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Lecture Comments (11)

2 answers

Last reply by: Peter Ke
Thu Apr 28, 2016 12:50 PM

Post by Peter Ke on March 21 at 08:22:34 PM

At 22:50, how you get 2pi/lambda(x) - 2pi/T - 2pi.
Specifically I don't understand where you get the 2pi at the end from.
Please explain..

2 answers

Last reply by: Anna Ha
Sun May 31, 2015 8:29 PM

Post by Anna Ha on May 31, 2015

Hi Professor Selhorst-Jones,

Thank you for your wonderful videos! They have been very helpful :)

I was just wondering do transverse and longitudinal waves require a medium?
I also wanted to check that: mechanical waves require a medium and can be transverse and longitudinal waves. And electromagnetic waves do not require a medium and are only transverse waves. Right?

Thank you!

1 answer

Last reply by: Professor Selhorst-Jones
Mon Sep 9, 2013 11:28 AM

Post by HARRISON IGWE on September 9, 2013

hi professor I've been finding difficulties in  solving this problem .
The equation Y=55m(3x-4t),where Y is mm,X is in m,T is in seconds represent a wave motion . Determine (i)frequency (ii)period (iii)speed of wave?

2 answers

Last reply by: Goutam Das
Tue Jun 11, 2013 3:23 AM

Post by Goutam Das on June 7, 2013

Hi professor, as far as I know,
All waves need a medium to propagate.

The medium of sound is a fluid, and fluids have no shear strength, therefor they can only be longitudinal (pressure) waves.

Electromagnetic waves are transverse (shear) waves, so their medium (ether) has to be solid. The formula for the speed of a transverse wave in a solid is
c = √(G/ρ), where G is the shear modulus (stiffness) and ρ is the inertial density of the medium. If we knew the density of the ether, we could calculate its shear modulus from the known speed of light, and vice versa. Evidently, the ether is ultra dense—perhaps a googol times denser than a neutron star. That's inertial density; the ether probably has no gravity of its own, as (I believe) it is also the medium of gravity, and all the other forces. What do you think professor?

Intro to Waves

  • There are three main categories of waves:
    • Mechanical Waves: These travel through a material medium.
    • Electromagnetic Waves: EM waves do not require a material medium to exist.
    • Matter or Quantum Mechanical Waves: These describe the motion of elemental particles (electrons, protons, etc.) on the atomic level. We won't investigate them in this course.
  • We also classify waves based on how they move:
    • Transverse Waves: The particles of the wave move perpendicular to the motion of the wave.
    • Longitudinal Waves: The particles of the wave move parallel to the motion of the wave. This is done through compression and rarefaction (expansion), i.e., the wave is transmitted by pressure changes.
  • We describe a wave with the following characteristics:
    • Amplitude (A): How tall the wave is at its maximum height.
    • Wavelength (λ): The distance between "repeating" points on the wave, such as top-to-top.
    • Wave speed (v): How fast the wave is moving.
    • Period (T): The time it takes to go through a full oscillation.
    • Frequency (f): The number of oscillations that occur per second. [The unit for this is the hertz (Hz) where 1 Hz = [1/1s]. Thus, f = [1/T] and T = [1/f].]
  • Because speed, frequency, and wavelength are all related, v = λf .
  • We can find the height (or pressure differential if it's a longitudinal wave) with the following equation:
    y(x,t) = Asin(kx − ωt).
    • x is the horizontal location we are considering.
    • t is the time we are looking at the wave.
    • k is the angular wave number and is connected to the wavelength:
      k =

    • ω is the angular frequency and is connected to the period (which is connected to the frequency):
      ω =

          ⇔     ω = 2π·f.

