The front that steepens until it cannot

A wave in a linear medium keeps its shape for ever. That is the whole content of the statement that a disturbance travels: the medium supplies a speed, and every part of the profile moves at it, so the shape at a later time is the shape at an earlier one, moved.

Nothing in the world quite works that way. In air, the speed of sound rises with the local temperature, and the compression at a crest is warmer than the rarefaction at a trough; on top of that the medium itself is moving forward at the crest. Both effects push the crest along faster than the trough. The correction is small — a part in a thousand for ordinary sound — and it is cumulative, which is the point.

The one term that changes everything #

Write the local propagation speed as $c_0 + \beta u$, where $u$ is the disturbance and $\beta$ collects whatever makes the medium’s response amplitude-dependent. For air $\beta$ is $(\gamma+1)/2$, about 1.2, multiplying the particle velocity. The equation of motion is then

$$\frac{\partial u}{\partial t} + (c_0 + \beta u)\frac{\partial u}{\partial x} = 0$$

which is the ordinary travelling-wave equation with the constant speed replaced by one that varies from point to point along the wave.

That single change destroys two things at once. Superposition is gone, because the equation is no longer linear: adding two solutions does not give a solution, and two tones passing through the same medium interact. And shape preservation is gone, because the different parts of one profile no longer keep step.

In the linear case the same shape appears at two times, moved and otherwise untouched. Everything in this essay is what happens when the speed at one part of the profile differs from the speed at another — the crest travelling faster than the trough, so the front leans forward and the shape is no longer preserved. One term does that, and losing shape-preservation loses almost everything else with it.

It is worth being clear that neither loss is a small correction to a mostly-linear picture. Superposition either holds or it does not, and once it fails there is no sense in which a wave has a spectrum that propagates: the spectrum at the far end is a function of the whole waveform at the near end rather than of each component separately. Every instrument that measures a frequency response is making a linearity assumption, and the assumption is checked by turning the level down and seeing whether the answer stops changing.

The remarkable thing is how completely solvable the result is. Each value of $u$ simply travels at its own constant speed, so the solution is written down immediately in implicit form: $u = u_0(x - (c_0 + \beta u)t)$, where $u_0$ is the initial profile. Solving that for $u$ at each point is a root-find, and the profiles in the opening figure are exactly that root-find, checked afterwards against the differential equation itself, which they satisfy to three parts in ten million.

The lines that have to meet #

The clearest way to see what goes wrong is to plot, in the plane of position against time, the paths along which each value of $u$ travels.

Each line is straight, because each carries a fixed value of $u$ and therefore a fixed speed. They are not parallel: the crest’s line leans further forward than the trough’s. Straight lines that are not parallel meet.

Where two of them meet, the solution is being asked to take two values at the same place at the same time. That is not a difficulty with the arithmetic; it is the statement that no single-valued profile satisfies the equation past that moment.

The time at which it first happens is set by the steepest descending gradient of the initial profile, and by nothing else:

$$t_{\text{b}} = \frac{1}{\beta\,\lvert \mathrm{d}u_0/\mathrm{d}x\rvert_{\max}}$$

A wave of any shape whatever breaks after one over $\beta$ times its steepest downward slope. The amplitude enters only through that slope, which is why the life of a wave is inversely proportional to how loud it is.

Two numbers make that concrete. A conversational sound at sixty decibels has a particle velocity of about a ten-thousandth of a millimetre per second, and its shock formation distance is thousands of kilometres — the wave is attenuated to nothing long before nonlinearity does anything at all. A sound at 190 decibels, which is what a metre from a large explosion produces, has a particle velocity comparable with the sound speed itself, and it steepens within its own wavelength. Everything between is a matter of how far the wave has to travel before the accumulation matters, and the accumulation is what the figure above is a measurement of.

What steepening does to the sound #

Before it breaks, the wave is doing something audible: it is manufacturing frequencies the source never emitted.

