Chapter 47: Sound and the Wave Equation

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Welcome back to the Deep Dive.

Today we're getting into something, well, really fundamental across physics mechanics, light, quantum theory,

you name it.

It's the wave equation.

That's right.

We're using a classic text to guide us, focusing on how disturbances, like sound, actually travel through a medium.

It's all about propagation.

And our mission really is to get past just seeing the equation.

We want you, the listener, to grasp the logic behind it.

How does it emerge from basic physics?

Exactly.

It's not magic.

It comes directly from things like conservation of mass and Newton's laws.

We want to show you how those pieces connect to prove that waves have this characteristic speed.

Sound is just our perfect sort of tangible example here.

Okay, so by the end of this, you should have a solid feel for why things like air particle acceleration are linked to, say, how stretched or squeezed the air is.

Let's start with the basics of wave motion then.

First things first, let's separate two ideas.

You've got oscillation that's just something moving back and forth or up and down in one spot over time.

Think of maybe a single point on the guitar string vibrating.

Okay, just local movement?

Right.

Then you have propagation.

That's when the pattern of oscillation travels through space.

The shape of the disturbance moves.

That moving pattern is the wave.

Got it.

And when these waves travel, they interact, don't they?

The text mentions a couple of key ways.

Yes, two important ones, both based on what we call the principle of superposition.

First, there's interference in time.

You probably know this as beats.

Ah, right.

Like when you tune an instrument, two notes are almost the same frequency.

Precisely.

They go in and out of so the sound gets louder and softer, louder and softer.

That's interference happening over time because the peaks don't always line up.

That rhythmic throbbing.

Okay, what's the second type?

The second is interference in space.

This happens when waves are bouncing around, maybe reflecting off walls in a room.

So the wave meets itself coming back.

Exactly.

And they combine.

This can create really complex patterns,

the most famous being standing waves.

These are stationary patterns, fixed points of high and low amplitude.

Super important for acoustics, musical instruments, all sorts of things.

Okay.

And we also talk about different types of mechanical waves.

Sound in air is?

Sound in a gas -like air is a longitudinal wave.

This means the little parcels of air move back and forth parallel to the direction the sound wave is traveling.

It's a sequence of compressions where air is squeezed and rarefactions where it's stretched out.

Slinky toy analogy, basically.

Pushing one end.

That's a good visual, yeah?

Contrast that with a transverse wave.

Think of ripples on a pond or shaking a rope up and down.

The movement, the displacement, is perpendicular to the direction the wave travels.

Up and down motion, but the wave goes sideways.

Correct.

Now, mathematically, the source points out something really elegant.

Any wave, longitudinal or transverse, that keeps its shape as it travels, can be described as some function, let's call it f, that depends only on the combination x minus ct.

Where x is position, t is time, and c is the speed.

Exactly.

fxxt just means take whatever shape the function f has and shift it along the x -axis by an amount ct.

It's literally a pattern moving rightward at speed c.

If it were 5x plus ct, it'd be moving leftward.

Simple, yet powerful.

Which brings us back to that superposition idea.

Yes, the principle superposition.

It's solution to the physics, and another pattern, maybe gx plus ct, that also works.

Then just adding them together, 5xat plus gsct, is also a valid solution.

That's the key.

It's why different sounds can coexist in the same space without destroying each other.

Your voice and my voice pass right through each other.

Their effects add up where they overlap, but they continue on unchanged.

This linearity is what makes the world sound the way it does.

Okay, general picture

Let's drill down into sound in a gas.

What's the actual physical mechanism?

What's really happening to the air?

Right, so sound starts with some mechanical disturbance of vibrating speaker cone, your vocal cords, whatever.

This disturbance pushes on the air right next to it, momentarily compressing it.

Increasing its density and pressure locally.

Exactly.

Now, this little region of higher pressure pushes outwards onto the air next to it, causing that air to compress, and that region pushes the next, and so on.

The compression, the disturbance,

propagates outwards.

But the air molecules themselves aren't flying across the room, are they?

No, not at all.

That's crucial.

Each little bit of air just moves back and forth a tiny amount around its average position.

It bumps into its neighbors, transfers the momentum, and then maybe moves back.

It's the disturbance that travels, not the air itself over long distances.

Okay, so to describe this mathematically, what do we need to track?

The source mentions three key things.

Yes, three variables, and importantly, we're usually interested in the small changes or deviations from the normal quiet state of the air.