Intro to Waves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Pulse 1:00
    • Introduction to Pulse
  • Wave 1:59
    • Wave Overview
  • Wave Types 3:16
    • Mechanical Waves
    • Electromagnetic Waves
    • Matter or Quantum Mechanical Waves
    • Transverse Waves
    • Longitudinal Waves
  • Wave Characteristics 7:24
    • Amplitude and Wavelength
    • Wave Speed (v)
    • Period (T)
    • Frequency (f)
    • v = λf
  • Wave Equation 16:15
    • Wave Equation
    • Angular Wave Number
    • Angular Frequency
  • Example 1: CPU Frequency 24:35
  • Example 2: Speed of Light, Wavelength, and Frequency 26:11
  • Example 3: Spacing of Grooves 28:35
  • Example 4: Wave Diagram 31:21

Transcription: Intro to Waves

Welcome back to Today we’re going to be talking about an introduction to waves.0000

We’re going to finally get an understanding of how waves work and they have an incredibly large presence in our daily lives.0005

Waves are way for one object or location to have energy transferred to another object or location.0011

That seems like a really large definition and it is. It’s not the only way for energy to be transferred but it a motion of energy from location to location.0018

It also has a lot of other effects and we’ll be investigating some of those.0028

There are many different kinds of waves and you’re constantly around them.0032

Some basic examples that you’re being currently being exposed to: sound and light.0036

Also any vibrating strings, waves in water, and the list goes way on.0041

There’s all sorts of other things, seismic waves in the ground, vibration along a steel pipe.0046

There’s quantum motion in waves, though we won’t be getting into that.0052

Waves make up a fundamental part of the universe and nature around us.0055

To begin with, let’s imagine the idea of a pulse. Imagine you’ve tied one end of a string to a wall, so it’s tied over here and you pull the string taunt.0061

Then you whip it, really suddenly, once. You whip it hard, what’s going to happen?0072

Well you’re going to create a pulse of energy that’s going to be sent down the string.0077

As the energy might be here, we’ll have a sort of raised up area and then as time goes on, not long time probably, it’s going to move to here and then it’ll move to here then it’ll move to here until eventually it hits the wall.0081

So that pulse of energy will be sent down the string and it’ll keep moving down the string towards the wall.0095

We whip it once and we move the energy, the energy that we put into our whip gets transmitted down the string.0099

Some of it goes into heat and other things, we won’t actually be investigating the energy, we will just be investigating the motion of waves.0105

There is definitely a change of energy from a location to another location.0111

You transmit that motion down the line into the wall.0119

Do the same situation but this time you start whipping the string up and down regularly, creating multiple pulses at even time intervals.0122

Some of these pulses are pointing up and some of these pulses are pointing down. What’s going to happen then? You’re creating a wave.0128

Instead we’ve got one pulse here and one pulse here but they form together to make a continuous wave shape. The whole thing makes a wave.0133

Once again, as time moves on, the things going to slide over and the whole thing will move farther along the line and get closer to the wall.0141

As time goes on, the thing shifts forward an amount. It has a constant velocity moving towards wherever it’s moving towards.0150

We’ve created a wave, there is a number of characteristics about the wave. Some of these drawings aren’t quite perfect but the important things is that waves have regularity.0159

It’s the fact that we can always trust it to get to the same height or pressure differential as we’ll talk about much later in sound.0168

We can always trust it to get to the same levels every amount that it goes.0173

Once it goes through one oscillation, one period of itself, it manages to repeat all the same effects.0178

So looking at one snapshot, the right chunk snapshot, it can look just like another chunk snapshot, can look just like another chunk snapshot.0184

That’s when the important ideas of waves, the fact that we’ve got regularity, something we can trust is going to be regular coming down the line.0191

There are three main categories of waves.0199

Mechanical waves are the waves that you probably have the most familiarity with at this point.0202

These are the waves that travel through any sort of material medium.0206

Common examples, the most common one that we’re all used too, sound waves.0209

Every time you hear something, if you hear a big bass drum get kicked at a concert and you feel those vibrations in your chest, in your body, that’s actually another form of sound waves.0212

They are just a very low frequency, so they’re able to vibrate your entire body.0223

Another type would be water waves. Waves on the top of waters is another form of waves.0228

Seismic waves, waves in the motion of the Earth. Things that cause earthquakes, these are all different forms of waves and different propagations of energy.0234

