10.5 Shock formation and the second-order equations

When wave steepening progresses far enough, the wave front becomes so steep that the linear theory predicts a multi-valued profile — the pressure would have to take three different values at the same point in space. This is unphysical. Real fluids resolve the contradiction by forming a shock: a finite-but-very-thin region across which pressure, density, and velocity jump discontinuously.

The Rankine–Hugoniot jump conditions

Across a propagating shock, mass, momentum, and energy must each be conserved. Applying these conservation laws to a thin control volume straddling the shock gives the Rankine–Hugoniot relations:

• Mass: $\rho_1 (u_1 - U_s) = \rho_2 (u_2 - U_s)$
• Momentum: $p_1 + \rho_1 (u_1 - U_s)^2 = p_2 + \rho_2 (u_2 - U_s)^2$
• Energy: $h_1 + \tfrac12 (u_1 - U_s)^2 = h_2 + \tfrac12 (u_2 - U_s)^2$

where subscripts 1 and 2 refer to upstream and downstream of the shock, $u$ is the fluid velocity, $U_s$ the shock velocity, and $h$ the enthalpy per unit mass.

Together with an equation of state, these three relations determine the downstream state given the upstream state and the shock strength.

Shock Mach number

A key dimensionless parameter is the shock Mach number $M_s = U_s / c_1$ (ratio of shock speed to upstream sound speed). For weak shocks $M_s \to 1$ (the shock moves at the sound speed); for strong shocks $M_s \gg 1$ (the shock outruns the linear sound speed).

For a weak shock ($M_s = 1 + \delta$, with $\delta \ll 1$), the Rankine–Hugoniot relations linearise back to the linear-acoustic limit: pressure jump $\Delta p \propto \delta$, density jump $\Delta \rho \propto \delta$, velocity jump $\Delta v \propto \delta$. Weak shocks are sound waves at finite amplitude.

For a strong shock, the post-shock state is dramatically different: pressure may jump by orders of magnitude, density by a factor of up to $(\gamma + 1)/(\gamma - 1) = 6$ for diatomic gas. The temperature behind the shock can rise to thousands of K — leading to dissociation, ionisation, and the regime treated by hypersonic gas dynamics.

Second-order acoustic equations

For weak nonlinear sound — where the wave is still recognisably acoustic but the linear theory is starting to fail — the relevant framework is the second-order acoustic equations: the linear wave equation plus terms quadratic in the perturbation. The Burgers equation from lesson 10.4 is the simplest second-order propagation equation.

More generally, the second-order theory of sound includes:

• Wave steepening (Burgers nonlinearity).
• Acoustic streaming: a steady fluid flow generated by the time-averaged momentum carried by an acoustic wave. The flow appears in viscous media as the absorbed momentum drives a slow circulation.
• Radiation pressure on obstacles (revisited from chapter 5.6, now with the right systematic expansion).
• Sum and difference frequency generation: when two finite-amplitude tones $\omega_1$ and $\omega_2$ propagate together, nonlinear mixing creates components at $\omega_1 + \omega_2$, $\omega_1 - \omega_2$, $2\omega_1$, $2\omega_2$, etc. This is exploited in parametric arrays — focused-beam ultrasound sources that generate audible-frequency components purely by nonlinear interaction in the medium.

What “shocked sound” sounds like

Spectrally, a steepened/shocked sound wave is rich in odd harmonics — the periodic sawtooth profile that linear and dissipative effects approach in the limit. The energy at the fundamental decreases; energy at the harmonics increases. The “harshness” or “edge” of very loud music is partly this nonlinear harmonic enrichment.

A pure tone propagating in air can, at sufficient amplitude and over sufficient distance, develop into a near-sawtooth. The condition is roughly that the Reynolds number for acoustic nonlinearity — the ratio of nonlinear steepening rate to viscous diffusion rate — exceeds unity. For typical audio frequencies and amplitudes this happens only for the loudest sources (sirens, jet engines, sonic booms).

Looking ahead

The transition from linear to nonlinear acoustics is also the transition from signal to flow to phase transition. The next and final lesson bridges from this regime to the planned Cavitation book, where bubble formation, growth, and collapse drive nonlinear acoustic phenomena well beyond anything in standard acoustics.