A pure sine, distorted, is no longer a pure sine, and decomposing the distorted profile gives harmonics whose amplitudes grow with distance. The closed form for them was found by Fubini in 1935 and is a series of Bessel functions: the $n$-th harmonic has amplitude $2J_n(n\sigma)/(n\sigma)$, where $\sigma$ is the fraction of the way to the shock. That the series and the point-by-point solution agree to two per cent of the amplitude at nine tenths of the way to breaking is a check that both describe the same thing.

A linear medium cannot do this at all. In a linear medium the frequencies present at the end are the frequencies present at the beginning, which is why a wave keeps its note however far it goes. Harmonic generation is therefore a direct measurement of nonlinearity, and it is the effect a doubling crystal in a laser exists to exploit.

There is one more reading of that figure worth taking. The harmonics do not appear all at once: the second grows first and roughly linearly with distance, the third more slowly, and the fifth is negligible until the wave is most of the way to breaking. So a nonlinear medium at short range sounds like a slightly harsher version of its input, and only at long range like a sawtooth. Measuring the ratio of the second harmonic to the fundamental is therefore a way of measuring how far a wave has come, or how nonlinear the medium is, and it is the basis of tissue harmonic imaging in ultrasound — where the image is formed from the second harmonic the tissue generated rather than from the frequency the probe emitted.

It also has an unwelcome side. An overdriven loudspeaker produces harmonics not present in its input, and some of them are the air’s rather than the cone’s; at high enough level, a horn’s throat is a nonlinear medium and its output is measurably distorted before it has left the device.

Superposition, and what replaces it #

The loss of superposition deserves more than a clause, because it is the property most of wave physics is built on.

Adding is what a linear medium does and the only thing it does, which is why superposition is so much more than a convenience. Every technique in wave physics that decomposes a signal into components — Fourier analysis, normal modes, Green’s functions — assumes that the components do not interact. In a nonlinear medium they do, so none of those techniques applies, and there is no substitute of comparable power.

In a linear medium a complicated signal can be broken into sinusoids, each followed independently, and reassembled at the far end. That is the licence behind Fourier analysis, behind the whole idea of a frequency response, and behind the decomposition of a note into a fundamental and its overtones. It is a licence granted by the equation, not by the waves.

Nonlinearly, two tones do not pass through one another untouched. They produce their sum and their difference frequencies, and then combinations of those, so a medium driven at 40 and 41 kilohertz emits at 1 kilohertz — which is exactly how an acoustic parametric array makes an audible beam as narrow as an ultrasonic one, something no linear loudspeaker of any size can do. The narrow beam is not built; it is inherited from the ultrasound that made it.

The same mechanism, run the other way, is the reason a nonlinear medium can amplify: energy pumped in at one frequency appears at another, and if the geometry is right the transfer is one-way. That is a parametric amplifier, and it exists in acoustics, in optics and in electronics with the same arithmetic in each.

The shape it is heading for #

Follow the harmonics to $\sigma = 1$ and the profile becomes a sawtooth: a slow linear rise and a vertical drop. That is not a coincidence of the sine’s shape; almost any initial profile ends the same way, because the steepening is dominated by the front.

Past the shock the wave loses energy at a rate no linear theory predicts, because the discontinuity dissipates. A sawtooth in air decays as one over the distance rather than exponentially, so a very loud sound loses its excess quickly and then propagates almost linearly at a level set by the distance rather than by the source. That is the reason a distant explosion sounds like a thump rather than a bang, and it is why doubling the charge does not double the noise at a kilometre.

The shock nobody can draw #

The equation stops describing the wave at $t_{\text{b}}$, and the natural question is what happens instead.

A Mach cone is a shock of a different origin: a source outrunning its own wavefronts. The steepening in this essay produces the same object with nothing moving faster than anything — the front assembles itself out of a smooth profile — and the two look identical once formed. That is worth knowing because it means a shock is a statement about the flow rather than about what made it.