Electrometric waves, EM waves, don’t require material medium to exist.0244

We’re also exposed to these constantly, but you might have less of an intuitive understanding of them.0248

Examples of this is radio waves, the visible light that you’re currently seeing from the screen, X-rays.0254

Anything that has, that is, is light, is a part of the electrometric spectrum, but we’ll talk about that more.0258

That’s a little bit difficult to understand, because light has a whole bunch of special properties that really affect the nature of the universe in incredibly strong ways.0267

That’s going to be its on special section. We’ll very lightly touch the surface of that.0277

But for now, we can just think of it as light.0281

Finally, matter or quantum mechanical waves. These are waves that describe the motion of elemental particles, like the electron, the proton, or even smaller, the quark or the photon on the atomic level.0284

They’re very important to modern physics. They have a whole big impact and what we understand now and where our research is going currently.0296

We’re not going to be investigating them in this course. They behave a little bit differently than the classical waves we’re going to be studying.0302

They’re going to be more challenging to understand, so that’s something we’re going to save a future course.0307

Alright, in addition to the previous categories, the mechanical waves, EM waves, and mater waves.0313

We won’t be talking about mater waves. Mechanical waves are the ones we most understand.0320

They’re the ones that have motion medium. EM waves, they sort of come with their own medium.0325

In addition to the previous categories, we can also classify waves based on how they move.0329

Transverse waves. The particles of the waves move perpendicular to the motion of the waves.0334

If we got something that would normally be a flat line like the surface of the sea.0339

Then we’ve got a wave, an undulating curve on top of it that causes the surface of that sea to move up and down depending on the location of the wave.0344

That’s going to be a transverse wave. It’s happening transverse to the motion, it’s happening perpendicular to the motion of the wave.0353

Examples include a vibrating string, EM waves, any waves in water; not any waves, sorry you can also have sound waves in water.0361

Waves that we’re used to seeing in water. There is a huge varieties of things like this.0369

If you were to knock on a pipe, you’d also once again…if you were to shake a pipe, you’d get waves. If you were to knock on a pipe, you’d get sound waves.0375

We’ll talk about this more later.0384

Longitude in the waves. The particles of the wave move parallel to the motion of the waves.0387

So this a little bit harder to see and we’ll get the chance to get a better visual understand of this once we get to sound.0390

The particles of the wave move parallel to the motion of the wave. This done through compression and rarefaction, expansion.0395

So if we got two particles like this and remember they’re charged particles so they don’t like getting to close to each other normally.0401

The wave is going to be transmitted by pressure changes. If one particle pushes towards the other one for a moment, a brief moment, they’ll be together and push back away from one another.0407

That pressure wave will wind up getting transmitted all the way through.0418

A pressure front and once they push away from each other, we’ll have a rarefaction of area and they’ll wind up getting pushed back together again by the next pressure front coming through.0423

Pressure changes will transmit the information, will transmit the energy. The motion is being caused by pressure changes.0431

Most common examples is sound. But it also exists in certain kinds of seismic waves.0439

Alright, let’s investigate the wave more deeply. Here’s an okay drawing of how a wave looks.0446

This is a general sinusoidal wave and we can treat most waves as sinusoidal.0451

Meaning it comes from sin, as in the sin function that looks like just like this when you graph it out.0456

First off, amplitude. How tall the wave is at its maximum height.0464

In this case, the amplitude would show up right here. We’d get A there.0469

Somewhere else we’d see it here, we’d see it also right here. Anytime we see the maximum height.0472

What about down here? Well down here, it’s also going to be a height of A.0477

Instead if we were to look at it from an absolute point of view it’s going to show up at –A, although the length will be A.0482

Amplitude is the absolute value of you location from the medium location.0489

Its how far up or down you go, but it’s always going to wind up being a positive number.0495

How tall the wave is at its maximum height is amplitude. A.0499

Wave length, the distance between repeating points on the wave, such as top to top.0504

Remember, one of the important qualities of the wave is that it repeats itself after a certain amount of time.0508

If we go from here to here, notice that this entire area here looks just like this entire area here.0513

