Chapter 15: Mechanical Waves

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All right, welcome back to the Deep Dive.

Today we're going to be tackling mechanical waves.

Okay.

And we've got, you know, quite a bit of material here that covers this in detail.

It's a big topic.

So today I think our mission is going to be to make sure that people listening have like a really solid understanding of the fundamentals and mechanical waves.

The different types.

Yeah, the different types.

The properties they have.

The properties and how we can describe them.

Mathematically.

Yeah, mathematically.

So are you ready to get wavy?

Let's do it.

Okay, let's jump into it.

So first things first, what exactly are we talking about when we say mechanical waves?

Like what's the basic definition?

Well, I think the way that this material really lays it out, it really gets at the core of it by saying that a mechanical wave is a disturbance that propagates energy through a medium.

And the distinction it makes right away is that it's not necessarily the bulk matter that's moving.

It's the energy that's moving.

So, you know, think about a pebble dropped in a pond and you get those ripples going outwards.

The water molecules themselves aren't actually moving outwards.

They're just kind of oscillating up and down.

Right.

But the energy that disturbance is propagating outwards is going outwards.

No, okay.

So we've got energy on the move.

So we're not just talking about one kind of wave, are we?

No.

So the material points out that there are three main types of mechanical waves.

Okay.

And the first kind is a transverse wave.

Okay, a transverse wave.

And the best way to think about that is like a wave on a string.

Okay.

Where the string itself is going up and down.

Right.

But the wave is traveling horizontally along the string.

So it's like perpendicular.

Right.

The motion of the string is perpendicular to the direction of the wave.

Okay.

So that's transverse.

So wiggle is sideways, but the wave moves forward.

Right.

So that's one type.

What's the second type?

So the second type is what's called a longitudinal wave.

Longitudinal.

And a good way to visualize this is like a spring or a slinky.

You know, if you push on one end, you create a compression.

Right.

And that compression will travel down the spring.

Right.

So the spring itself or the coils are moving back and forth parallel to the direction the wave is traveling.

Okay.

So it's like a push and pull kind of a thing.

Exactly.

A push and pull.

And sound waves are a good example of this.

Okay.

You know, a speaker will vibrate back and forth.

And it creates these compressions and rarefactions in the air, which is what we perceive as sound.

Okay.

So transverses is a side to side disturbance.

Longitudinal is a push and pull.

Right.

And then this material mentions a third category.

Right.

So the third one is a little more complicated because it's actually a combination of both transverse and longitudinal.

And the example given here are surface waves in water.

Yeah.

Okay.

You know, if you imagine a cork bobbing up and down in the water, it doesn't just move up and down.

It kind of moves in this circular or elliptical pattern.

Right.

Which means that there is both a transverse component and a longitudinal component.

Okay.

So we've got three different ways that the medium can be disturbed.

Exactly.

So but what are the fundamental characteristics that connect all of these?

Right.

Right.

So the material points out four main ones.

Okay.

They all require a medium to travel through.

Okay.

You can't have a mechanical wave in a vacuum.

Right.

Secondly,

they all propagate with a certain speed.

Okay.

That's determined by the medium.

By the medium.

Okay.

The third one is that the particles in the medium.

Yeah.

Oscillate around a fixed point, but there's no net displacement.

Right.

Of the matter itself.

Okay.

And lastly, they all carry energy.

Right.

And that energy originates from some mechanical work that created the disturbance.

Okay.

That makes sense.

So now the material dives into this idea of periodic waves.

Right.

What are those?

So a periodic wave is just a wave where the disturbance repeats itself.

Okay.

Regularly over time.

Uh -huh.

So if you were to, you know, keep tapping that pond.

Right.

At a very regular interval, you'd create a periodic wave.

Okay.

And a very important type of periodic wave is a sinusoidal wave.

Okay.

Which basically means if you plotted the displacement,

it would look like a sine or cosine curve.

Okay.

And this arises when the source of the wave.

Yeah.

Is undergoing what's called simple harmonic motion.

Right.

Which is basically just a very smooth back and forth oscillation.

Okay.

And you get this really nice symmetrical wave with these well -defined crests and troughs.

Okay.

So now we're getting into some specifics about ways we can describe these waves.

Right.

Like frequency and period and wavelength.

Can you explain those?

Yeah, absolutely.

