The wave a surface is enough to hold

A pipe guides a wave with walls, and refuses anything below its cutoff. A fibre guides with a slower core, and binds at least one mode however weak the contrast. A solid with a free surface guides with neither, and the wave it binds is not a bulk wave in disguise.

The distinction matters because the obvious account is wrong in a checkable way. Total internal reflection needs two media and an angle of incidence. A free surface has one medium and no second one to be internal to, and the wave that runs along it has no angle: it is a solution in its own right, with a speed of its own, that happens to satisfy the free-surface condition.

The condition, and what it leaves #

The equations of an elastic solid carry two waves. A pressure wave, in which the material is compressed along the direction of travel, and a shear wave, in which it is displaced across it. Everything else in an isotropic solid is a superposition of those two.

At a free surface neither is a solution on its own, because neither leaves the surface free: a pressure wave arriving at a surface produces a shear traction on it and a shear wave produces a normal one. What can satisfy the condition is a particular combination of the two, and the combination turns out to require both of them to decay into the material rather than propagate.

Write each with its own exponential decay rate. Requiring the shear traction to vanish at the surface fixes their ratio. Requiring the normal traction to vanish is then a condition on the speed and on nothing else, and it has one root.

The root for a Poisson solid, $\nu = 0.25$, is $0.9194$ shear speeds. It is always less than one, and it has to be: a disturbance travelling along the surface faster than the bulk shear wave could shed energy into the bulk as a shear wave and would not be confined.

What is remarkable is how little the number moves. Poisson’s ratio is the only material parameter in the problem once speeds are measured in shear speeds, and over its whole range the answer stays within seven per cent of $0.92$. A surface wave’s speed is a shear speed, to the accuracy most purposes need, whatever the rock.

The reason two decaying solutions are needed rather than one is worth pausing on, because it is where the wave’s existence comes from. A single decaying pressure disturbance leaves a shear traction on the surface and a single decaying shear disturbance leaves a normal one. Neither can be cancelled by making it smaller; they have to be cancelled against each other, which requires both to be present in a fixed ratio. The surface wave exists because there are exactly two bulk solutions and exactly two conditions to satisfy, and the count works out.

Change that count and the wave disappears. A fluid has no shear wave, so there is only one decaying solution and only one traction condition, and the equation has no root: there is no Rayleigh wave on the surface of a liquid. What runs on water is held up by gravity instead, obeys an entirely different dispersion relation, and is not a relative of this at all.

The skin, and whose length sets it #

Both are evanescent — real exponentials rather than oscillations — and both decay into the material at rates proportional to the wavenumber, so the depth to which the wave reaches is a fixed number of its own wavelengths.

At one wavelength down the vertical displacement is $19.3$ per cent of its surface value. At two it is under four per cent. The wave lives in a skin, and the thickness of the skin is a property of the wave rather than of the ground.

That has a consequence which turns the whole thing into an instrument. A wave of long period has a long wavelength and therefore samples deep rock; a short-period one samples only the near surface. In a uniform half-space there is no dispersion at all — every period travels at the same $0.9194$ — but the Earth is layered, so different periods sample different average material and travel at different speeds. The dispersion of a surface wave is a direct measurement of how the material changes with depth, and it is the standard means of determining crustal structure from earthquakes nobody arranged.

How many modes a layered crust supports is the same question as how many bound states a well holds, and it has the same answer: it depends on how deep the well is, and each state reaches to a different depth. A layered ground therefore carries several surface-wave modes at once, each sampling a different thickness of it, and their separate dispersion curves are exactly what is inverted for structure.

The horizontal displacement’s sign change at $0.193$ wavelengths is a second feature of the profile and it is not a node. A node is a place where the displacement is zero at all times; this is a depth below which the horizontal motion is in the opposite direction. What lies either side of it is a reversal of the particle’s path.

It is worth putting a number on how large the effect is at a scale people meet. Over the depth a two-second surface wave reaches — about six kilometres, since a typical crustal shear speed of three kilometres a second gives a six-kilometre wavelength — the rock’s stiffness varies by a factor of several. A twenty-second wave reaches ten times further and averages over material that includes the base of the crust. The two therefore travel at measurably different speeds, and the difference between them is read as a depth profile without anybody drilling.

Which way the ground goes #

Because the horizontal and vertical displacements are a quarter cycle out of phase, each point of the ground travels round an ellipse as the wave passes. The ellipse is not drawn as an ellipse; it is what the two components make.

Near the surface the sense is retrograde: at the top of its swing the ground moves backwards, against the direction the wave is travelling. That is the opposite of what a deep-water wave does, where the orbit is prograde, and it is the single most useful identification a three-component seismogram offers. A record showing a large, slow, retrograde elliptical motion is a Rayleigh wave and cannot be anything else, in the way that the direction of the shaking identifies a transverse wave without measuring anything else about it.