Just like this entire area to the front of it and this entire area to the front of it looks the same.0523

And if fact, if we were to go all the way through, we’d wind up getting all the same information.0528

It’s going to look like the exact same thing from the wave length point of view.0532

When you get a one wave length, the distance that it takes to do an oscillation in the wave, you’re going to see repeating points.0537

Everything will wind up repeating. If we go from top to top, we’ll have a wave length.0545

The guy that we use to call out wave length is lambda. Lambda is another Greek letter, he gets thrown around now in waves, he becomes very important.0553

He’s pronounced lambda and spelled in case you’re curious, lambda. Lambda. Lambda is the guy.0561

What some other repeating points on this? What about from here to here?0569

Well that isn’t going to wind up being lambda because notice how this section right here is doing something different than this section right here.0573

If we wanted to say, look from this point, we’d have to go all the way to here because know we’ve got just to the right of it.0582

It’s going up a bit and just to the right of it over here it’s going up a bit.0588

So from here to here would also be another example of lambda.0592

We could do to this from anything as long as we wound up being at the same height and we went through the right amount to wind up having it experience the same affects inside the wave.0597

We’re able to get one wave length. The important part about a wave length is that after you go that distance everything repeats in the wave.0607

More characteristics. Waves speed, how fast the wave is moving.0615

Remember this whole thing is moving forward at some speed. We’ve got some velocity that it’s got.0619

After one second it will be that many meters farther ahead.0625

For velocities and meters per second, we take that and one second later it will be that much velocity meters ahead.0630

Remember the wave is moving along. And as this moves forward we’re going to wind up seeing more of it coming out of it.0638

There is always more wave that’s going to be put in. Either it’s too the left outside of where we’re looking or it gets whipped into place by the motion of whatever is creating the wave.0645

The wave continues out on either side, and the whole thing moves forward.0656

Period. The period is the time it takes to go through a full oscillation.0664

Remember if we got some speed V, that this whole thing has, we’re going to be able to get from here to here and we’ll have a repartition.0668

Now one way is we could change the location we’re looking, but we could also fix the location that we’re looking here.0678

We’d be able to see that T later. We start here, say we start here, but T later because of the velocity will wind up showing up there.0684

If you notice, the velocity times the period is always going to wind up equaling the wave length.0698

Because the amount of time that it takes, if the velocity goes this way and T later is how long it takes to go through a single oscillation, the period is how long it takes from wave peak to get to another wave peak.0708

Or one point on the wave to get to its corresponding twin point on the wave.0721

That’s going to wind up having to be the wave length right? That’s what the definition of the wave length is.0727

We can think of it as a movement, the time that we’ve allowed it to move, and has given us a repetition, a period.0732

Or we could think of it as how far different we’ve seen in the distance that we’re looking in the wave, has given us a repetition on the wave.0740

So we’ve got the velocity times the time that it takes for the period is going to equal that wave length because they’ve got to be equivalent.0747

Frequency, the number of oscillation’s that occur per second.0754

Say we’ve got some period. T is equal to the period right? So in here…it’s gets from here to here if we’re looking from some single place.0759

If we fix our gaze here, so fix our gaze on this blue line, this blue dash line and we look up, we’re going to see this point here.0775

T time later will see the red dot above us. T time later the red dot will have moved to where the blue dot is.0783

If that’s the case, we can ask, alright so, it takes T many seconds and T could a larger number, greater than one or T can be a small number that’s well less than one.0791

We could ask, if it takes us T many seconds to get a repetition in the wave, how many repetitions are we going to get every second?0805

How many oscillations are we going to have per second? How many wave peaks are we going to have per second?0814

Well if that’s the case we’re going to need something that measures per second and that’s where the hertz comes in.0820

That’s one hertz equals one per second. One over one second. So that means the frequency is equal to one second divided by the period.0825

The period, so if we have something is the period of T=0.2 seconds then that means the frequency is going to be 5 because in one second we managed to have 0.2 seconds occur five times.0836

Frequency is the inverse of the period. And the period is equal to the inverse of the frequency.0854

