The shaking a uniform Earth could not carry

The shadow that shows the core is liquid read the Earth’s interior from body waves — compressions and shears travelling through the planet along curved rays — and in particular from where they failed to arrive. Body waves are only the beginning of a seismogram. After them, and much larger, come the surface waves, which spread over the surface rather than through the volume and so fade far more slowly with distance, as how a wave thins out worked out: as one over the square root of the distance rather than one over the distance itself.

Rayleigh had predicted one surface wave in 1885, and the wave a surface is enough to hold computed it: a rolling motion in the vertical plane, bound to the free surface of a uniform solid, travelling at a little over nine-tenths of the shear speed and with no dispersion at all. When good horizontal seismographs became common in the early twentieth century, they showed something else as well — a large shaking at right angles to the direction from the earthquake, horizontal and transverse, arriving before the Rayleigh wave. A Rayleigh wave has no transverse motion. Nothing in a uniform Earth could produce this one.

Augustus Love explained it in 1911, and his explanation needed one thing a uniform Earth does not have: a layer. This essay computes Love’s wave for a continental crust 35 km thick, with a shear speed $\beta_1$ of 3.5 km/s, lying on a mantle whose shear speed $\beta_2$ is 4.5 km/s — and then shows how the wave turns the crust’s thickness into a pattern of arrival times that any seismograph records.

A wave a free surface cannot hold #

Shear waves come in two polarisations. Picture a shear wave travelling along the surface and slightly downward. If its particles move in the vertical plane containing the direction of travel, it is called SV; if they move horizontally, at right angles to the travel, it is SH. At a free surface the two behave completely differently.

An SV wave striking the surface has to make two tractions vanish — the push across the surface and the drag along it — and it cannot do so alone. It reflects as a mixture of SV and compression, and a suitable combination of the two, with both parts decaying downward, is exactly the Rayleigh wave. The coupling between kinds is what makes binding possible.

An SH wave has only one traction to satisfy: the horizontal drag on the surface, $\mu\,\partial v/\partial z$, where $v$ is the sideways displacement. It can make that vanish by itself, and it does, reflecting completely as SH at every angle, with no conversion. Total reflection sounds like the beginning of a trap, but there is nothing on the other side to make it one. A wave that reflects perfectly from the top of a uniform half-space and then travels on downward is simply a wave going back into the ground. No combination of SH waves in a uniform solid decays with depth while satisfying the free-surface condition, so a uniform Earth carries no surface wave that shakes sideways. The observed transverse shaking was a sign the Earth is not uniform near its surface.

Two mirrors and a zig-zag #

Put a slow layer on top of the half-space and an SH ray inside the layer now meets two surfaces. The top is the perfect mirror it always was. The bottom is the boundary with faster rock, and there an SH wave meeting it obliquely enough cannot get through: by Snell’s law, a ray at angle $\theta$ from the vertical in the crust continues into the mantle at an angle $\phi$ with $\sin\phi = (\beta_2/\beta_1)\sin\theta$, and once $\sin\theta$ exceeds $\beta_1/\beta_2$ there is no such angle. The wave is totally reflected, as light is in the core of a fibre and for the same reason, which the angle at which total reflection stops being total took apart in the optical case.

For this crust the critical angle is $\arcsin(3.5/4.5) = 51.1°$ from the vertical. A ray steeper than that leaks part of itself into the mantle at every bounce and dies out; a ray flatter than that zig-zags between the two mirrors without loss. Along the surface, the crests of a trapped zig-zag move at $\beta_1/\sin\theta$, which runs from 4.5 km/s at the critical angle down to 3.5 km/s for a ray skimming the surface. Every Love wave travels between the two shear speeds, and a speed outside that range is impossible.

Not every flat angle works. A wave that zig-zags must come back into step with itself after each round trip from top to bottom and back, or its successive reflections cancel. The phase it gains on the way down and up, plus the phase shift of total reflection at the base, must be a whole number of cycles. That condition, worked out exactly, is Love’s dispersion relation:

$$\tan\!\left(kH\sqrt{\frac{c^2}{\beta_1^2} - 1}\right) = \frac{\mu_2\sqrt{1 - c^2/\beta_2^2}}{\mu_1\sqrt{c^2/\beta_1^2 - 1}},$$

where $c$ is the phase velocity along the surface, $k$ the wavenumber, $H$ the layer’s thickness and $\mu$ the two rigidities. It is the slab waveguide of the channel with no walls with one change. Because the free surface is a perfect mirror with a peak of motion on it, the crust behaves as the upper half of a slab twice as thick, mirrored about its own surface, and only the patterns that are symmetric about that mirror survive. The channel essay’s central result carries over: the symmetric fundamental mode of a slab has no cut-off. However thin the crust and however small the contrast, there is always a Love wave, and at long enough periods it always exists.

