Every compression becomes a shock in the end

Acoustics is the theory in which a small disturbance travels at $a_0$ and keeps its shape for ever. It has no time scale in it, and that absence is the sign that something has been thrown away: nothing in the linear equations says how long for ever is.

The discarded term is that a disturbance moves at $u + a$ rather than at $a$, and that $a$ itself is raised where the gas is compressed. For a simple wave

$$a = a_0 + \tfrac12(\gamma-1)u\quad\Rightarrow\quad u + a = a_0 + \tfrac12(\gamma+1)u,$$

so the coefficient of nonlinearity is $\beta = \tfrac12(\gamma+1) = 1.2$ for air. Each point of the waveform travels at its own speed, the compressions catch the rarefactions ahead of them, and the profile leans forward.

The distance #

Setting the front vertical gives

$$x_{\mathrm{shock}} = \frac{a_0^2}{\beta\,\omega\,u_0},$$

which goes exactly as the reciprocal of the amplitude at fixed frequency — checked to twelve decimal places, which is a check on the arithmetic rather than on the physics, and worth having because the inverse dependence is the whole practical content.

Nothing on that list is exempt. Ordinary conversation would shock after six hundred kilometres, and it does not only because atmospheric absorption removes it first. The ratio of the absorption length to the shock-formation distance is the one number that decides whether a sound wave ever becomes a shock, and for everyday sound it is very small.

Why the compressions and not the rarefactions #

The asymmetry is the whole mechanism and it deserves to be stated as a piece of physics rather than as a sign in an equation.

A point of the waveform at positive $u$ is moving forward relative to the undisturbed gas and is in gas that has been compressed and is therefore hotter, so its local sound speed is higher. Both effects push it forward. A point at negative $u$ is moving backward relative to the gas and is in cooler gas, and both effects hold it back.

So the compression half of a wave is always catching up with what is ahead of it and the rarefaction half is always falling behind. Over a full cycle, the compression’s leading edge runs into the rarefaction that precedes it and the profile steepens there, while the compression’s trailing edge runs away from the rarefaction behind it and that part spreads.

That is why a compression can become a discontinuity and a rarefaction cannot. An expansion fan spreads and stays smooth for ever; a compression steepens until it is vertical and then becomes a shock. It is the same asymmetry expansion-fan records as a thermodynamic statement — compression through ten degrees costs total pressure and expansion through ten degrees costs nothing — and here it is the same fact seen kinematically, before any entropy has been mentioned.

The one-and-a-fifth #

The coefficient $\beta = \tfrac12(\gamma+1)$ is worth taking apart, because it is a sum of two effects of comparable size and the split is instructive.

Of the total, $\tfrac12$ comes from the fluid velocity: a compressed region is moving forward, and the disturbance rides on it. The other $\tfrac12(\gamma-1) = 0.2$ comes from the sound speed: the compressed gas is hotter, so signals in it travel faster.

For air the two are in the ratio 5 to 2, so the convection term dominates and the thermodynamic one is a forty per cent correction to it. For a gas with $\gamma$ near one — a heavy polyatomic vapour, or hot dissociated air — the second term nearly vanishes and $\beta\to1$, so the steepening is purely convective and slower.

And in a liquid the same expansion has $\beta = 1 + B/2A$, with $B/A$ a measured nonlinearity parameter of the substance rather than a ratio of specific heats — about 5 for water, so $\beta = 3.5$, three times air’s. That is why nonlinear acoustics is easier to demonstrate in water than in air at the same intensity, and why medical ultrasound has to account for it.

The exact solution, and the check #

Before the front breaks, the distortion is not a qualitative story — it is a closed-form solution. Inverting $u = \sin(\varphi + \sigma u)$, with $\sigma = x/x_{\mathrm{shock}}$, gives the waveform at any distance short of breaking, and Fubini wrote down its harmonics in 1935:

$$B_n = \frac{2}{n\sigma}J_n(n\sigma),$$

a Bessel function of its own order.

Computed here two ways that share no arithmetic — the waveform by bisection on the implicit relation and then quadrature, the Bessel functions from their integral representation — the two agree to nine parts in a thousand million million across eight harmonics.

That is the strongest agreement anywhere in this group, and it is worth having because the alternative to it is a sequence of pictures with no number in them.

Where the energy goes #

The harmonics are not a technicality; they are what the distortion sounds like.

The fundamental falls monotonically and everything above it grows. By the breaking point the second harmonic is thirty-five per cent of the fundamental and the sixth is a tenth.