The physical answer is a shock: a very thin region across which the pressure, density and velocity jump. What sets its thickness is exactly the thing the equation above was written without — viscosity and heat conduction, which are negligible for gentle gradients and are not negligible for a gradient approaching vertical. The front steepens until the dissipation is strong enough to balance the steepening, and then it stops, at a thickness of a few mean free paths.

That number is remarkable and worth holding onto. A shock in air at ordinary conditions is a few hundred nanometres thick — comparable with the distance a molecule travels between collisions — which means the continuum description is only just still valid inside it, and for strong enough shocks it is not valid at all.

The free path between collisions is what decides how thin a shock can be. A front cannot be sharper than the distance over which the medium exchanges momentum, so the mathematical discontinuity is replaced in practice by a layer a few mean free paths thick — a few hundred nanometres in air at atmospheric pressure. The equation predicts a jump and the medium supplies a thickness the equation knows nothing about.

The other thing a shock does is make the propagation irreversible. Entropy is generated across it, the jump conditions that connect the two sides are not the linear ones, and the shock travels faster than the sound speed ahead of it and slower than the one behind — which is what makes it stable and keeps it sharp.

Where the same term turns up #

The equation at the top of this essay is not about air. It is what happens whenever a wave’s speed depends on its own amplitude, and that is common.

A water wave’s speed depends on its wavelength, which is dispersion rather than the nonlinearity this essay is about — and in shallow water it depends on the depth as well, so a crest sitting in deeper water than its trough travels faster. Both effects are present in the same wave, they act against each other, and where they balance exactly the result is a solitary wave that neither steepens nor spreads.

Water waves in shallow water obey it with $c_0 = \sqrt{gh}$ and the correction coming from the crest sitting in deeper water than the trough. A wave approaching a beach therefore leans forward, and it breaks: the front goes vertical and then over. The shape it takes just before is the same sawtooth.

Traffic obeys a version of it, with the density in place of the disturbance and the drivers’ speed choice in place of the medium’s response. A jam is a shock, it travels backwards along the road, and it is thin compared with the traffic around it for the same reason.

Light in a medium whose index depends on intensity obeys it in time rather than in space, and the harmonics it generates are the whole business of nonlinear optics.

A crowd obeys it too, badly but visibly: a dense region of people moves more slowly, so the back of a moving crowd catches the front, and the resulting compression waves in a marathon start are shocks in the same sense.

And blood in an artery obeys it well enough that the pressure pulse steepens measurably between the heart and the periphery — the peripheral pulse is sharper than the aortic one, which is a clinically visible instance of a wave running out of linear behaviour on a journey of about a metre.

The discontinuity that was withdrawn #

The steepening was worked out in 1848, the discontinuity it produces was written down, and the man who wrote it down retracted it under pressure from the two most authoritative physicists of the age, on the basis of an objection that was wrong.

Stokes published a short paper titled On a difficulty in the theory of sound, in which he showed that a wave of finite amplitude must steepen, that the equations then admit a discontinuous solution, and that the discontinuous solution is what has to be adopted once the smooth one has ceased to exist. That is the whole of this essay’s argument, arrived at almost a century and a half ago.

Kelvin and Rayleigh objected on grounds of energy. Impose conservation of mass and of momentum across a discontinuity and work out the mechanical energy on each side, and it does not balance: energy appears to vanish. Since energy is conserved, they argued, the discontinuous solution is inadmissible.

Stokes accepted the objection. When his collected papers were reissued in 1883 he added a footnote withdrawing the discontinuous solution and describing it as untenable, and he never reinstated it.

The objection is arithmetically correct and physically incomplete, and the missing quantity is heat. Energy is conserved across a shock; what is not conserved is mechanical energy, because the shock converts some of it into internal energy of the gas — it heats it — and generates entropy in doing so. The mechanical theory of sound as it stood in 1848 had no term for that, so a budget written in that theory’s own variables came out short, and the shortfall looked like a violation.

