The wave that arrives first by going the long way

Every front is a source set out Huygens’s rule: treat each point of a wavefront as a source of wavelets and take their envelope to find where the front will be. It derived straight-line propagation, reflection and refraction from that one rule. Later essays gave the wavelets amplitudes and phases and followed the rule to diffraction at a rim, to the backward wave Huygens had to remove, to the plane-wave fan inside a beam and, in the mirror that sends a wave back to where it began, to a recording played backwards.

This essay uses the rule in its oldest form, the envelope of wavelets, on a problem where it predicts something the ray picture of refraction misses: a wave that exists only because a boundary carries a disturbance faster than the medium above it can. It is the wave by which the thickness of the Earth’s crust was first measured, and by which engineers still find the depth to bedrock before they build.

A front drawn by a racing boundary #

Fire a shot at the surface of a layer of rock, of thickness h and wave speed $v_1$, lying on rock with a higher speed $v_2$. The wave spreads into the layer as a hemisphere. Where it meets the boundary it refracts into the faster rock, bending away from the vertical, as the bend at the boundary found. At one angle of incidence, the critical angle $\theta_c = \arcsin(v_1/v_2)$, the refracted ray runs exactly along the boundary, and beyond it there is no transmitted ray at all, only total reflection.

The ray picture stops there. Huygens’s construction does not. The disturbance at the boundary below the critical point is running along the boundary at the faster speed $v_2$, and every point of the upper layer that it passes is set moving, so every point of the boundary becomes in turn a source of wavelets in the upper layer, spreading at $v_1$. A source moving at speed $v_2$ through a medium whose waves travel at $v_1 < v_2$ leaves behind it a cone whose half-angle has sine $v_1/v_2$ — exactly the cone the source leaves behind for a supersonic aircraft, and exactly the critical angle. The envelope of the wavelets is a plane front leaning at that angle, travelling up towards the surface. That is the head wave.

The head wave is the Mach cone of a refracted disturbance. Nothing in the upper layer moves faster than $v_1$; what moves at $v_2$ is the place along the boundary where the wavelets are being launched, as the bright line where a breaking wave meets a sea wall can run faster than any water in it.

Three arrivals, and one that overtakes #

A line of receivers laid out from the shot records three waves. The direct wave travels through the layer just below the surface and arrives at a time x/$v_1$, a straight line through the origin. The wave reflected from the boundary follows a longer path, down and back up, and arrives on a hyperbola. The head wave travels down to the boundary at the critical angle, along it at $v_2$, and back up at the critical angle, and its arrival time is

$$t_{\text{head}} = \frac{x}{v_2} + \frac{2h\cos\theta_c}{v_1},$$

a straight line with the slope of the deeper layer’s speed and an intercept set by the thickness of the upper one. It exists only beyond the critical distance $2h\tan\theta_c$, where it touches the reflection’s hyperbola, since the critical ray first returns to the surface there.

Close to the shot the direct wave is first. The head wave starts later, burdened by its detour down and back, but it runs most of its path at the faster speed, and beyond the crossover distance

$$x_c = 2h\sqrt{\frac{v_2 + v_1}{v_2 - v_1}}$$

it arrives first and stays first, its lead growing with distance. At three hundred kilometres from the shot in the figure it beats the direct wave by almost five seconds, although its path is twenty-six kilometres longer. The first signal at a distant receiver is not the one that went straight there. It is the one that took the least time, which is the path that does not change, Fermat’s principle, choosing a route with a long stretch of fast travel over a short route that is slow throughout.

The crust in two P waves #

On 8 October 1909 an earthquake struck the Kupa valley south of Zagreb, and the seismologist Andrija Mohorovičić collected the records from stations across Europe. At stations within a couple of hundred kilometres the first compressional wave arrived at times that grew steeply with distance; beyond that, the arrivals came earlier than that trend predicted, on a flatter line, and at intermediate stations two separate compressional arrivals could be seen, one following the other. He interpreted the flatter line as a wave that had travelled through faster rock deep below and the steeper one as a wave through the slower rock above, computed the speeds from their slopes and the depth of the boundary from where they crossed, and found a crust about fifty-four kilometres thick resting on denser material.