Below $0.193$ wavelengths the horizontal component has changed sign and the sense reverses. Nothing discontinuous happens there; the ellipse flattens to a vertical line and opens out again the other way.

Why it is the one that does the damage #

The last property is not about elasticity at all, and it is the one that decides what an earthquake feels like.

A body wave leaves the source in all directions and its energy is spread over an expanding hemisphere, so the energy per unit area falls as the inverse square of the distance and the amplitude as the inverse first power. A surface wave is confined to a skin, so its energy is spread round an expanding circle: the area grows as the first power of distance, and the amplitude falls only as the inverse square root — the same counting that decides how any wave thins out, applied to one dimension fewer.

The ratio between them therefore grows without limit. At ten kilometres a surface wave is three times larger than a body wave of the same source strength; at a thousand kilometres it is thirty-one times larger.

That is the whole of why a distant earthquake is felt as a slow roll that goes on for a minute, arriving well after the sharp arrivals that a seismograph records first, and why the damage in a distant city is done by the last waves to arrive rather than the first. It is geometry acquired along the way rather than anything about the rupture.

Confining a wave to fewer dimensions is worth more, over distance, than anything that can be done to the source. A wave spreading in three dimensions loses amplitude as $1/r$, one spreading in two loses it as $1/\sqrt{r}$, and over four decades of distance that difference is a factor of a hundred in amplitude — far more than any plausible change in how the energy was put in.

The same wave, made small #

The identical solution runs on a polished piece of quartz at a hundred megahertz, where the wavelength is tens of micrometres and the skin is tens of micrometres deep.

A surface acoustic wave device works by launching such a wave with an interleaved metal comb on the surface, letting it travel a few millimetres, and picking it up with another comb. Because the wave travels at about $3{,}000$ metres a second rather than $3\times10^8$, a delay of microseconds occupies millimetres instead of kilometres, and a filter whose response is set by the geometry of a printed pattern becomes possible. Several billion of them are made a year.

The reason the surface wave is used rather than a bulk one is exactly the confinement: it stays where the electrodes can reach it and does not disappear into the substrate. The reason the frequency response is so precisely controllable is that the speed is nearly independent of the material’s compressibility, so it depends on the pattern rather than on the batch.

Nothing about this is a different physics from the earthquake. It is the same root of the same condition, at a wavelength eight orders of magnitude smaller.

What it takes to make one on purpose #

A wave that a surface holds is one thing to find on a seismogram and another to launch deliberately, and the second turns out to be surprisingly easy for a reason worth stating.

Anything that pushes on a free surface periodically in space and time excites the wave whose wavelength matches the pushing. Because the surface wave is the slowest solution at a given frequency, its wavelength is the shortest, so a comb of electrodes at a given spacing couples to it and not to the bulk waves, which at that frequency have longer wavelengths and do not fit the pattern. Selectivity comes free from the ordering of the three speeds.

The same ordering is why a hammer blow on a large block of metal puts most of its energy into the surface wave rather than into the bulk. A point source on a free surface delivers roughly two thirds of its energy to the Rayleigh wave, and that figure is nearly independent of the material — which is the laboratory version of the statement about earthquakes and is measured with a hammer rather than inferred.

What the coupling cannot do is reach a wave whose wavelength is much longer than the source. A hammer excites nothing at seismic wavelengths and an earthquake excites everything, which is why the two ends of this subject use entirely different instruments to look at the same solution.

How sharp a surface has to be #

Everything above assumes a surface: a plane at which the material stops. Real interfaces are gradational, and there is a question about how gradual is too gradual.

A transition zone that is thin compared with a wavelength behaves as a sharp boundary, and one that is thick compared with a wavelength barely reflects at all. That is a statement about how the wave sees the interface, not about the interface, and it means the same physical surface can be sharp for one wave and invisible to another.

For a Rayleigh wave the relevant surface is the ground, which stops within a few centimetres at most, so for seismic wavelengths of hundreds of metres it is as sharp as a boundary can be. For a surface acoustic wave at gigahertz frequencies the wavelength is a micrometre, and the top micrometre of a polished wafer is no longer obviously a plane — which is why the roughness specification on such a substrate is quoted in nanometres.

The abrupt case is the limit rather than a separate theory. What an interface reflects, when it has no thickness at all, depends only on the ratio of the two media — a curve with no length in it — and the graded calculation approaches that curve as the transition zone is made thin. Nothing new is introduced at the sharp end; a length simply stops mattering.

What a soft layer does to it #

The geometric argument above says the surface wave arrives largest because it has spread over a circle. It is not the only thing that happens to it on the way, and the other thing is capable of undoing all the essay’s reassuring insensitivity to material.

