A Slinky, a Rope and a Stone: Seeing How Earthquake Waves Move

Earthquakes send out two main kinds of wave through the body of the Earth.

• P waves push and pull. They squeeze and stretch the rock along the direction they travel.
• S waves shake side to side. They shear the rock at right angles to the direction they travel.

P waves can cross solids and liquids. S waves can only cross solids. That single difference is how scientists worked out, from instruments at the surface, that the Earth has a liquid outer core nearly 3,000 kilometres down.

This piece builds that idea from three everyday things: a Slinky, a rope and a stone. No equations are needed to follow it. I include a couple anyway, because they show why the idea works.

What does a Slinky show about P waves?

Lay a Slinky along a table. Squeeze together a few coils at one end and let go.

A bunched-up patch races down the spring. Behind it, the coils relax. Ahead of it, coils get pushed together in turn.

Here is the part that is easy to miss. No coil travels down the table. Tie a scrap of ribbon to one coil and watch it. It jerks forward a little, springs back, and stays roughly where it started. The squeeze travels. The coils only wobble.

That is a P wave:

• The material moves back and forth along the line of travel.
• The wave is a moving pattern of squeezed and stretched regions.
• Nothing is carried along with it.

The "P" stands for primary, because these waves arrive first at a seismometer. Some books say "pressure" or "push-pull". All three help you remember it.

What does a rope show about S waves?

Now take a rope, tie one end to a fence, and pull it fairly taut. Flick your wrist once, sideways.

A hump runs down the rope to the fence. Again, tie a ribbon on and watch. It moves left and right across the rope's length, then settles. The hump travels along the rope while the rope moves across it.

That is an S wave. The material moves at right angles to the direction of travel.

"S" stands for secondary, because these waves arrive second. Some books also say "shear" or "shake". It is not because they are weaker. A common mix-up, and I held it myself for longer than I'd like to admit.

A small, honest caveat about the toys. A Slinky is a spring, and it holds its shape, so you can flick it sideways and send a side-to-side wave down it too. A good Slinky can do both jobs. I find that handy. Do the push-pull trick first, then the flick, and watch how differently the two waves feel in your hands.

Why can't a liquid carry the rope trick?

Now for the thinking. Picture a rope made of water.

You can't. Even in a thought experiment, it falls apart. Flick the end, and the next bit of water has nothing pulling it sideways. Water layers slide past one another and stay wherever they end up. There is no sideways restoring pull, and so no sideways wave.

The stone makes the contrast clear. Imagine two blocks of the same size: one of solid rock, one a tank of water.

Rest a flat plate on top of each and push it sideways, gently.

• The stone: the top layer shifts a tiny amount, then springs back when you let go. Rock resists being sheared, and it remembers its shape.
• The water: the top layer slides, the layers below drag along, and nothing springs back. It does not resist a change of shape for long. It only resists a change of volume.

Now squeeze each block instead of pushing it sideways. Both fight back hard. Water is very reluctant to be compressed, which is why you can't squash a full bottle by pushing the cap.

So we have two kinds of stiffness:

Kind of stiffness	What it resists	Solid rock	Liquid
Squeezing (bulk)	Change of volume	Yes	Yes
Shearing (rigidity)	Change of shape	Yes	No

A P wave only needs the first kind. An S wave needs the second.

That is the whole secret. The rest is detective work.

Why are P waves faster than S waves?

Here is where a little arithmetic earns its place.

For a wave in a solid, the speed depends on stiffness divided by density. Let's call:

• K the resistance to squeezing (the bulk modulus),
• G the resistance to shearing (the shear modulus),
• ρ the density.

Then:

• P wave speed = √( (K + 4G/3) ÷ ρ )
• S wave speed = √( G ÷ ρ )

Notice what this says. A P wave in a solid feels both kinds of stiffness, because pushing along a solid also tries to change its shape a little. An S wave feels only the shear stiffness. The P wave has more stiffness helping it along, so it is quicker.

A worked example. Many rocks behave roughly like an idealised solid where K = 5G/3. (Geologists call this a Poisson solid. Real rocks vary, so treat this as a handy approximation, not a law.)

• K + 4G/3 = 5G/3 + 4G/3 = 9G/3 = 3G
• P speed = √(3G/ρ)
• S speed = √(G/ρ)

Divide one by the other and the G and ρ cancel. The P wave is √3 times faster, which is about 1.7 times. In many real rocks, P waves do run about 1.7 times faster than S waves. I find that tidy: a single ratio, from a short bit of algebra, roughly matches the ground.

Now try water. Water has no shear stiffness, so G = 0.

• S speed = √(0 ÷ ρ) = 0. There is no S wave at all.
• P speed = √(K ÷ ρ).

With K about 2.2 billion pascals and ρ about 1,000 kilograms per cubic metre, that gives √(2,200,000) ≈ 1,480 metres per second. That is about 1.5 kilometres per second, which is the speed of sound in water. Sound is a P wave. It's a pleasing link between a seismometer and a swimming pool.

How do the two waves help locate an earthquake?

Because P waves are faster, they race ahead. S waves trail behind. The gap between the two arrivals tells you how far away the earthquake was.

Here is a worked example with round, made-up numbers. Say P waves travel at 6 km/s and S waves at 3.5 km/s through the crust. Real rocks vary, so these are for illustration only.

For an earthquake 100 km away:

• P arrives after 100 ÷ 6 ≈ 16.7 seconds.
• S arrives after 100 ÷ 3.5 ≈ 28.6 seconds.
• The gap is about 12 seconds.

