The thunder that passes over the listener's head

Count the seconds between a flash and its thunder, divide by three, and the answer is the distance to the lightning in kilometres. The rule works because light arrives at once and sound at about a third of a kilometre a second. It also stops working at a fairly definite distance. Lightning thirty kilometres away is easily visible, especially at night, as the flickering of a distant storm, and it is usually silent — so usually that the silent flashes have a name, heat lightning, and a folk explanation, that they are a different kind of lightning. They are the same lightning, and the thunder is being made as loudly as ever. It simply does not reach the ground where the listener is.

The distance that takes the treble out explained why distant thunder rumbles instead of cracking: the air absorbs high frequencies far more strongly than low ones, so the crack is stripped away within a few kilometres and the rumble is what remains. It closed with a caution that nothing in its account had any weather in it, and that real outdoor sound is dominated by refraction long before absorption matters. Thunder’s horizon is the plainest case of that caution. Absorption takes the crack; refraction takes the rest.

A wavelet that shrinks with height #

The speed of sound in air depends on its temperature and on nothing else of importance: $c = 20.05\sqrt{T}$ metres a second with $T$ in kelvin, 346 m/s at 25 °C and 331 at 0 °C. On a summer afternoon the air near the ground has been warmed by the ground and the air above is cooler, falling by about 6.5 K for every kilometre of height. Sound therefore travels about 3.8 m/s slower for every kilometre it climbs.

Every front is a source set out Huygens’s rule: each point of a wavefront sends out a small wavelet, and the front a moment later is their envelope. In uniform air the wavelets are all the same size and a spherical front stays spherical. In air that cools upwards, each wavelet’s radius is the local sound speed times the time step, and the wavelets near the top of a front are smaller than those near the bottom. The envelope is no longer a sphere; its upper part falls behind, and the whole front leans back.

A ray is the normal to a front, and the normal to a front leaning back points upwards. The sound that set out along the ground is turned away from it: by the time each front has travelled a few kilometres its foot has lifted, and the ground beneath is no longer on any front. That is refraction, the same refraction as at a boundary between two media in the bend at the boundary, spread continuously through the air instead of concentrated at a surface. The direction of the turn follows a single rule that holds for light, sound and seismic waves alike: a ray curves towards the side where the wave is slower.

The curvature is large. For a nearly horizontal ray, the radius of curvature is the sound speed divided by its gradient, $R = c/|dc/dz|$, which comes to about 91 km for an afternoon’s lapse. The ray that bends without a surface computed the same quantity for light in the same atmosphere and found its rays curved with a radius of about 38,000 km, gently enough that the Earth’s own curvature is larger. Light’s refractive index barely notices temperature; sound’s speed is half a temperature law. An afternoon bends sound rays four hundred times more sharply than it bends light.

The ray that grazes the ground #

Put a source high in the air — a stretch of lightning channel three kilometres up — and trace the rays leaving it in every direction. Each one follows an arc of radius 91 km, bending upwards. Rays aimed steeply down reach the ground nearby, reflect and climb away. Rays aimed less steeply reach the ground further out, and the shallowest of them meets it tangentially, just touching the surface before curving up again. No ray at all reaches the ground beyond that point.

Beyond the grazing point lies the acoustic shadow. A listener inside it hears nothing of that part of the channel by direct propagation: the sound has passed overhead and is still climbing. The geometry is the same as the geometry of a horizon. A circle of radius $R$ descending from a height $H$ to become tangent to the ground covers a horizontal distance $\sqrt{2RH}$, exactly as the distance to the horizon from a height $H$ on a planet of radius $R$; here the curved thing is the ray, not the ground. Add the small extra reach to a listener’s ear at height $h$, another $\sqrt{2Rh}$, and the shadow begins at

$$x_s \approx \sqrt{2RH} + \sqrt{2Rh}.$$

For a source 3 km up that is 23.7 km. The figure’s rays were traced by integrating the ray equation through the actual temperature profile, and the traced distance agrees with the formula to within a fraction of a per cent, because a linear temperature profile makes every ray almost exactly a circle.

Which part of the channel is heard #

A lightning channel is not a point. It runs from the ground to the cloud, five kilometres or more, branching on the way, and each part of it is a source of thunder. The shadow distance grows as the square root of the source’s height, so the parts of the channel high up are heard further away than the parts near the ground.

