10.4 When linearity breaks — wave steepening

The linearised wave equation is excellent for small amplitudes: $|p'| / p_0 \ll 1$. Beyond that, second-order terms — the ones we threw away in chapter 4 — start to matter. The most important effect is wave steepening: the crest of a finite-amplitude wave propagates faster than the trough, and the wave gradually distorts.

Why crests travel faster than troughs

For a finite-amplitude compression wave, the local speed of sound depends on the local pressure and density (because $c$ is a derivative of pressure evaluated at the local state, not at the equilibrium state). At a crest where $\rho > \rho_0$, the local sound speed is slightly higher than at a trough where $\rho < \rho_0$. To lowest order in the perturbation,

where $v'$ is the local particle velocity (positive in the direction of propagation in a compressive cycle, negative in a rarefactive cycle). For an ideal diatomic gas, $\gamma = 1.4$ and $(\gamma + 1)/2 = 1.2$.

The crest of the wave (positive $v'$) propagates faster than the equilibrium $c_0$; the trough (negative $v'$) propagates slower. As the wave travels, the crest catches up with the next trough ahead of it. The wave shape steepens.

The Burgers equation

Including this nonlinear correction plus viscous dissipation, the propagation equation for a 1-D wave reduces (to leading order) to the Burgers equation:

where $\beta = (\gamma + 1)/2$ is the coefficient of nonlinearity and $\nu$ a viscous diffusivity. Without the viscous term ($\nu = 0$), the equation predicts wave steepening leading to a true discontinuity in finite time — a shock. With viscosity, the steepening is balanced by diffusion and the wave settles into a finite-width steady profile.

How fast does steepening happen?

For an initially sinusoidal wave of amplitude $v_0$ at frequency $\omega$, the distance over which a shock forms (in the inviscid limit) is

where $M = v_0/c_0$ is the acoustic Mach number. For conversational speech, $M \sim 10^{-5}$ and the shock distance is $\sim 10^7$ wavelengths — millions of metres. The nonlinearity is invisible.

For very loud sounds: $M \sim 10^{-2}$ (140 dB), shock distance $\sim 10^4$ wavelengths — still many kilometres at 1 kHz.

For extremely loud sounds: $M \sim 10^{-1}$ (160 dB+), shock distance $\sim 10^3$ wavelengths. Shocks form within a few hundred metres.

For jets and sonic booms: $M \sim 1$, shock distance $\sim$ a few wavelengths. Shocks form essentially at the source.

What shocked sound sounds like

A steepened sound wave has the spectral content of a sawtooth: rich in odd harmonics. The “harshness” of very loud music or close-up jet engines is partly this nonlinear harmonic enrichment — energy that started at the fundamental frequency leaks to higher harmonics as the wave propagates.

This is also why the sonic boom from a supersonic aircraft sounds like a “boom” rather than a “tone” — the steepened pressure wave at the Mach cone has been converted into a near-discontinuity in pressure, with a broadband Fourier spectrum.

Stabilisation by dissipation

The Burgers equation has an exact solution for the steady-state shock profile — a hyperbolic tangent that smoothly transitions from upstream to downstream values, with width $\delta \sim \nu / (c_0 M)$. For air at $M \sim 0.1$, this gives a shock width of order microns to millimetres — small enough that for most purposes the shock is treated as a true discontinuity, but finite in real fluids.

The width of natural shocks is set by the balance between nonlinear steepening (which sharpens the front) and viscous + thermal dissipation (which smooths it). Beyond about $M = 5$, additional physics enters (chemical dissociation, ionisation) but that is far outside the scope of acoustics.

Looking ahead

The next lesson — 10.5 — works out the jump conditions across a shock (the Rankine–Hugoniot relations), which describe what happens to pressure, density, and velocity when a shock passes through a fluid element. Then 10.6 bridges to the planned Cavitation book, where the nonlinearity goes even further: into the regime where the medium changes phase from liquid to vapour and back.