Anytime we want one for the other, we just flip it and we’re able to get the answer.0859

One divided by the other and we take reciprocal and we’ll get it.0863

That’s because the fact we’re looking for long it takes to get from one like point on the wave to another point.0866

How long it takes for a single oscillation to occur is going to also be if we divide one second by how long it takes we’ll get the number of times we get an oscillation.0873

If we have the number of times oscillation happens in a second and we divide one second by those numbers of oscillations, we’ll get how long it takes each oscillation to occur.0882

Finally because speed, frequency, and wave length are all deeply related, we’re going to have the fact that velocity is equal to lambda times frequency.0892

Which is equivalent to velocity times time equals lambda. Remember we know that velocity that we’re moving, times the time it takes to from peak to peak has to be equal to lambda.0900

We’ve got the same thing going on here since V=λf. Well if we divide by F on both sides, we get v/f=λ. Which 1/f=t, so we get vt=λ.0914

So these are equivalent expressions. Why is V=λf make sense? Well if we’ve got lambda distance here, this is lambda…distance.0926

And we know that time, in one second we’re going to see the frequency occur. So if we see ten oscillations in one second, then how much distance have we covered?0939

We’ve covered ten oscillations to each oscillation as one wave length, then ten times the wave length.0949

So the frequency, the number of oscillations we occur, that occur in one second, we have in one second, times how long each one of those oscillations is, is going to be the velocity since we can just divide by one second.0955

Velocity is equal to lambda, the distance, times frequency, which comes in one of our seconds. So we’ve got meters per second.0967

Great. Wave equation, finally we can talk about if want to know the height of a point on a wave.0975

We’ll need to know two inputs. The horizontal location x and the time we’re looking at the wave.0982

Here’s just a quick sketch of a wave. If we got some wave like this, we could either look at this location x or maybe we want to look at this location x.0987

We’re going to get totally different results for that. However, what happens if then we also say, one of them is T=1?0999

What happens when we look at T=2? Well then the entire wave is going to have slid over some amount.1006

We’re going to have the same wave, but it’s going to have slid over and we’re going to get totally different answers for each one of those x’s.1013

That means, the time that we’re looking and also the location on the wave that we’re looking matters.1021

We’ve got a two variable function, we’ve got an equation that’s going to be based off of two variables that give us that dependent variable, that why, what output value.1027

Notice that one other thing about this, if we only care about the point where the wave originates, the very beginning point, x=0, we can simplify this because we can just knock out that x term.1037

We can make it simple at A times sin of omega T. Now, at this point you’re probably wondering what k and omega is, that’s a great question, we’re about to answer it.1046

Our equation is Y of x,t equals a, amplitude, times sin of kx minus omega t and quantity.1056

So before we can use our equation we’re going to define what k and omega mean.1065

K is the angler wave number and it causes our function to give the same value for every wave length lambda.1070

Now remember, in case you don’t remember some of the stuff from trig, every time sin of 2π times any number, n.1075

Where n equals -2, -1, 0, 1, 2… on either side. Any integer number.1085

Sin of 2π repeats, the whole thing repeats, no matter what we’ve got added to that inside of it, we’re going to wind up getting the same answer.1096

Because what we’ve done, whenever we do 2π, we’ve lapped the circle. And we’ve winding up starting here and we lap the circle, then we’re still going to get the same answer.1103

If we lap the circle in the other direction twice it doesn’t matter, we’re going to the same answer as long as we land at the same point.1113

As long as we change by things of 2π we’re going to have the same thing show up.1119

Which makes a lot of sense because on our wave, as long as we change by some wave length distance, we’re going to have all the same stuff show up.1124

If we change by one wave length and we’re using sin, we’re going see inside of it, a factor of 2π show up.1132

So for every wave length we change, we show up by 2π, that’s why we get k here.1140

Because if k is equal to 2π/λ and then we move over by 2λ amount, well the lambdas cancel out and we get 4π, which is equal to 2π times some n.1146

So we’re going to see the exact same thing as if we hadn’t shifted over, which is exactly what makes sense.1159