Rankine supplied the correct jump conditions in 1870 and Hugoniot completed them in 1887, by writing the energy equation with the internal energy included. The three conditions that result — for mass, momentum and energy — determine everything about the jump, and Stokes’s discontinuity is a perfectly good solution of them. Rayleigh acknowledged in 1910 that the objection had been mistaken.

The shape of the episode is worth having. An argument that a phenomenon is impossible almost always proceeds by writing down a conserved quantity and showing it is not conserved. What has to be checked is whether the list of places the quantity can go is complete, and here it was not: the entire discrepancy went into a form of energy the theory had left out. A conservation argument is only as strong as its inventory, and this one was defeated by a term that a thermometer would have found.

The number that says how nonlinear a material is #

The coefficient $\beta$ has been carried through this essay as a property of the medium, and it is a measured one, tabulated for materials in the same way a sound speed or a density is.

The convention is to expand the medium’s equation of state about ambient conditions, writing the pressure as a series in the density change with coefficients $A$, $B$ and so on. The dimensionless ratio $B/A$ is the nonlinearity parameter, and $\beta = 1 + B/2A$.

For a gas the expansion can be done in closed form and gives $B/A = \gamma - 1$, so air’s is 0.4 and $\beta$ is 1.2 — which is the $(\gamma+1)/2$ quoted at the top of this page, arriving by a different route as a check.

Liquids are more surprising. Water’s $B/A$ is about 5, ethanol’s about 10.5, and fatty oils reach 11. So a liquid is more nonlinear than a gas by this measure, by a factor of three to six, despite being the medium in which sound is usually treated as most nearly ideal. The reason is that the nonlinearity here is about how stiffly the medium resists further compression, and a liquid’s stiffness rises with pressure much more sharply than an ideal gas’s does.

The parameter is measured two ways that ought to agree and do. It can be got acoustically, by driving a pure tone and measuring how fast the second harmonic grows — which is the essay’s own harmonic figure used as an instrument. Or it can be got thermodynamically, by measuring how the sound speed changes with pressure and with temperature, which requires no large amplitudes at all.

Its practical interest is that it distinguishes materials that other acoustic quantities do not. Soft tissues differ in sound speed by a few per cent, which is why an ultrasound image of soft tissue has poor intrinsic contrast. They differ in $B/A$ by a factor of nearly two — fat sits near 10 and liver near 6.8 — and that difference has been pursued as a contrast mechanism in its own right, and as a way of grading how much fat a liver has accumulated. A property that exists only because the medium is nonlinear turns out to be the one that tells two tissues apart.

What the pictures cannot show #

Every profile drawn stops at the breaking time. Past it the implicit solution is triple-valued, and the figure would have to show a curve folding back on itself. That fold is a real feature of the mathematics and not of any wave; it is where the model is asked a question about physics it does not contain.

There is no dissipation anywhere in the equation drawn. A real medium attenuates, and attenuation competes with steepening: a wave that is too quiet dies before it breaks. The ratio of the two — the Gol’dberg number — decides which happens first, and attenuation is the other half of the story.

The harmonic amplitudes are the lossless ones. Real harmonic generation is limited by the fact that higher harmonics attenuate faster, which is why a distorting medium does not simply accumulate an unlimited sawtooth.

And the characteristics are drawn for a one-dimensional wave. In two or three dimensions the fronts curve, converge and cross in ways a plane picture cannot follow, and focusing can produce a shock long before this arithmetic says one is due.

The ladder from here #

Later rungs on this anchor: the Rankine–Hugoniot conditions, which say what jumps across a shock and by how much; the weak-shock theory that gives the sawtooth’s one-over-distance decay; Burgers’ equation, which puts the dissipation back and has an exact solution containing both the steepening and the shock thickness; and the N-wave, which is the far field of every impulsive source and is what a sonic boom actually is by the time it reaches the ground.

The neighbouring ladders are the cone the source leaves behind, where a shock is made by a moving source rather than by amplitude, the packet that will not keep its shape, where a wave is destroyed by dispersion instead, and how a wave thins out, which is the amplitude falling for a reason that has nothing to do with the medium at all.