So first, there's the displacement, often called six dollars, which is how far a tiny parcel of air has moved from its equilibrium spot.

How far it's nudged.

Second, the excess pressure, let's call it P dollars.

That's the difference between the actual instantaneous pressure and the normal atmospheric pressure, P dollar.

So P dollars is P dollar, P P dollars.

Right.

And third, the excess density, maybe shores, same idea.

The difference between the actual density toward and the normal equilibrium density toward is toward.

These three things, displacement, excess pressure, excess density, all depend on both position and time.

And the goal of the derivation in the text is to link these three,

somehow get one equation that governs the whole process.

That's exactly it.

We use three fundamental physical relationships to connect them and ultimately eliminate the pressure and density bits to get an equation just for the displacement.

OK, let's walk through those relationships.

What's the first one?

The first is conservation of mass.

Think about a tiny, thin slab of air, initially maybe of width delta X door.

When the sound waves passes, the front face of the slab might move a distance six dubs and the back face moves a distance six plus delta X door.

So the width of the slab changes, it can get stretched or squeezed.

Right.

If the displacement six doll increases as you move along, six ea, meaning the back face moves more than the front face, the slab gets stretched.

If it decreases, the slab gets compressed.

And if it stretches, the density inside must go down, assuming the mass inside is constant.

If it compresses, density goes up.

Precisely.

The change in density, P pourers, turns out to be directly proportional to how much displacement changes with position.

Specifically, it's related to the negative of the spatial derivative, partial X partial X for that links density change to displacement change.

OK, step one, mass conservation links density to the slope of the displacement.

What's step two?

Step two is Newton's second law.

Force equals mass times acceleration, five in a boulder.

We need to find the net force on our little slab of air.

Where does the force come from?

Pressure.

Yes, the pressure difference between the two phases.

If the pressure on the left face is slightly higher than the pressure on the right face, there's a net force pushing the slab to the right.

This net force is proportional to how steeply the excess pressure P tall changes across the slab.

It's related to the negative pressure gradient, partial P partial in Gascaway.

OK, so force depends on the pressure gradient.

And Newton's law says this force must equal the mass of the slab, which is density times its volume, multiplied by its acceleration.

And acceleration is just the second time derivative of the displacement, partial two X partial T two.

So now we have a link.

The pressure gradient causes acceleration, partial P partial X or is proportional to partial two X partial T two two.

Got it.

Mass conservation linked density and displacement.

Newton's law linked pressure and acceleration, which is based on displacement.

We need one more link.

The third piece is the pressure density relationship for the gas itself.

How does the pressure change when you For the small rapid changes in a sound wave, it's essentially direct proportionality.

The excess pressure gap is proportional to the excess density partial.

Pawpropto and the constant of proportionality, let's call it kappa, sort of measures the stiffness or elasticity of the gas, right?

How much pressure you need to cause a certain density change.

Exactly.

Gap is defined as the rate of change of pressure with density evaluated at the equilibrium state dpdx.

Now we have all three pieces.

So how do we put them together?

We play a little substitution game.

We use the pressure density link to replace Pater and Newton's law, and we use the mass conservation relation partial X partial X to replace Pater ball.

So you substitute density out using displacement, then substitute pressure out using density, which ultimately means you substitute pressure out using displacement too.

You got it.

When the dust settles, both people and partners are gone, and you're left with a single equation relating the displacement sex by two doll.

And that is the famous one dimensional wave equation for sound.

That's the one.

And look what it says.

The second time derivative of displacement acceleration is proportional to the second spatial derivative of displacement.

What does that second spatial derivative represent physically?

Like the curvature.

Exactly.

Partial two X partial by two two measures how bent the displacement pattern is in space where the displacement curve is most bent, like at the peak of a compression or the trough of a rare faction.

The acceleration is the greatest.

It tells you how the shape of the wave in space drives his change in time.

That's actually quite intuitive when you put it that way.

The sharper the bend, the bigger the push or pull.

Right.

And that constant factor, kappa who dollar connects them.

Notice it's units.

Kappa is pressure density.

The Harker dollar is the density.

So kappa hood has units of pressure density too, which if you work it out as velocity two.

Uh huh.

So we define that constant as two deco.

Two there goes copper.

We do.

We do.

And the equation becomes partial two X partial T two equals C two partial two X partial by two.

This mathematical structure demands that any solution representing a traveling disturbance must move at this specific speed.

And that the fact that the equation is linear and no terms like six by two or partial X partial T two, that's important to you, right?