When we shift over by some wave length amount, it shouldn’t have an effect on what we would have seen otherwise.1164

Shifting over by a wave length is no different from the point of view of what value we’re going to see for the height or the pressure differentials that we’ll talk about later in sound.1168

Same basic ideas going for omega. Omega is the angler frequency and it causes our function to give the same value for every period time. So every period, big T.1178

Omega, not sure if I said, I don’t think I said, this guy right here, is once again another Greek letter and its pronounced omega. O M E G A.1187

You’re probably seen his bigger brother at some point. Capital Omega is that guy but this is lowercase omega.1201

Back to the thing at hand. Omega is the angler frequency and it cause our function to give the same value for every period.1209

Remember once again, every time we pass one period of time, as long as we’re looking at the same location, we should experience no difference right?1216

Every time you go through one period you’ve gone through one entire oscillation, so the wave just looks the same to you from that point of view.1224

As long as you wait 1T to look, everything is going to be the same.1230

So if we plug in omega for 2π over T. And then we hit it with some time that is factor of a period, 3t. Those t’s cancel out and we get something that is just going to get canceled out by sin.1234

We’re going to care about the other information inside of there. So as long as we wind up shifting some amount by a period, it’s not going to have an effect.1246

Let’s look at a slightly more complex example. We’re not going to see the time be just a factor of the period.1256

What if we look at t=1 and t=1+t? So if we look at that then we can say, let’s make it easy on ourselves, we’ll look at just generally…we’ll look at the same location x.1260

X is going to be fixed, otherwise we wouldn’t wind up seeing the periodic nature of the oscillations reoccur.1276

So our two values is going to be, our function is going to be y=a(sin)kx. And what’s k? We’ve got as 2π/λx - ω2π/. x t. The time.1282

If plug in t=1, we’ll plug in t=1 over here so t=1, we’re going to get y is equal to a(sin)2π/λ whatever x we’d chosen minus 2π/tx1, so there we are. That’s what we’re going to wind up seeing.1305

What happens when we plug in 1 plus one more period of time? Well we’re going to hope that we’re going to see the exact same value.1328

We’ll plug that one over here, we’ve got y=a times sin of 2πλ times x minus, now what…times one plus t, we’ll get 2π over t plus just 2π right?1334

So if that’s the case then we’ve got y=a sin 2π λ x - 2π/t - 2π.1358

This part right here winds up being the exact same as this part right here.1371

The only part that’s different is this here, but remember since we’re dealing with sin, since we’ve moved just one more 2π, we’re some location.1379

Then we just spin it around and boom we wind up being at the exact same location because we’ve moved by -2π.1389

So this location here, this stuff all here, is our real location. So if wind up shifting things by one whole period, even if we start at some time that isn’t a whole thing of a period, the way we’ve got this set up.1396

Since we’re using sin, a regular periodic function, we’ve got the fact we’re able to make those oscillations occur in our mathematics, our algebra, is able to support our visual oscillations.1409

We move a period temporally in algebra, we wind up moving that same full length time, so we see the exact thing.1422

The exact same would happen if we used k, if we used…if we moved x around and this sort of thing, but this is just to illustrate that the math here is really working and why it is.1430

We’ve got to make sure that 2π over t is able to handle a motion of a period as causing no effect to the values we’ll get out. Same thing with the motions of a wave length.1439

That’s why that 2π comes in, is because sins, sins natural period is 2π right? Its period before it repeats is 2π.1448

So we have to have a way to have those two periods, the period of the wave we’re working with and the period of the natural function we’re working with to be able to communicate with each other and that’s what this is all about.1456

Then finally, if we have 2π/t since over t is the same thing as times frequency, we get omega is also equal to 2π times the frequency, as simple as that.1465

Finally we’re ready to work on some examples. If we have a CPU on some device that has a frequency of 1 gigahertz, then we’ve got a frequency equals 1 gigahertz.1476

