Chapter 51: Waves – Shock, Bow & Surface Phenomena

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Welcome back to the Deep Dive.

Today we're doing something a bit different.

We're setting aside the heavy maths for a moment.

Our mission is to get into the, well, the feel of the physics in chapter 51 of the Feynman Lectures.

Purely the qualitative logic behind waves.

We're really embracing that Feynman spirit, you know, trying to understand the why behind the phenomena first.

Because if you get that intuition, the math just becomes the description and the thread connecting everything today.

It's this idea that wave speed is rarely just a simple number.

It's almost always complicated, depends on the medium, the size, the source, lots of things.

That's a great way to frame it.

This chapter dives into four really distinct complex wave types.

We've shock waves, waves and solids, and surface waves.

And they all challenge that simple picture of waves we might have.

Our goal here is to pull out the core ideas for each one.

Get that intuitive grasp Feynman was always aiming for.

Okay, sounds good.

Let's unpack this.

Where better to start than with that classic image, the bow wave, that V shape you see behind a boat?

So what happens geometrically when the thing making the waves, the source, is actually moving faster than the

waves it creates spread out at CW.

If V is bigger than CW, will you get that characteristic V shape or really a cone in three dimensions?

Think about it like this.

The source emits waves continuously,

like expanding circles or spheres.

But because the source keeps moving forward faster than those circles expand, it's always ahead of where the wave started.

The waves kind of pile up along a line.

Each wave pulse, no matter when it was emitted, ends up touching a common tangent line.

And that line forms the edge of the cone.

Okay, so it's purely about the geometry of motion, the source outpacing its own disturbance.

Exactly.

And there's a precise relationship.

The angle of that cone, specifically the half angle, let's call it theta, is directly tied to the speeds.

The sine of that angle, sin theta, is simply the wave speed, CW, divided by the source speed, V.

So sin theta equals CWV, which means obviously V has to be greater than C do for this to even form a real angle, right?

Otherwise, sine would be greater than one.

Precisely.

That condition, VCW, is built right into the geometry.

It applies universally boats, jets, anything moving faster than its wave speed.

Okay, and this is where Feynman throws in that really cool kind of mind -bending example.

We usually think sonic booms, right, airplanes.

But he talks about light creating a bow wave.

How does that even work?

Isn't light the ultimate speed limit?

Right, that sounds impossible at first, but the key detail is where the light is.

We're talking about light moving through a dense medium, like glass or water.

In a medium, light slows down.

Its speed, what we call the phase velocity in that specific material, is less than C, the speed of light in a vacuum.

Okay, so phase velocity is just light speed in the material.

Got it.

Exactly.

Now, it is possible for a very high energy charged particle, say an electron, to travel through that glass faster than the light's phase velocity in that glass.

Crucially, the particle is still moving slower than C, the vacuum speed limit, so no laws of relativity are broken.

Wow.

So the particle outpaces the light waves it's generating within the glass.

Yes.

And when that happens, it creates an electromagnetic bow wave, a cone of light.

It's the light equivalent of a sonic boom.

That's incredible.

The same fundamental geometry applies to a speed boat and a subatomic particle moving through glass.

It really shows the power of these basic principles.

And this phenomenon, it's related to Cherenkov radiation, which is actually used in particle physics.

How so?

Well, by measuring the angle of that light cone, experiments can determine the speed of those super fast particles.

It's a practical application of this wave geometry.

Amazing.

Okay, so that's the relatively well -behaved bow wave, all neat geometry.

But what happens when the disturbance is much more violent, when that neat cone idea breaks down?

Ah, then you step into the realm of the shock wave.

Think sonic boom, but often much more intense.

A shock wave isn't just a bunch of sound waves piling up nicely.

It's fundamentally different because it's a highly non -linear process.

You get these incredibly sharp fronts, almost instantaneous jumps in pressure, temperature, and density.

How sharp are we talking compared to, say, a normal sound wave?

The difference is huge.

Normal sound involves tiny pressure fluctuations,

maybe one part in a million of atmospheric pressure, or even less, a strong shock wave.