So the frequency is a measure of how many complete cycles of the wave occur per unit time.

Right.

And it's usually measured in hertz.

Yeah.

Which is cycles per second.

Hertz.

Okay.

And the period is simply the inverse of that.

Oh.

It's the time it takes for one complete cycle.

Got it.

So if the frequency is f, the period is one over f.

All right.

And then wavelength.

Yeah.

Is the spatial distance over which the wave pattern repeats.

Yeah.

So you could think about it as the distance from one crest to the next.

Okay.

Or from one trough to the next.

Right.

Or for a longitudinal wave.

Yeah.

It's the distance between two successive compressions.

Okay.

So now we've got frequency, period, wavelength.

Right.

And this material points out a key equation that relates these to the speed of the wave.

Right.

Can you tell us what that is?

Yeah.

So this is a very important equation.

Yeah.

The wave speed denoted by v.

Right.

Is equal to the wavelength.

Lambda.

Lambda.

Times the frequency.

So v equals lambda times f.

V equals lambda f.

Okay.

And this equation is true for all periodic waves.

Right.

Regardless of whether they're transverse or longitudinal or a combination of both.

Okay.

There's something really interesting here.

Yeah.

The speed of a mechanical wave is usually only determined by the medium itself.

Okay.

So things like the elasticity, the density.

Right.

The temperature of the medium will affect the speed.

Okay.

But generally speaking,

waves of different frequencies will travel at the same speed in a given medium.

So it's like a speed limit.

Yeah.

Kind of like a speed limit.

Based on the medium.

That's set by the medium.

Okay.

Cool.

Now to get a little more detailed.

Right.

The material introduces this idea of a wave function.

Right.

Which sounds kind of complicated.

But it's not.

But what is it?

So the wave function, usually denoted by y of x and t.

Okay.

It's basically just a mathematical description of the wave's displacement as a function of position and time.

Okay.

So it tells you for any given point in space and at any given time.

Right.

How far that point is displaced from its equilibrium position.

Okay.

So it gives you like a snapshot.

Yeah.

Of the entire wave at any given moment.

Okay.

So now the material provides a few different forms.

Right.

Of this wave function.

Can you walk us through those?

Yeah.

So for a sinusoidal wave traveling in the positive x direction.

Okay.

One common form is y of x and t equals a time cosine of omega t minus kx.

Okay.

Can you break that down for us?

Yeah.

Absolutely.

So a is the amplitude.

Okay.

Which is maximum displacement from equilibrium.

Right.

Omega is the angular frequency.

Okay.

Which is related to the regular frequency by a factor of 2 pi.

Right.

And k is the wave number.

Okay.

Which is 2 pi divided by the wavelength.

Okay.

So we've got a, omega, and k.

Right.

What about this whole thing inside the cosine?

So the expression inside the cosine.

Yeah.

Is called the phase of the wave.

The phase.

Okay.

And it basically tells you.

Yeah.

You know, where in the cycle the wave is.

Right.

At a given point in time.

Okay.

Now there's another form that's often used.

Okay.

Which is y of x and t equals a times cosine of 2 pi times the quantity x over lambda minus t over t.

Okay.

So that one uses wavelength and period.

Right.

It uses wavelength and period explicitly.

Okay.

And the material points out that if the wave is traveling in the negative x direction.

Right.

The sign in there changes.

Yeah.

The sign between the x and the omega t terms becomes a plus sign instead of a minus sign.

Okay.

So the wave function can really tell us a lot about the wave.

Yeah.

It's a very powerful bit.

But we can actually use it to figure out things like the velocity and acceleration of the particles in the medium.

Right.

You can use calculus.

Okay.

To figure out the velocity.

Yeah.

By taking the first derivative with respect to time.

Okay.

And the acceleration.

Yeah.

By taking the second derivative with respect to time.

Okay.

And something really important here.

Yeah.

Is that sinusoidal wave functions.

Uh -huh.

And many other wave functions.

Okay.

Satisfy something called the wave equation.

Okay.

Which is a partial differential equation.

Okay.

That governs the propagation of waves.

Right.

And it basically relates the spatial curvature.

Okay.

Of the wave to its time evolution.

Okay.

So it's a very fundamental equation in wave physics.

Sounds complicated.

It is.

But it is a very important equation.

Okay.

Let's get back to something a little simpler.

Like this specific formula.