How many hertz is that? Well it goes, kila, mega, giga. So 10^3, 10^6, 10^9.1487

So 1 gigahertz becomes 1x10^9 hertz.1494

What’s the time that it would take to complete a single cycle? So if we want to know what the period is, then we remember the fact that frequency equals 1 over t.1501

T equals 1 over f. If want to know the period, t equals 1 over 1 gigahertz. 1x10^9.1512

Which becomes 10^-9, which is the same thing as mila, micro, nano. So we get one nanosecond is how long it takes.1523

There are 10^9 nanoseconds in a single second, which makes sense because 10^9, there has to be 10^9 nanoseconds because each of those cycles has to be complete for it to be able a whole frequency of 10^9 cycles completing.1524

So CPU, we’ve got the same, it’s not directly a wave in the same way, but we’ve got the fact that it’s having repetitive things happen.1539

Its going through cycles, it’s cycling through data. We’re able to talk about it in terms of frequency and in terms of periods just like we do with waves.1566

If the speed of light is velocity 3x10^8 meters per second, mighty fast. And you receive a wave with a wave length of 500 nanometers. Very small wave length. What is its frequency?1573

Speed of light is v=3x10^8 meters per second and you receive a wave with a wave length of 500 nanometers. We already know what v is, v is here.1588

So 500 nanometers is the wave length, right? The wave length equals 500 nanometers, which is the same thing as nana, 10^-9, 500x10^-9 meters.1598

If we want to know what its frequency is, we remember that velocity is equal to the frequency times lambda.1611

The frequency times lambda has to give us the velocity because that’s how much distance has been covered.1620

Wave length is a chuck of distance and frequency is how many times you have those chunks distance in one second to we cover. Frequency times wave length.1625

We cover fxλ and that gives us our v.1634

v=fxλ, so v/λ=f, so if we want to know what f is. F is going to be equal to 3x10^8/500x10^-9.1637

We plug that into a calculator and we get 6x10^14 hertz. That’s a giant frequency compared to the stuff that we’re used to seeing in sound.1655

For light though that’s pretty reasonable and you’re probably seeing a color pretty close to that.1666

6x10^14 hertz is a receivable for the human eye, it’s something in the visual spectrum and I think, don’t hold me to this, it’s probably pretty close to either yellow, green, or blue, somewhere in that sort of range.1671

Probably a little bit closer to the blue, green or blue. Anyway, that’s something that your eye is actually really able to see and so as opposed to seeing a numerical frequency data when we look at something.1687

We don’t say “Oh, that frequency is that”. We see a color. We’ve got these other ways of interpreting the information that the universe is sending to us.1697

It is a real thing, we are getting real information here just like when we go to some height, we’re really at a height but we can measure that height and periodically we’re able to measure the frequency we see in periodically.1704

If you’re driving a car going 30 meters per second and the car beings to run over a rumble strip, rumble strips are these evenly spaced grooves used to alert drivers, so little down grooves in the road like this.1716

So when a tire rolls over it, the tire falls in the up and down motion, winds up jostling the driver and they notice something, or they’re accidentally swerving off to the side, they notice that they’re swerving off to the side or if they’re coming up on some toll, they notice that they’re coming up to some toll.1733

It’s just something to alert drivers. If the strip vibrates your car at 98 hertz, what’s the spacing of the grooves?1747

First off, I’d like to point out that this isn’t technically a wave. Just like in #1 we weren’t technically working with a wave, but we can still apply many of the concepts.1754

In this case, the wave speed, we know the velocity of the car is equal to 30 meters per second.1762

While the wave might not be moving, the way it’s experiencing the wave is moving. In another way we could consider the tire is still, and the wave is the thing moving.1771

From your point of view you can’t really tell who’s moving when you’re inside of the car, although we look at our scenery and we can easily tell.1783

But you know, we can’t be sure who’s the thing that’s moving. It’s possible the road is the thing moving. So you can take it as the wave moving underneath you.1789

Once again, we’re not getting full repartition quite the same, maybe. But we’re not necessarily thinking it purely in terms of wave lengths, but that isn’t the issue.1796