Speeds that depend on the period #

Since the dispersion relation involves $kH$ and $c$ together, each value of $c$ between the two shear speeds picks out a wavenumber, and so a period, for each mode. The figure below draws the result for the fundamental mode and for the first overtone, whose zig-zag fits one extra half-wavelength of shaking into the thickness.

At short periods the fundamental travels at 3.51 km/s, essentially the crust’s own shear speed, and at long periods it approaches the mantle’s, 4.48 km/s at 200 s. The overtone enters at a cut-off, as every higher mode of every guide does: it appears at 12.6 s, at the mantle’s shear speed, where the crust holds exactly one half-wavelength of vertical variation for a ray at the critical angle, and at longer periods it does not exist. Its cut-off period is $2H\sqrt{\beta_2^2/\beta_1^2 - 1}/\beta_2$, which contains the thickness directly. A higher overtone enters at a half, a third, a quarter of that.

The dispersion is the whole of the wave’s use. A Rayleigh wave on a uniform half-space was the same speed at every period, because a half-space contains no length. A Love wave is bound by a layer, the layer has a thickness, and every number on the chart is set by how the wavelength compares with that thickness. The curves are the crust’s thickness written as a function of period.

Deeper waves run faster #

Why long waves travel faster becomes clear when the shaking is drawn against depth. In the crust the motion is a standing cosine across the thickness, with its maximum at the free surface, where the drag must vanish; below the base it dies away exponentially, the evanescent tail every totally reflected wave carries.

The 8-second wave keeps 15 per cent of its surface motion at the base and below that dies away within 7 km: it is almost entirely a crustal wave, and travels at almost the crust’s speed, 3.56 km/s. The 80-second wave keeps 89 per cent at the base and penetrates 276 km into the mantle before falling by a factor e: most of its energy is in faster rock, and it travels at 4.41 km/s. The wave’s speed is a weighted average of the speeds it samples, and the period chooses the weights.

This is the most important fact about surface waves as instruments. Body waves sample the Earth along rays, and a ray’s depth is set by where it was aimed. A surface wave samples a depth range set by its period, so that a single recording, decomposed by period, holds a profile of the ground with depth. Short periods report on the crust, longer ones on the upper mantle, and the longest — Love waves of several minutes’ period, which circle the planet repeatedly after a great earthquake — on the top few hundred kilometres of the mantle everywhere they pass. Those waves are slower than the formula for a flat crust predicts, because the Earth’s curvature and the mantle’s own increase of speed with depth both matter at such lengths, but the principle is the same.

The order of arrival on a seismogram #

A seismograph does not record phase velocity. A shaking with energy over a range of periods breaks up into its periods as it travels, each arriving at its own group velocity — the speed of energy, which the packet that moves at another speed distinguished from the speed of crests. The group velocities on the dispersion chart are lower than the phase velocities at every period, and at 19 s the fundamental’s group velocity falls to a minimum of 3.39 km/s, below even the crust’s shear speed: energy at those periods travels more slowly than any shear wave in either rock, because a wave whose speed is changing rapidly with period is, in effect, rebuilding its shape across the depth as it goes.

The figure synthesises what a transverse seismograph records 3,000 km from a sudden sideways slip at the surface. Each period between about 8 and 120 seconds is sent along the surface with the wavenumber the computed dispersion gives it, and the results are added.

Nothing arrives before 11.0 minutes, the travel time at the mantle’s shear speed, which is the fastest any Love wave can go. Then the long periods come in — slow, gentle swings, 80-second waves at 11.8 minutes — and the period of the shaking shortens steadily through the train as the slower energy arrives: 30-second waves at 14.3 minutes. The largest motion comes last, near 14.8 minutes. At the minimum of the group velocity a whole band of periods travels at nearly the same speed and so arrives at the same moment, piling their energy into one burst. Seismologists call such an arrival an Airy phase, after the mathematics of the stationary point. Water waves have an extremum of their own where gravity gives way to surface tension, which the speed below which nothing on water leaves a wake found in their phase velocity; the extremum that makes this burst is in the group velocity, and it is the slowest energy rather than the slowest crest.

