it’ll go on for just a little bit and then it will slide down to 715.8 hertz.3033

This is gives you some idea of what you might actually hear. Alright, ready for that example sound? Go.3043

– 5055 [Pitch sounds varying]3048

Alright, sounds pretty good. There you go, there’s an idea of what it sounds like.3055

Keep in mind a couple things, real sounds in real life are not just pure tones.3059

Those that you just heard now was a pure tone. It was a pure tone of 800 hertz.3065

It was a pure tone of 906.7 hertz. In real life, you are not going to actually hear pure tones because there is many, many tones.3069

If you hear an actually cop car siren, it goes way between all these different things. It’s trying to catch your attention by sliding through a huge variety of tones.3079

In real life, we hear many, many tones at once compacted together, interacting with each other.3088

Which is actually how we experience light too.3094

The way we experience is different than breaking it up in single values that we get here.3099

This still gives you a good idea. It’s much easier to work these single values, it’d be hard to describe the whole range of possibilities.3106

Which is why when you get a very high level of physic, when you are really trying to describe it as opposed to just understanding what’s going fundamentally.3113

Which this does, it gives you the chance to understand it fundamentally. It gets hard to deal with all the things happening at once.3120

In real life there’s many frequencies going on. There is many things happening. The real sound is compounded by a number of things.3127

It doesn’t just directly between these two, it slide by you.3135

The angle that you are from the car, I mean the distance that you are from the path of the car is taking as it goes by you.3139

If maybe it turns around you as it goes by, all these different things will change the sounds that we’ll hear but this is one possibility that gives you a reasonable approximation of what it’s really like.3144

I hope that made sense, I hope that waves are beginning to come together.3154

It’s a really, it’s a huge can of worms that we’ve opened and we only a little bit of time to experience some of it.3157

There is so much more stuff in waves that we can talk about and go in depth but we’re just sort of scratching the surface so we can at least get an idea of what’s going… on here before moving to the next thing.3163

Alright, hope it made sense and I’ll see you in sound which will be a great use of this stuff that we’ve been talking about.3171