Absolutely critical.

Because it's linear, if you have two different solutions, say seven hundred dollars and seven to two, there's some sex by dollar plus by 32 is also a solution.

This mathematically proves the principle of superposition we started with.

It falls right out of the equation.

Okay.

So the equation itself tells us sound travels at a constant speed of dollars where somebody too is the stiffness to density ratio kappa ho dollars.

Now, how do we actually calculate that speed?

There was some historical nuance here.

Yes.

A very important bit of physics history.

Kappa is DPD shoe.

The question is, how does pressure change with density in a quick sound compression?

Newton made the first serious attempt.

He assumed the process was isothermal, meaning the temperature stays constant.

He figured the compressions and expansions happen slowly enough for heat to flow and keep everything at the same temperature.

Seems reasonable, but it didn't work.

It gave a value for the speed of sound and air that was consistently too low, about 15, 20 percent off from measured values.

Nothing was missing.

So who fixed it?

It was Plosu corrected it.

He realized that sound waves oscillate very quickly.

There's simply no time for heat to flow in and out of the compressed or expanded reasons.

The process isn't isothermal.

It's adiabatic, meaning no heat exchange occurs.

Ah, so the temperature does fluctuate slightly within the sound wave.

Compressions get momentarily hotter, rarefactions momentarily cooler.

Exactly.

And when you calculate DPD dollars under adiabatic conditions instead of isothermal, you get an extra factor involved, usually called gamma gamma, the ratio of specific heats for the gas CPCV.

So the corrected stiffness is actually gamma P zero galler under adiabatic conditions.

Well, using the ideal gas law, the adiabatic relationship leads to two dollars gamma P zero galler.

That factor of gamma, which is about 1 .4 for air, fixed Newton's calculation and matched the experimental speed perfectly.

That's a fantastic piece of physics detective work.

Just changing the thermal assumption makes all the difference.

It really does.

And you can take it a step further.

Using the ideal gas law, P dollar S goody key new, where K Boltzmann's constant, T is absolute temperature and whole and can't lose the average molecular mass, you can substitute for TPU.

What do you get then?

You find two dollars and others.

Look at that speed of sound squared is proportional to the absolute temperature.

It doesn't depend on the overall pressure or density, just the temperature and the type of gas through gamma and moves, which matches what we observe and connecting this back to the molecules themselves.

The final insight is beautiful.

If you calculate the typical speed of the air molecules themselves due to their random thermal motion, the root mean square speed, you find it's related to scored in P mu, the speed of sound scored in team is very similar, just multiplied by scored enough.

So the speed of sound is directly related to, but a bit different from the average speed of the molecules carrying it.

Exactly.

It confirms that sound is fundamentally carried by the motion and collisions of the gas molecules.

The disturbance propagates at a speed determined by how fast those molecules are buzzing around and how they transfer momentum during collisions.

Wow.

All right, let's recap quickly.

We started with the general idea of waves and superposition.

Then we focused on sound in a gas identifying displacement, excess pressure and excess density as key variables.

Then we use three core principles, mass conservation, Newton's AFA and the pressure density relation that gets the elasticity to combine everything.

Eliminating pressure and density led us straight to the wave equation acceleration proportional to spatial curvature.

Partial two X partial T two equals C two partial two X partial by two.

The constant E two two involves the gas properties.

And finally, we figured out that calculating an hours correctly required the places insight about the process being adiabatic, not isothermal, bringing in the factor gamma and linking the speed of sound directly to temperature and molecular speeds.

A complete journey from basic principles to a quantitative understanding of sound speed.

Fantastic.

Now for a final thought to leave you with, we derived this elegant wave equation for sound for air molecules bumping into each other.

But think about this, the very same mathematical form partial T two tech something partial T two constant T two partial by two two also describes how light waves propagate through empty space.

Indeed.

In that case, the something isn't particle displacement,

but the electric and magnetic fields and the constant T two two isn't gamma P zero rule, but one dollar Exelon group involving fundamental constants of electromagnetism.

But the underlying mathematical structure, the relationship between time change and spatial change is identical.

It really makes you wonder about the fundamental unity in the way physics describes propagation across vastly different phenomena, doesn't it?

It certainly does.

Are we just finding different physical manifestations of the same core mathematical truths?

Something to ponder.

Definitely food for thought.

Thank you for joining us on this deep dive into the wave equation and the physics of sound.

Keep looking for those connections and we'll see you next time.