We can expand a lot of these ideas to more things.1806

So we’re running over some rumble strips and it’s moving by us at 30 meters per second, either because the cars moving, or because it’s moving relative to us.1810

If the strip vibrates your car or bounces you up and down 98 hertz or 98 times per second, what the spacing of the grooves got to be?1818

It’s the exact same thing, if it manages to bump us up and down 98 times in a second and the wave length is the spacing of the strips. The spacing of the grooves is going to be the wave length.1827

How far they spread apart, then the 30 meters per second that we experience has got to be that number of times that the wave length times how many times they show up in a second.1838

Velocity is equal to the frequency times lambda. So we plug in our numbers and if we want to know what lambda, is we’re going to have the velocity divided by the frequency.1849

We plug in our numbers 30 meters per second divided by a frequency of 98 hertz and we get 0.306 meters.1860

There we are, the spacing of those grooves is about a third of a meter which makes a lot of sense if you’ve ever looked at them on the street.1872

Example four. Use the following diagram to give an equation describing the wave in the diagram.1882

Now to begin with, lets point out, we just want to remember y equals a, the amplitude, times sin, the function that allows us to have periodicity, allows us to have oscillations occur algebraically.1887

kx - ω x t. So x, the location we are in the wave. T the time that we’re looking at the wave. Omega is the factor that allows us to handle the fact that periods cause repetitions.1901

K is the fact that allows us to handle that wave lengths cause repetitions. To begin with, we know that the period is equal to 0.02 seconds.1915

We know that amplitude is equal 0.5 meters. Okay, great. What is this here?1927

Well that is not equal to the wave length. Remember, if we look just to the right of this point, and look just to the right of this point. We should see the exact same thing.1935

We don’t see the exact same thing, it’s going down over. If we go over here though, we will see it.1944

And because its sin wave, it’s evenly spaced out throughout, we know that 3.5 meters isn’t going to be the wave length, but it’s going to be half the wave length because it’s in one of the dips.1950

So the wave length over 2, so that means that our wave length is equal to 7 meters.1961

That’s all the data that we wind up needing. Now we want to solve for what k has to be.1968

K is equal to 2π/λ, so k is equal to 2π/7. Omega is equal 2π over the period.1972

By the way, k, we’re throwing around k a lot. I never mentioned this but k is not the same k as when we’re dealing with springs. It’s a different k, we’re using it to mean a totally different thing at this point.1985

The k that we’re using in this stuff is different than the k we used before. Just like how little t and big t don’t necessarily mean the same thing. We can even wind up using the same letter for different things, and we have know contextually what we’re talking about.1999

Sorry, I made that assumption, but that’s actually a really important thing. You don’t want to get confused and think that springs and waves necessarily have to do with each other every time. It’s not that same spring constant we were talking about before.2011

Totally different use of k. Anyway back to the problem. Omega is equal to 2π divided by the period.2024

If we know the period is, 2π/0.02 seconds and that will wind up giving us 100π.2030

Simple as this at this point, we just plug in all of our numbers. Y is equal to that amplitude, 0.5 meters times sin, of the numbers 2π/7x-ω100π times the time that we’re looking.2039

That right here is our answer. Simple as that. So we just want to be able to analyze the diagram.2065

One of the most important things to pay attention to the fact that wave length has to be not just some distance where you get the same point, but some distance where you get a point that means the exact same thing.2071

That if you look just a little bit further on and a little bit further behind you’re going to see a full repetition.2083

It’s not just the same point, because the same point occurs at any horizontal thing, except for the very tops and bottoms.2089

These pairs of points are not enough to determine a wave length. What you need is to reach is to reach a little bit farther and look here.2097

The very top to top because tops themselves only occur once every wave length.2104

Whereas middles occur twice. Same with bottoms, you can go from the bottom to the bottom and the top to the top.2111

That’s normally the easiest thing to measure, but if you’re measuring from the middle to the middle, this isn’t enough.2117

You need to also go to here. Okay, great. I hope you enjoyed that, I hope that made sense, waves are a whole bunch of big ideas, but we’re going to…2123