The arrival order is the signature by which Love waves are recognised: a transverse train whose period shortens with time, beginning at the mantle’s shear speed and ending a little below the crust’s. The shaking is entirely horizontal, which is why Love waves are among the most damaging motions in an earthquake: structures are designed first to carry their own weight, vertically, and are weakest against being dragged from side to side.

A thickness read from timing #

The only length in Love’s equation is the crust’s thickness, so changing it rescales the whole dispersion curve along the period axis. Double the thickness and every feature of the curve moves to periods twice as long.

At a period of 30 s the group velocity is 3.94 km/s through a 20 km crust, 3.49 through 35 km and 3.40 through 70 km. Over 3,000 km those speeds differ in arrival time by more than two minutes, an enormous difference on a seismogram that is timed to a fraction of a second. The minimum of the group velocity, the Airy phase, moves from 11 s to 19 s to 38 s as the crust thickens. Measuring the arrival time of each period between two stations, and fitting a layered model whose dispersion matches it, is the working method of surface-wave seismology. Its first great result was the difference between oceans and continents: surface waves crossing the Pacific arrive earlier at a given period than those crossing a continent, which showed the crust beneath the ocean to be only a few kilometres thick against the continents’ thirty to fifty. The same measurement finds the 70 km crust beneath Tibet, the thickest on Earth, as a wave that comes late at every period.

The inversion is not unique in its details. A dispersion curve constrains averages over depth ranges rather than sharp boundaries, so a gradual increase of speed and a sharp step can give nearly identical curves; and a curve measured between two stations is an average along the path. Seismologists combine many paths into maps, and Love waves with Rayleigh waves, whose dispersion depends on the same structure through different weights. The thickness is not read off one figure, but the figures show where the information sits.

Where a flat layer is too simple #

The model computed here is a single uniform layer on a uniform half-space, flat, perfectly elastic and isotropic. The real crust has layers within it — sediments that are much slower still near the surface, an upper and a lower crust — and the mantle’s speed changes with depth. Each addition changes the curve’s details, and modern inversions use dozens of layers, but none changes the physics: total reflection at a speed increase, a perfect mirror at the free surface, and a period that sets the depth sampled. A soft sedimentary basin, a few kilometres of slow material, traps short-period Love waves so strongly that cities built on such basins shake longer and harder than those on rock nearby, which is the reason seismic building codes take the ground beneath a site into account.

The flat model also omits attenuation, which damps the short periods more than the long ones and is why distant seismograms are dominated by long-period surface waves, and the sphericity of the Earth, which matters for periods above about a minute. And it treats the source as a simple sideways slip; a real earthquake radiates Love and Rayleigh waves in a pattern set by its fault’s orientation, which is how the direction of slip on a distant fault is found from its surface waves.

Reciprocity, the principle of swap the ends and nothing changes, guarantees one useful thing about all this: the dispersion measured from an earthquake at A to a station at B is the dispersion from B to A, whatever the layering in between. It is what allows the background seismic noise — ocean waves pounding coasts, which keep the ground trembling everywhere at periods of several seconds — to be cross-correlated between two stations to recover the surface waves that would have travelled between them, so that crustal structure can be mapped from the noise without waiting for earthquakes.

Still open: the speed that depends on the direction of shaking #

When Love waves and Rayleigh waves are measured along the same path, a strange discrepancy appears. Each can be fitted with a layered isotropic model, but not with the same one: the shear speed that fits the Love waves, which shake horizontally, is consistently a few per cent faster in the upper mantle than the one that fits the Rayleigh waves, which shake in the vertical plane. The mantle carries horizontally polarised shear faster than vertically polarised shear. This radial anisotropy is generally attributed to olivine crystals aligned by the slow flow of the mantle, and is now built into reference Earth models; how deep it extends, how it varies between ocean and continent, and how much of it comes from aligned crystals and how much from fine layering of different materials, are still argued from the same dispersion measurements.

On other worlds, the question is whether there is a crust at all, and how thick. The seismometer landed on Mars in 2018 recorded surface waves from meteorite impacts and marsquakes, and their dispersion put constraints on the Martian crust beneath the lander — a few tens of kilometres, layered — from the same arithmetic as this essay’s, applied to a planet with a different gravity and different rock.

The habit worth carrying away is to ask what a boundary can hold by itself and what needs a second boundary. A free surface reflects horizontal shear perfectly and binds none of it; add a slower layer and total reflection at its base completes the trap, with no cut-off for the fundamental, a speed always between the two rocks’ shear speeds, and a period that chooses how deep the wave reaches. Long waves sample the mantle and arrive first, short ones sample the crust and arrive later, and the timing of the train, period by period, is the thickness of the ground it crossed.