A surface wave arriving under a basin of soft sediment enters a layer whose shear speed may be fifty metres a second where the rock beneath it is two thousand. Two consequences follow immediately. The wave slows down, and since the energy flux is conserved the amplitude must rise to compensate — a factor of several, from the impedance contrast alone. And the layer, having a boundary at the bottom and a free surface at the top, is a resonator with its own natural period, namely four times its thickness divided by its shear speed, in exactly the way a stopped pipe resonates.

A forty-metre layer at sixty metres a second resonates near two and a half seconds. That is squarely inside the band a distant large earthquake delivers most of its surface-wave energy in.

The consequences were demonstrated in 1985, when a magnitude-eight earthquake off the Pacific coast of Mexico did modest damage near its epicentre and destroyed a large part of Mexico City three hundred and fifty kilometres away. The city’s centre sits on the clays of a drained lake, with shear speeds among the lowest ever measured in a populated place. Ground motion in the lake-bed zone was ten to fifty times that measured on the hills a few kilometres away, and the shaking went on for over three minutes rather than the few tens of seconds the rock recorded.

The damage was also selective in a way that names the mechanism. Buildings between about six and fifteen storeys have natural periods near two seconds; they failed, while shorter and taller buildings on the same streets did not. Three resonances had been brought into coincidence — the source’s dominant period, the basin’s, and the buildings’ — and none of the three had anything to do with the others.

The instrument the skin makes #

At the other end of the scale the confinement is a virtue rather than a hazard, and it is the basis of a standard industrial inspection.

A surface wave carries essentially all of its energy within one wavelength of the surface. That makes it the natural probe for a defect that lives there — a fatigue crack starting at a machined fillet, a grinding burn on a rail head, a weld toe — because a bulk wave sent into the part spends nearly all of its energy where nothing is wrong.

The tuning is the useful part. Launch a Rayleigh wave in steel at two megahertz and its wavelength is about a millimetre and a half, so it interrogates the top millimetre and a half and is nearly blind to anything deeper. Drop to two hundred kilohertz and the same wave reads fifteen millimetres down. The depth being inspected is chosen by choosing a frequency, with no change to the geometry at all, and the reason is the same statement made three sections above: the skin is a fixed number of the wave’s own wavelengths, never a fixed distance in the material.

A surface-breaking crack is then an unusually strong reflector, because it interrupts the whole of the region the wave occupies rather than a small part of it — which is the opposite of the situation for a bulk wave, and is why rail inspection trains, turbine-blade inspection and aerospace fastener-hole checks all use this mode. It also travels several metres along a curved surface without leaving it, so a component can be examined from one accessible spot.

Where the model runs out #

The half-space is uniform and the Earth is not. Everything computed here would be exactly right for a planet of homogeneous rock, and the deviation from it is the entire content of surface-wave seismology. In a layered medium the single Rayleigh wave becomes a family of modes with separate dispersion curves, and the fundamental mode is only the first of them.

There is a second kind of surface wave and it is not here at all. A Love wave is a horizontally polarised shear wave trapped in a layer over a faster half-space, and it needs the layer: it does not exist on a uniform half-space. Both arrive on the same seismogram and they are distinguished by polarisation rather than by speed.

The solid is isotropic. In a crystal, or in rock with a preferred fabric, the speeds depend on direction, the root of the surface condition depends on which way the wave runs, and the particle orbit is not confined to the vertical plane. Surface acoustic wave devices are cut from crystals along specific orientations for exactly this reason.

And nothing here dissipates. Real ground attenuates, and it attenuates the higher frequencies more, which is another reason a distant earthquake arrives as a long roll: the treble has been taken out on the way, by absorption, on top of everything geometric spreading has already done.

What happens at a change of medium is that part of a pulse returns and part continues, with the division set by the mismatch. A surface wave is what remains when the second medium is removed entirely: there is nothing to continue into, and the condition at the boundary stops being a matching of two solutions and becomes a statement that the tractions vanish. That is a different kind of boundary condition, and it is what allows a solution that exists only near the surface.

The ladder from here #

Later rungs on this anchor: the Love wave and the layer it requires; the dispersion curves of a layered half-space and their inversion for structure, which is the working method of crustal seismology; Lamb waves in a plate thin compared with a wavelength, where the two surfaces interact and the single wave becomes an infinite family; and Stoneley waves at the interface between two solids, which exist only for a restricted range of contrasts.

The neighbouring ladders are the channel with no walls, which binds by a slower core rather than by a surface, the pipe that will not carry a low note, which binds by walls and has a cutoff this wave does not, and how a wave thins out, where the geometry of spreading is worked out for its own sake.