The boundary is the Mohorovičić discontinuity, the Moho, and it is found everywhere: about thirty-five kilometres under continents on average, deeper under mountain ranges, which have roots, and only five to ten under the oceans. The reduced-time figure shows why it was visible in records that were only a few seconds accurate: subtracting a time proportional to distance makes the mantle’s arrivals run nearly flat and the crust’s climb steeply, and the break in slope between them is as plain as a kink in a line.

Seismologists call the head wave along the Moho Pn, and the direct compressional wave through the crust Pg, and the two names are a working vocabulary for every regional network. Locating a small earthquake, or a suspected underground explosion, from stations a few hundred kilometres away depends on identifying which of the two each station recorded first, because they travel at different speeds; a station beyond the crossover that is assumed to have recorded Pg will put the event far too close. The speed of Pn itself varies from region to region, from about 7.8 to 8.3 kilometres a second, with the temperature and composition of the uppermost mantle, and maps of it are one of the ways the mantle beneath continents is charted.

How far the receivers must reach #

The same arithmetic tells a survey how to plan. The crossover distance is proportional to the depth of the boundary, and the constant depends on the contrast in speed: for a modest contrast, ten per cent, the head wave takes over only at nine times the depth; for a large contrast, as between loose soil and bedrock, at under three times. A survey to find bedrock under a building site a few tens of metres down lays out a line of geophones a hundred or two hundred metres long and strikes a plate with a sledgehammer; a survey of the crust sets off explosions or uses earthquakes and records them on stations hundreds of kilometres apart.

From the line of first arrivals the method reads three things. The slope of the first segment gives the upper layer’s speed, the slope of the second the lower layer’s, and the intercept of the second at zero distance gives the thickness of the upper layer: $h = t_i v_1/2\cos\theta_c$. Several layers of increasing speed give several segments, each a head wave from a deeper boundary, and the same reading peels them off one at a time. Reversing the survey, shooting from both ends of the line, separates a tilted boundary from a change in speed, since a boundary that slopes makes the apparent speed different in the two directions.

From salt domes to the Earth’s core #

The method was put to industrial use within fifteen years of Mohorovičić’s paper. In the oil fields of the American Gulf Coast in the 1920s, oil collected against domes of salt pushed up through the sediments, and salt carries sound about twice as fast as the sediments round it. Survey crews laid out receivers in a fan around a shot and looked for those whose first arrivals came early, having found a fast path through salt: fan shooting found dozens of domes in a few years, and the oil beside them. Refraction gave way to reflection surveys as the main tool of exploration once electronics could record the weaker, later reflected arrivals well, but it remains the standard way to map the shallow ground and the base of the crust.

At the scale of the whole planet, the same reasoning about which wave arrives first, and from which direction, found the core. Waves from a large earthquake recorded round the globe arrive on curves that fan out through the mantle, whose speed increases with depth, and between about 104 and 140 degrees of arc from the epicentre the direct compressional waves are missing, a shadow cast by a core in which they travel more slowly. Beno Gutenberg located its boundary at about 2,900 kilometres in 1914. In 1936 Inge Lehmann noticed faint arrivals inside the shadow that could only be explained by a solid inner core, faster again, reflecting and refracting waves into the zone the outer core had shadowed.

A layer the method cannot see #

The method has a blind spot that follows from the same construction. A head wave needs a boundary below which the speed is higher. A layer slower than the one above it bends waves towards the vertical, never reaches a critical angle, and sends nothing back along the surface. A slow layer sandwiched between faster ones — water-saturated sand under a dry crust, a coal seam in sandstone, a zone of fractured rock — is invisible in the first arrivals.

Worse, it is not merely invisible but misleading. The waves that do reach the deeper, faster rock still have to cross the slow layer, and they take longer to do it than they would at the upper layer’s speed. That extra time shows up in the intercept of the deepest segment, which a two-layer reading interprets as a thicker upper layer. In the figure, rock two kilometres down is placed at 2.72. Nothing measured along the surface reveals the error, because the travel times are exactly what a thicker two-layer section would give; only a borehole, or a reflection survey of the kind the layer too thin to see followed, shows the slow layer for what it is.

Why the first arrival is a faint one #

The construction gives the head wave’s timing and says nothing about its strength, which is the limitation the spiral that says how much arrives exposed in Huygens’s original method: an envelope is a locus and carries no amplitude. Adding the wavelets with their phases, as that essay did for diffraction, shows that the head wave is weak and grows weaker fast. It is fed only by the narrow bundle of rays that met the boundary at precisely the critical angle, and its energy is shared along an ever longer stretch of boundary, so its amplitude falls off much more steeply with distance than a direct wave’s, and it is weaker still for a small contrast in speed.