A listener ten kilometres away hears the whole channel. A listener twenty kilometres away hears only the parts more than about two kilometres up, and the thunder heard there is a shorter, weaker rumble from the top of the channel alone. Further away, less and less of the channel contributes, until the listener is in the shadow of all of it. Where that happens depends on how fast the air cools: under the dry adiabatic lapse of 9.8 K per kilometre — a hot afternoon over dry ground — the shadow of a source three kilometres up is reached in under twenty kilometres; under a moist 4 K per kilometre it is past thirty.

The meteorologist Robert Fleagle made this calculation in 1949 and found that it accounted for the observed limit of audibility, about twenty-five kilometres in typical conditions, better than any account based on attenuation. The numbers bear him out. Air absorbs sound near 100 Hz, where most of thunder’s energy lies, at a few tenths of a decibel per kilometre — thirty times less than at a kilohertz. Twenty-five kilometres of air costs thunder’s rumble perhaps five or ten decibels by absorption, against more than a hundred decibels at its source. The shadow costs it everything at once.

The wind’s share #

Sound travels through the air, so a wind carries it: downwind its speed over the ground is the sound speed plus the wind, upwind the sound speed minus the wind. A uniform wind would shift everything sideways and bend nothing. What bends rays is the wind’s change with height, and near the ground wind almost always increases with height, held back by the surface’s friction.

The effective sound speed, the sound speed plus the wind along the ray’s direction, plays the part that the sound speed alone played before. Upwind, the wind’s shear adds to the cooling, the rays curve up twice as sharply, and the shadow of a source a kilometre up moves in from 14.0 km to 9.8 km. Downwind, a shear of 4 m/s per kilometre slightly outweighs the cooling’s 3.8 m/s per kilometre; the rays curve down rather than up, and the sound reaches the ground at every distance in the figure. A breeze aloft too light to notice at the surface is enough to make the same storm loud in a town downwind of it and silent at the same distance upwind, and the boundary between audible and inaudible can lie tens of kilometres from where still air would put it.

The same asymmetry is why shouting into the wind is so ineffective, and why it is not because the wind blows the sound back. A wind of ten metres a second changes the sound’s speed by three per cent and its travel time across a field by the same, which no ear could notice. What it changes greatly is the rays’ direction: a few metres a second more wind at head height than at the ground turns the upwind sound over the listener’s head, and the sound that does reach them has diffracted down into the shadow.

A night that folds sound back down #

On a clear, still night the ground radiates its heat to the sky and cools faster than the air above it. By morning the air near the ground can be several degrees colder than the air a hundred or two hundred metres up — a temperature inversion — and the gradient that bent sound upwards in the afternoon is reversed.

Sound now travels faster higher up, and the rays curve down. A ray launched at a shallow angle climbs into the warm layer, turns, and comes back to the ground, where it reflects and repeats the arc. The limiting angle follows from Snell’s law in the stratified air, which keeps $\cos\theta/c$ fixed along every ray: a ray turns where the sound speed has risen to $c_0/\cos\theta_0$, so the eight-kelvin inversion, raising the sound speed from 337.4 to 342.1 m/s, traps every ray launched below 9.5°. The sound that the afternoon sent into the sky is folded back along the ground, and spreads in two dimensions rather than three, falling off as the distance rather than as its square.

That is why sound carries at night and over cold water — a train heard five kilometres away at dawn and not at noon, voices across a lake in the evening. The fog that often comes with such nights has a reputation for carrying sound, and the distance that takes the treble out pointed out that it is not the fog: droplets far smaller than a wavelength scatter almost nothing. It is the inversion under the fog. The trapped layer is the same idea as the channel with no walls: a region of slower propagation between faster ones binds a wave without any boundary to reflect from, and here the ground supplies one wall and the warm layer the other.

The zone of silence #

Large explosions showed something stranger. When Queen Victoria’s funeral was marked by cannon salutes in 1901, the guns were reported heard well over a hundred kilometres away, while places at intermediate distances heard nothing. The explosion of the Oppau chemical works in 1921 was heard at distances of several hundred kilometres in rings separated by bands of silence. Seismologists had long been comfortable with waves that do not arrive in certain zones — the shadow that shows the core is liquid is the most famous of them — but a zone of silence surrounded by a ring of sound had to mean that the atmosphere, far above the ground, was bending sound back.