Okay.

For the speed of a transverse wave on a string.

Right.

So the material gives us this formula.

It does.

So the speed of a transverse wave on a string.

Uh -huh.

Is given by the square root of the tension in the string.

Okay.

Divided by its linear mass density.

Okay.

Which is just the mass per unit length.

Okay.

So it's based on how tight the string is.

Right.

And how heavy it is.

Yeah.

Okay.

And that makes intuitive sense.

Yeah.

It does.

And if you have a tighter string, the wave's gonna travel faster.

Right.

And if you have a heavier string.

Yeah.

It's gonna travel slower.

It'll be fluggish.

Exactly.

Okay.

And a really interesting point here.

Yeah.

Is that the speed doesn't depend on the amplitude or the frequency of the wave.

So those don't matter.

They don't matter.

It's just about the properties of the string itself.

Properties of the string.

Exactly.

Okay.

Now we know that waves can carry energy.

Yep.

So the material talks about the power transmitted.

Right.

By a wave on a string.

Can you talk about that?

Yeah.

So the average power.

Uh -huh.

Transmitted by a sinusoidal waves on a string.

Okay.

Is given by one half.

Okay.

Times the square root of mu, which is the tension, times the linear mass density.

Okay.

Times omega squared, times a squared.

Okay.

Hold on.

So f and mu are the properties of the string.

Yeah.

Omega is the angular frequency.

Right.

And a is the amplitude.

That's the amplitude, right?

Okay.

So it depends on all those things.

And is it saying that the power is related to the square.

Yes.

Of both the frequency and the amplitude.

Exactly.

It's proportional to the square of both.

So if you double the frequency.

You quadruple the power.

Okay.

And same with the amplitude.

Same with the amplitude.

Okay.

So bigger and faster oscillations mean more energy.

More power.

Okay.

Now what about intensity?

So intensity is just the average power per unit area.

Okay.

So it tells you how much power is flowing through a given area.

Right.

And this is particularly useful when you're talking about waves that spread out in three dimensions.

Okay.

Like sound.

Like sound.

Yeah.

Coming from a speaker.

Exactly.

Okay.

And for a point source that radiates uniformly in all directions, like a light bulb.

Right.

The intensity obeys something called the inverse square law.

Okay.

What is that?

So it means that the intensity is inversely proportional to the square of the source.

Okay.

So if you double the distance.

You quarter the intensity.

So the further you are from the source.

Yeah.

The weaker the signal is going to be.

Okay.

That makes sense.

It does.

Like how a sound gets quieter as you move further away.

Exactly.

It's the same amount of energy, but it's spread out over a larger area.

Over a larger area.

Exactly.

Okay.

Now what happens when waves meet each other?

So when waves meet each other.

Yeah.

They interfere.

Okay.

And the

superposition.

Okay.

Big word.

What does it mean?

It's a very simple concept actually.

Okay.

It just says that when two or more waves overlap.

Yeah.

The resulting displacement.

Right.

Is just the sum of the individual displacements.

So they just add together.

They just add together.

That's it.

Yeah.

Now this can lead to two main types of interference.

Okay.

Constructive interference.

Okay.

Where the waves reinforce each other.

Right.

And destructive interference.

Okay.

Where cancel each other out.

Okay.

So we've got waves meeting.

Right.

What about when a wave hits a boundary?

So when a wave hits a boundary.

Yeah.

It can be reflected.

Okay.

And the way it's reflected.

Yeah.

Depends on what's called the boundary conditions.

Right.

So for example, if you have a transverse wave on a string.

Go.

And the end of the string is fixed.

Yeah.

The reflected wave will be inverted.

Okay.

It'll flip.

It'll flip upside down.

Right.

But if the end of

So fixed and flips it.

Exactly.

Free end keeps it the same.

Keeps it the same.

Okay.

I get it.

Now this idea of reflection brings us to standing waves.

Right.

How can a wave be standing still?

So a standing wave is formed when a wave interferes with its own reflection.

Okay.

And it creates this pattern that looks like it's standing still.

Okay.

But it's actually just the result of two waves traveling in opposite directions.

Okay.

And there are certain points called nodes.

Nodes.

Okay.

Where the displacement is always zero.

Right.

And other points called antinodes.

Antinodes.

Where the displacement is maximum.

Right.

And the distance between two successive nodes or two successive antinodes.