Double the distance and the gap doubles. Each second of gap works out to roughly 8 km of distance in this example. Real seismologists use travel-time tables built from many recordings, and they combine readings from several stations to pin down where the earthquake began.

If you have ever felt a small sharp jolt followed a moment later by a rolling sway, you have probably noticed this yourself. The jolt is roughly the P wave. The sway is roughly the S wave. A longer gap means a more distant source.

One more detail, so the picture is honest. After P and S, a third group of slower waves travels along the surface. They do not pass through the deep Earth, so they do not feature in the core story. They are a separate topic.

How did S waves reveal a liquid core?

Now we put the pieces together.

When a large earthquake strikes, seismometers all over the world record it. Some stations are close and some are on the opposite side of the planet. Each one sees waves that took a different path through the Earth's interior.

Early in the twentieth century, seismologists noticed something odd in the records.

• S waves vanished. Beyond roughly 103 degrees of arc from the earthquake (measured around the Earth's centre), no direct S waves arrived at all.
• P waves had a shadow zone. Between roughly 103 and 142 degrees, direct P waves were also missing or very faint. Past that, they came back, but later than expected.

Think about the S waves first. A wave only goes missing if something stops it. Waves that skim through the outer layers reach nearby stations without trouble. The waves that dive deepest hit something at the centre that will not let a shear wave through. A liquid will not carry S waves. So somewhere deep inside there is a large region that behaves as a liquid.

Now the P waves. A wave bends when its speed changes, the same way light bends entering water. When P waves meet the core, their speed drops sharply. From roughly 14 km/s at the base of the mantle, the speed falls to something like 8 km/s just inside the core. Our formula explains much of why: the shear stiffness term (the 4G/3 part) disappears in a liquid, so the wave loses a big helping hand. The waves bend sharply downward on entry, and the bending leaves a patch of the surface they never reach. That patch is the shadow zone.

Here is how I understand the history, though I'm still learning it, so treat names and dates as a rough outline:

• Richard Oldham, in the early 1900s, recognised that records pointed to a distinct central region that slowed the waves.
• Beno Gutenberg later worked out how deep that boundary sits, at about 2,900 km below the surface.
• Harold Jeffreys, in the 1920s, made the case that the core is liquid, leaning heavily on those missing S waves.

The core's outer edge is at about 2,900 km depth. Since the Earth's radius is about 6,371 km, the core has a radius of roughly 3,470 km. It is a bit over half the width of the planet.

Is the Earth's mantle liquid too?

No, and this is a common misconception worth clearing up.

S waves cross the mantle without trouble. So on the timescale of a seismic wave, which is seconds, the mantle is a solid. It resists shearing and springs back.

But rock can also creep, very slowly, over millions of years. Think of cold honey or a glacier: it holds its shape when you tap it, yet it flows if you wait long enough. The mantle is a bit like that. The same material is "solid" to a wave and "slowly flowing" to the long, patient forces that move continents.

So there is no contradiction. The answer depends on how fast you push. That connects the seismometer to the slow motion of plate tectonics, which I find quietly wonderful. One ground, two behaviours.

What about the solid inner core?

There is one more twist.

The shadow zone was not completely empty. Faint P waves showed up in places where the simple picture said there should be none. Inge Lehmann, in the 1930s, proposed that these came from a small, distinct region at the very centre that reflected waves back out. In other words, inside the liquid outer core sits a smaller inner core, and it behaves as a solid.

That seems strange at first, since the centre is the hottest place of all. The usual explanation, and it is textbook-standard, is pressure. The pressure at the centre is enormous, and higher pressure raises the temperature at which iron-rich material melts. So the inner core can be hotter than the outer core and still be solid.

The story so far, from the surface inwards:

1. Crust and mantle: solid. Both P and S waves pass.
2. Outer core: liquid. P waves pass (slowed), S waves stop.
3. Inner core: solid. Waves can reflect from it, and it is thought to be iron-rich.

Notice the method as well as the result. Nobody has drilled anywhere near the core. The deepest boreholes barely scratch the crust. Everything we know here came from listening to how waves behaved, and from understanding why a rope and a pond behave differently.

Try it yourself

You don't need a laboratory.

• Slinky: stretch it along the floor. Squeeze a few coils and release, then flick it sideways. Tie a ribbon on and watch how the coil moves in each case.
• Rope: make a hump, and see how it flips when it reaches a fixed end.
• A tray of jelly: tap the side and watch it wobble. It carries both kinds of wave, because it is a very soft solid. It is a fun way to see a shear wave that is slow enough to follow with your eyes.
• A bath or basin: tap the wall underwater and listen. You are hearing a P wave through a liquid. Trying to shake the water sideways and send a "rope wave" down it will not work, which is the point.

I'll admit my first attempt with a Slinky was clumsy. I flicked it so hard the whole thing flew off the table. A gentle push is enough.

What is still unknown about the Earth's core?

Plenty. The broad picture above is well established. The details are still being worked out.

• Scientists are still refining how the inner core formed and how fast it grows.
• They are still working out what exactly the core is made of beyond iron, and in what proportions.
• Waves seem to travel at slightly different speeds through the inner core depending on direction, and the full explanation is still debated.

I don't know enough to say which answers will win. I do find it a good sign that after more than a century, a handful of wiggly lines on paper still raise fresh questions.

So here is something to wonder about next time you pick up a stone. It resists both a squeeze and a twist, and you can feel that in your hand. Down there, under thousands of kilometres of the same kind of rock, there is a place where one of those two abilities simply switches off, and the Earth's shaking told us so from the surface.

If you spot a mistake in anything here, I'd be glad to hear it, and thank you for reading.