Nothing has been lost. There is no dissipation anywhere in the model. The energy has simply moved to higher frequencies, which is why a loud low note arrives sounding harsh and why a shock is the end state rather than an accident.

The consequence for real air is that the absorption, which rises steeply with frequency, is being fed by the nonlinearity. A wave loud enough to distort generates harmonics that are absorbed far faster than the fundamental, so the wave loses energy at a rate set by its own amplitude — an extra attenuation that linear acoustics has no term for.

The smallest measurable consequence #

At small distances the second harmonic’s amplitude is $\sigma/2$ exactly — the ratio comes out at $0.499983$ over the sweep — so it appears in direct proportion to the distance travelled, from zero, with no threshold.

That is the first measurable departure from linear acoustics and the one an experiment reaches first. It is also the basis of a practical technique: parametric arrays, in which two high-frequency beams are transmitted together and their difference frequency is generated in the water by exactly this nonlinearity, producing a low-frequency beam far narrower than a transducer of that frequency could make.

The distortion used as an imaging technique #

The parametric array is one application of the harmonics; there is a larger one, and it turns the whole of this essay’s mechanism into a clinical instrument.

Tissue harmonic imaging forms an ultrasound picture not from the transmitted frequency but from the second harmonic that the tissue itself generates, by exactly the process computed above. The nonlinearity coefficient of soft tissue is around four — comparable with water’s and three times air’s — and the amplitudes a diagnostic transducer uses are high enough that the second harmonic is substantial after a few centimetres.

Imaging on it rather than on the fundamental gives a markedly better picture, for three reasons that all come out of the arithmetic here.

The harmonic grows as the square of the fundamental’s amplitude, so it is generated only where the beam is intense — along its core — and hardly at all in the weak side lobes. The effective beam is therefore narrower than the transmitted one, and the lateral resolution improves without any change to the transducer.

And it is generated along the path rather than at the source, so it is nearly absent in the near field. Most of the clutter in a conventional image comes from reverberation and from phase aberration in the body wall, both of which afflict the transmitted beam on its way in; the harmonic has not been generated yet when the beam passes through that region, so it arrives comparatively clean.

The distortion is asymmetric, and that matters for two things other than the picture. A steepened waveform’s compressional peak grows much more than its rarefactional trough deepens, so the positive and negative peak pressures separate — which is why the safety index for cavitation is quoted on the negative peak specifically, that being the one that tears the liquid, and why it cannot be inferred from the transmitted amplitude once the wave has distorted.

The heating goes the other way and is exploited. Because absorption rises steeply with frequency, a wave that has moved energy up its own spectrum is absorbed far faster than a sinusoid of the same intensity — so the nonlinear steepening concentrates the heating near the focus of a high-intensity beam. Therapeutic ultrasound relies on it: a focused beam intense enough to distort deposits its energy in a smaller volume than a linear calculation predicts, and the treatment planning has to include the harmonics or it under-predicts both the temperature and how sharply localised it is.

Which is a satisfying place for the term linear acoustics discards to end up. It sets the resolution of the picture, the location of the heat and the definition of the safety limit — three quantities in one instrument, none of which a linear theory can express, all of them following from the fact that a compression travels slightly faster than a rarefaction.

What this is and is not #

Two distinctions worth drawing, because the mechanism is often confused with two others.

It is not dissipation. Nothing here removes energy. The waveform’s spectrum spreads and its total energy is conserved right up to the moment a shock forms; after that the shock dissipates and the wave decays, which is a different process starting later.

It is not instability. Nothing is growing exponentially and nothing is selecting a wavelength. The steepening is a deterministic distortion of a given waveform, present at every amplitude, and the timescale is inversely proportional to the amplitude rather than independent of it.

What it is, is the leading nonlinear term of a hyperbolic system doing what such terms always do: compressions steepen, expansions spread, and the asymmetry between them is the deepest fact in this field.

Where the boundary with the kinematic wave is #

The kinematic-wave essay makes the same argument with a flux law imported from outside — Manning’s formula, a fundamental diagram — and no momentum equation anywhere.

Here the propagation speed comes from the momentum and energy equations of a gas, and $\beta$ is a property of the molecule rather than of a correlation. The steepening is the same phenomenon and the two essays own different halves of why it happens.

Both end at the same place: a discontinuity whose speed is the chord of the flux curve, which for a gas is the Rankine–Hugoniot condition and for a river is the same rule applied to one conserved quantity.