That is why it is recognised by its timing rather than its strength. A seismometer beyond the crossover distance sees a small, sharp first motion, the head wave, followed by the larger direct wave and reflections, and finally, slowest and largest of all, the surface waves that the wave a surface is enough to hold followed, which carry most of an earthquake’s shaking. The order of arrival sorts the waves by the speed of the fastest part of their path; their sizes sort them almost in reverse.

Head waves in other media #

The construction does not care what is vibrating. A sound wave in water over a fast seabed sends a head wave along the bottom, which marine seismologists use to measure the speed of the sediments. In a well, a sonic logging tool fires a pulse of sound into the mud filling the borehole, and the first arrival at receivers a few metres along is a head wave travelling along the borehole wall at the rock’s own speed, from which the rock’s porosity and strength are inferred, metre by metre, down kilometres of hole. In medical ultrasound a pulse travelling along the surface of a bone creates a head wave in the soft tissue above it, and the speed of that wave is a measure of the bone’s quality, used to assess osteoporosis.

Light makes the same wave when it meets a boundary at the critical angle, though it is usually overwhelmed by the totally reflected beam. The disturbance along the boundary in that case is the evanescent wave that the reflection that happens where the glass is not found extending a fraction of a wavelength beyond the glass, running along the surface at the speed of light in the denser medium.

A rule that is Snell’s law read sideways #

It is worth seeing how little the head wave adds to what was already known. Snell’s law says that the wavenumber along a boundary is the same on both sides, which the law that only asks about one component made the whole of the argument. A disturbance running along the boundary at speed $v_2$ has a wavenumber along it of ω/$v_2$, and in the upper layer that same wavenumber belongs to a wave travelling at an angle whose sine is $v_1/v_2$ — the critical angle again. The head wave is the wave in the upper layer that matches, along the boundary, the motion in the lower one. Huygens’s envelope and Snell’s phase matching are the same statement in two languages, and the head wave was always implied by both; it took seismograms recorded far enough from the source to make it the wave that arrives first.

Where the envelope picture stops #

The fronts and the travel times are geometry: wavelets and rays in layers of uniform speed with flat, sharp boundaries. Real rock speeds increase gradually with depth, so a ray turns continuously instead of bending at a boundary, and the travel-time curve becomes a smooth curve rather than straight segments; a gradual increase of speed returns a diving wave to the surface by turning it round, without any head wave at all. Real boundaries are rough and tilted. The figures say nothing about amplitude, and head waves are weak: their amplitude falls off faster with distance than a direct wave’s, which is why in practice they are recognised as first arrivals rather than by their strength. And the figures treat one kind of wave, where solid rock carries both compressional and shear waves, each with its own speeds and head waves.

The domain is a layered medium with speed increasing downward in steps, which is a good first description of soil over bedrock and of the crust over the mantle, and a poor one of the inside of a single rock mass.

Still open: what the Moho is #

The Moho is defined by a jump in seismic speed, and the jump is not always sharp. In some places it is a transition kilometres thick, in others it appears doubled, and under some young mountain belts the seismic boundary and the boundary of rock composition — between the crust’s feldspar-rich rocks and the mantle’s olivine-rich ones — may not coincide, since a change of mineral phase under pressure can make crustal rock as fast as the mantle. Whether the Moho everywhere marks the base of the rocks that were once melted out of the mantle, or in places only a pressure-induced transformation, is argued from seismic surveys, from slices of ancient oceanic crust thrust up onto land, and from the rare deep drill holes that have tried, so far without success, to reach it.

The habit worth carrying away is to ask of any boundary what moves along it, and how fast. A wave meeting a faster layer at $\theta_c = \arcsin(v_1/v_2)$ drives a disturbance along the boundary at $v_2$, and Huygens’s wavelets from it form a plane front back up into the slow layer, arriving at $x/v_2 + 2h\cos\theta_c/v_1$ and overtaking the direct wave beyond $2h\sqrt{(v_2 + v_1)/(v_2 - v_1)}$ — 185 km for a 35 km crust. The first arrival is the quickest path, and a long detour through fast rock is quicker than a short one through slow.