You can have pressure jumps that are many times the atmospheric pressure happening across a microscopic distance.

Okay, that explains the destructive power, like from explosions, that sudden, massive pressure increase.

Precisely.

Now, understanding the physics inside that jump is fine.

Feynman uses a great analogy, the hydraulic jump.

If you have water flowing fast in a shallow channel and it suddenly piles up into a turbulent wall of water, a bore.

Exactly that.

Imagine a piston pushing water in a channel, creating that bore.

Analyzing what happens across that jump is very revealing.

You apply conservation laws, conservation of mass, momentum, and energy to the water flowing into the jump and coming out the other side.

What does that analysis show?

It shows something crucial.

There's an apparent loss of mechanical energy across the shock front.

The energy of the smooth flow going in seems higher than the energy of the turbulent flow coming out.

Apparent loss.

So the energy doesn't just vanish?

No, of course not.

Energy is conserved.

But that lost mechanical energy is instantly converted into other forms, mainly heat and the kinetic energy of turbulence.

You see that turbulence visually in the hydraulic jump, all that churning water behind the bore.

That's dissipated energy.

So the shock wave fundamentally involves dissipation.

It turns organized energy into disorganized heat and chaos.

Yes, and that makes it an irreversible process thermodynamically speaking.

You can't just run a shock wave backward.

This dissipation is what allows the shock to be stable.

It's not just fast sound, it's a distinct physical phenomenon.

Okay, that makes sense.

We've looked at fluids and gases, but what if the wave is moving through something rigid, something that can resist being pushed sideways?

What about solids?

Ah, solids introduce a whole new dimension, literally.

Because solids have elasticity, they resist not just compression, but also shear.

They resist twisting or sliding motions.

And fluids don't really resist shear, right?

They just flow.

Exactly.

And that ability to resist shear allows solids to support two distinct types of waves, unlike fluids, which mainly just support compression waves.

Two types.

Okay, what are they?

First, you have the familiar longitudinal waves, sometimes called P waves, for primary or pressure.

Here, the particles of the solid move back and forth parallel to the direction the wave is traveling.

It's a compression and expansion, just like sound and air or water.

Okay, that's like sound.

What's the second type?

The second type is transverse waves, also called S waves, for secondary or shear.

In these waves, the particle motion is perpendicular or transverse to the direction the wave travels.

So it's like shaking a rope up and down.

The wave travels along the rope, but the rope itself moves side to side.

Precisely.

It's a shearing motion.

And only something with rigidity, like a solid, can really support that kind of wave effectively.

Fluids can't sustain shear like that.

That distinction seems really fundamental.

What's the big implication of solids supporting both compression and shear waves?

It's enormous.

It's basically how we understand the inside of our own planet.

Think about earthquakes.

Okay, earthquakes generate waves, right?

Right.

They generate both P waves and S waves that travel out through the earth.

Now, these two types of waves travel at different speeds.

P waves are faster, and critically, they behave differently depending on the material they go through.

P waves can travel through solids and liquids, but S waves, the shear waves.

They can only travel through solids because liquids don't resist shear.

Exactly.

So seismologists set up detectors all around the world.

They measure the arrival times of the P waves and S waves from distant earthquakes.

By comparing the arrival times and noticing where the S waves don't arrive when they should, they can map out regions inside the earth that must be liquid.

Wow.

So the fact that S waves disappear when they hit the outer core tells us directly that the outer core is liquid metal, not solid.

That's the key insight.

The simple fact that solids support two wave types gives us this incredible tool to probe the deep earth, places we can never physically go.

That's a fantastic application.

All right, let's shift gears one last time.

We've done cones, shocks, solids.

What about the waves we see most often?

Water waves on the surface.

Right.

Surface waves, specifically water waves.

These are actually quite complex too.

The water particles near the surface don't just move up and down or back and forth.

They actually move in circles, combining both longitudinal and transverse motion.

But the most fascinating thing Feynman highlights here is dispersion.

Dispersion.

That sounds important.

What does it mean in this context?

It means the speed of the water wave is not constant.