Where the steepening has already been used #

Three places in this collection where the result is doing work under another name.

The sonic boom is this mechanism with spherical spreading laid over it. The N-wave that reaches the ground is the asymptotic form the steepening produces once both shocks have formed, and the whole of low-boom design is an attempt to arrange for the front shock to be still forming on arrival.

Nozzle design by characteristics is the reverse requirement. A minimum-length nozzle’s wall is shaped so that the expansion waves it produces are absorbed rather than reflected, and a wall that gets it wrong sends compression waves into the stream which coalesce — by exactly the mechanism here — into oblique shocks at the correct exit Mach number.

And the shock tube’s compression wave. When a diaphragm bursts the driver gas sends a compression into the driven section which is not initially a shock: it is a smooth compression that steepens over a short distance into one. The shock-formation distance is a design parameter of a shock tube, and a tube too short for it produces a wave that has not finished forming when it reaches the test section.

The nonlinearity that is measured rather than derived #

One last observation, because it changes what kind of quantity $\beta$ is.

For a perfect gas, $\beta$ follows from $\gamma$ and $\gamma$ follows from counting degrees of freedom, so the coefficient is derived. For any other substance it is not: $\beta$ is defined through the second derivative of the equation of state, and for a real fluid that has to be measured.

The measurement is usually made by this effect — the growth of the second harmonic is timed and $\beta$ read off the slope, using the exact relation this essay checks to fifteen digits. So the theory is being used as an instrument to measure the coefficient that appears in it, which is a common and respectable arrangement and is worth recognising as such.

It also means the coefficient carries information about the substance that nothing else in a fluid calculation does. Two liquids with the same density and sound speed can have $\beta$ differing by a factor of two, and a nonlinear measurement will distinguish them where a linear one cannot. That is the basis of a family of acoustic techniques for characterising materials, and all of them rest on the term linear acoustics drops.

What is not in the flow #

The amplitude.

Linear acoustics is a theory of infinitesimal disturbances, and an infinitesimal disturbance has an infinite shock-formation distance. Every real disturbance has a finite one, and the theory that describes it has no term in which the amplitude could appear.

That is a different kind of absence from the ones its neighbours here describe. Elsewhere the missing quantity is something outside the flow — an observer, a boundary, a gas, a history. Here it is a property of the disturbance itself, dropped by a linearisation that then has no way of saying when it stopped being valid.

A linear theory cannot report its own failure, because the quantity that decides the failure is the one it set to zero. The only diagnostic is to keep the next term and see how long it takes to matter, which is what this essay does.

Reading a distance #

Two habits follow.

Compare the two lengths. A nonlinear acoustic problem has a shock-formation distance and an absorption length, and their ratio — the Gol’dberg number — says which happens first. Above one the wave shocks; below it, it is absorbed while still nearly sinusoidal. Everyday sound has a Gol’dberg number well below one and a shock wave has one enormously above. It is the same shape of comparison every regime question on this site takes: two lengths, or two times, and a ratio that decides which physics is doing the work.

And do not treat the amplitude as a scale factor. Doubling the amplitude of a linear solution gives a solution; doubling the amplitude here halves the distance over which the solution is valid. A calculation performed at one amplitude and scaled to another has quietly changed the physics, and the scaling is exactly the failure this essay measures.

The model limit #

Three.

No dissipation. Real air absorbs, and the absorption is what stops most of the waves in the table above from ever shocking. Including it turns the equation into Burgers’, whose steady solution is the shock profile of the previous essay and whose travelling waves have a thickness rather than a discontinuity.

A plane wave. Spherical or cylindrical spreading reduces the amplitude as the wave travels, so the steepening rate falls and the shock may never form. The sonic boom is exactly that case with the spreading included, which is why its exponents are three-quarters and a quarter rather than one and zero.

And a simple wave, which means a disturbance travelling in one direction into uniform gas. Two waves travelling towards each other interact, and the interaction is a genuinely different problem — the one a shock tube’s diaphragm sets up, with three wave families at once.

The last limit is the one that bounds the whole calculation. Everything here holds up to the moment the front becomes vertical and not past it: at $\sigma = 1$ the implicit relation stops being invertible, Fubini’s series stops converging usefully, and the solution becomes multivalued. What happens afterwards needs the equal-area construction that the kinematic-wave essay and the sonic boom both use, and the harmonic content after breaking follows a different law — the Fay solution — with a spectrum falling as $1/n$ rather than following Bessel functions. This essay stops at the shock; the two that surround it take over there.