In 1923 the meteorologist Francis Whipple argued that the returns required the air some forty kilometres up to be warm — at least as warm as the ground — when the balloon soundings of the day reached only to the cold stratosphere below twenty kilometres. The same year Frederick Lindemann and Gordon Dobson reached the same conclusion from the heights at which meteors burned up. Both were right: ozone absorbs ultraviolet sunlight between about twenty and fifty kilometres and heats the air there, so the temperature climbs again from its minimum of −57 °C to nearly 0 °C at the stratopause. The sound speed climbs with it.

Temperature alone does not quite return the sound in the standard atmosphere, as the figure’s inset shows: the stratopause, at 271 K, is still colder than the ground at 288 K, so its sound speed of 330 m/s falls short of the ground’s 340. The rest is supplied by the stratospheric winds, which blow at fifty metres a second and more in winter. Along the wind, the effective sound speed near 45 km exceeds the ground’s, every ray launched shallowly enough turns there, and the sound returns 190 to 230 km out, having spent more than ten minutes on the way. Against the wind, there is no return. Because the stratospheric winds reverse between summer and winter, the ring of audibility around a fixed source moves from one side to the other with the seasons — an observation that puzzled the early investigators and is now one of the measurements used to check models of the upper atmosphere. The international network of infrasound stations built to detect nuclear tests relies on exactly these stratospheric returns, which carry the low-frequency sound of an explosion thousands of kilometres.

What the shadow is not #

The shadow is not perfectly silent. Huygens’s construction applied strictly says no ray enters it, but the spiral that says how much light arrives showed that a geometric shadow always has some wave in it: wavelets from the edge of the illuminated region spill across the boundary, falling off over a distance that grows with the wavelength. A sound shadow’s edge has the same property, and low frequencies leak further into it than high ones, so what is heard just inside the shadow is a muffled, low-pitched remnant of the sound, typically twenty or thirty decibels down. Turbulence in the air scatters some sound into the shadow too, and makes it fluctuate.

The ray picture is still the right one for asking where the sound goes, for the same reason it is right for the seismic head wave of the wave that arrives first by going the long way: the wavelength — three metres at a hundred hertz — is tiny compared with the ray’s radius of curvature of tens of kilometres. The rays and the fronts are perpendicular everywhere, the surface every ray is normal to proved that they must be, and either picture predicts the same shadow. The wavelets make the cause visible: a wavelet shrinking with height is a front tilting back, and a front tilting back sends its sound upwards.

What the figures leave out #

The atmosphere in the figures is horizontally uniform and still except for the winds drawn, its temperature changing linearly with height in the troposphere and following the standard atmosphere above; the ground is flat, perfectly reflecting and level, and the Earth’s curvature is neglected, which is reasonable over tens of kilometres and less so over hundreds. Wind is treated as an effective sound speed, accurate for nearly horizontal rays. Real afternoon air has a surface layer much steeper than its average lapse, which bends rays more sharply near the ground; real winds change direction with height; and a real lightning channel is a tortuous line that emits sound in all directions with a strength varying along it. Absorption, diffraction into the shadow and scattering by turbulence are all left out, so the shadow is drawn sharper than it is. The stratospheric wind is a single jet of 50 m/s centred at 55 km, chosen to be typical of winter; the thermosphere above 70 km, which returns sound at still greater distances, is not drawn.

Still open: hearing the weather in the sound #

Because the paths of sound through the atmosphere depend so strongly on its temperature and wind, the arrival times and directions of sound from known sources carry information about the air it crossed. The stratospheric winds between thirty and sixty kilometres are poorly observed — above the reach of weather balloons and below that of most satellites’ wind measurements — and infrasound from volcanoes, ocean waves and explosions is being used to constrain them, adding the returns of distant microbaroms to weather models. How much this can improve forecasts that depend on the stratosphere, and how well a network of microphones can map a wind field it sees only along curved, wind-dependent paths, is still being worked out. The path that does not change noted that a ray takes the path of stationary time; inverting the arrival times of many such paths to recover the medium they crossed is the problem seismology solved for the Earth’s interior, now posed for the air.

Air cooling 6.5 K per kilometre makes sound slow by 3.8 m/s per kilometre of height, so Huygens’s wavelets shrink upwards, every front tilts back and every ray curves skywards with a radius near 91 km; the ray that grazes the ground marks a shadow beginning at $\sqrt{2RH} + \sqrt{2Rh}$ — 14.0 km for sound from 1 km up a lightning channel and 23.7 km from 3 km — which wind shear moves in upwind and removes downwind, a night inversion reverses, and the warm, windy stratosphere answers 190–230 km away beyond a zone of silence. Distant thunder is not too faint to hear; it is too high.