It depends fundamentally on the wavelength of the wave.

Different wavelengths travel at different speeds.

Okay.

Unlike sound in air, where long and short wavelengths travel at roughly the same speed mostly.

Correct.

For water waves, this wavelength dependence is crucial and it forces us to distinguish between two different kinds of velocity.

Two velocities.

What are they?

First, there's a phase velocity.

That's the speed of an individual crest.

If you follow one specific peak of a wave, that's its phase velocity.

But then there's the group velocity.

This is the speed at which the overall group of waves, or more importantly, the energy of the wave packet,

travels.

Hmm.

Phase and group velocity.

That can be confusing.

How do they relate?

Are they the same?

In a dispersive medium like water, they're generally not the same.

The group velocity, the speed of the energy, is usually different from the speed of the individual crests.

Okay, Feynman uses is perfect.

Think about the V -shaped wake behind a boat.

That whole V, the whole disturbance pattern, moves forward at the group velocity.

Right.

The wake as a whole moves with the boat, more or less.

But if you look closely at the individual ripples within that V shape, you'll see something strange.

New crests seem to appear at the back of the V.

They then travel forward through the V shape, moving faster than the V itself, and eventually disappear off the front edge of the V.

Ah.

So the individual crests, phase velocity, are moving faster than the group velocity.

They travel through the energy packet.

Exactly.

The energy stays localized in the wake pattern, moving at the group speed, while the individual phases race through it.

If this didn't happen, if energy didn't travel at a distinct group velocity, things like signal transmission would be impossible.

Okay, that makes the distinction clearer.

Yeah.

And what determines these speeds in water?

Why does the wavelengths matter?

It comes down to the restoring force, what pulls the water back to flat when it's disturbed.

There are two main forces at play.

For very long waves, like ocean swells, the main restoring force is gravity.

Gravity tries to pull the high parts down and push the low parts up.

Makes sense.

What about short waves?

For very short waves, tiny ripples, the dominant force isn't gravity anymore, it's surface tension.

The skin of water, the capillary effect, tries to flatten the surface.

So, gravity for long waves, surface tension for short waves.

How does that affect speed?

Well, the equations show that waves dominated by gravity tend to move faster if they have longer wavelengths.

But waves dominated by surface tension move faster if they have shorter wavelengths.

If you plot wave speed versus wavelength, you find there's actually a minimum possible speed for water waves, somewhere in between the very long and very short wavelengths, where neither gravity nor surface tension is strongly dominant.

A minimum speed.

So, there's a certain wavelength that travels slower than any other.

Yes, around a couple of centimeters wavelength, actually.

It's a fascinating consequence of these competing forces.

It just underscores, again, how wave speed depends intimately on the properties of the system.

Wow.

Okay, so we've covered a lot of ground.

We went from the pure geometry of bow waves and that weird Cherenkov light.

Through the messy, irreversible energy loss in shock waves.

Into solids.

Yeah.

Seeing how their rigidity gives us P and S waves, which let us probe the Earth.

And finally, landed on water waves.

With their complex motion and the crucial difference between phase and group velocity driven by wavelength.

It really drives home that point of wave isn't just one simple thing.

The physics changes dramatically depending on the situation.

Whether it's light speeding glass or the structure of our planet, it's hidden in wave behavior.

Absolutely.

The qualitative understanding, the why, is so important.

So, the big takeaway for you, listening, is probably just how complex and context dependent wave speed really is.

It's not just one number.

Which leads to maybe a final thought to ponder.

We've seen wave speed depend on source speed, medium density, wavelength,

amplitude, energy dissipation, all sorts of things.

So, next time you just hear a single number quoted for, say, the speed of sound, maybe 343 meters per second in air, ask yourself,

what assumptions are baked into that number?

What does it assume about the air?

About the loudness of the sound?

About whether we're talking phase or group velocity?

About energy loss?

It's usually a simplification of a much richer reality.

It's a great point to end on.

Thinking about the assumptions behind the simple numbers.

Well, thank you for joining us on this deep dive into the physics of waves, Feynman style.

We'll see you next time.