Yeah.

Is always half the wavelength.

Half the wavelengths.

Okay.

And this material gives a mathematical representation.

Okay.

Of a standing on a string fixed at one end.

Right.

Which is Y of X and T equals A sub SW times sine of omega T.

Okay.

Where A sub SW is just the amplitude of the standing wave.

Okay.

So the energy is like trapped.

Right.

It's trapped.

And it's just vibrating in this fixed pattern.

Exactly.

Okay.

So this leads to this concept of normal modes.

Right.

Which is what happens when a string is fixed at both ends.

Right.

So if you have a string.

Yeah.

That's fixed at both ends.

Right.

Like on a musical instrument.

Okay.

Only certain frequencies.

Okay.

Will produce stable standing waves.

Okay.

And these frequencies are called the normal modes.

Normal modes.

Okay.

Or resonant frequencies.

Okay.

And the wavelengths.

Yeah.

Of these normal modes are quantized.

Quantized meaning?

Meaning they can only take on certain discrete values.

Okay.

Which are given by the formula lambda sub N.

Okay.

Equals two L over N.

Okay.

Where L is the length of the string.

Right.

And N is a positive integer.

Okay.

So one, two, three, and so on.

One, two, three, okay.

And each integer corresponds to a different normal mode.

Okay.

So that means each allowed wavelength.

Yes.

Has a specific frequency.

Exactly.

The frequency is given by F sub N.

Okay.

Equals N times V over two L.

Right.

Which is just N times F one.

Okay.

Where F one is the fundamental frequency.

Which is the lowest possible frequency for that string.

Right.

And the higher frequencies are called harmonics.

Harmonics.

Okay.

So the second harmonic is two F one.

Right.

The third harmonic is three F one.

Okay.

And so.

Okay.

And this fundamental frequency F one.

Yes.

This lowest note the string can make.

Right.

That depends on the properties of the string itself.

It does.

It's given by.

Yeah.

One over two L times the square root of F over mass.

So the tension and the linear mass density.

Right.

Exactly.

Okay.

So that's why string instruments can produce different pitches.

Right.

You can change the length of the string.

Uh -huh.

By pressing down on frets.

Yeah.

You can change the tension by tuning the pegs.

Right.

And you can use different types of strings.

Right.

With different materials and different thicknesses.

Exactly.

Yeah.

And the material points out.

Yeah.

That when a vibrating string produces sound.

The sound waves in the air.

Right.

Have the same frequency as the string.

Okay.

But a different wavelength.

Because the speed is different.

Right.

Because the speed of sound in air.

Right.

Is different from the speed of the wave on the string.

That's a really cool connection.

Yeah.

To real world music.

It is.

And to wrap things up.

Right.

This material includes a glossary of terms.

Yeah.

Which is always helpful.

Yeah.

I like glossaries.

The good reminder.

Yeah.

Of all the important definitions.

Yeah.

Because there were a lot of terms in there.

There were a lot of terms.

Okay.

So that was quite a deep dive.

It went right.

To mechanical waves.

Yeah.

We covered a lot.

We did.

We went from the basic definition to the different types.

Right.

We looked at the mathematical

description.

The speed.

The power.

The intensity.

The intensity.

Interference.

Standing waves.

Standing waves.

Normal modes.

Normal modes.

It's a lot.

It's a lot.

And it seems like that equation V equals lambda F is really fundamental to it all.

It is.

It connects everything together.

Yeah.

It's amazing how such a simple equation.

Right.

Can describe such a wide range of phenomena.

It's a good one to remember.

It is.

So as a final thought for our listeners.

Yeah.

To kind of take and ponder.

Right.

You know, we've talked about all this math.

All this physics behind mechanical waves.

Right.

Do you think there are other areas of science?

Yeah, I think so.

Where these same principles might apply.

I think so too.

Even if it's not obvious.

Right.

You know, like in systems that maybe don't even seem like they have waves.

Yeah.

Maybe they're analogies that we haven't even thought of yet.

Right.

Like are there hidden waves out there?

That we just haven't discovered.

That's a fascinating thought.

Yeah, it is.

I think that's a great place to leave it.

I think so too.

So thanks for joining me on this deep dive.

It was a good one.

Into mechanical waves.

It was.

And we'll catch you next time on the deep dive.

All right.

See you then.