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At finite amplitude, an acoustic waveform can change shape as it propagates. For a fluid with a positive coefficient of nonlinearity, compressions travel faster than rarefactions. The waveform steepens and energy transfers into higher harmonics. Dissipation competes with this steepening and gives real shock fronts a finite thickness.
Steepening of a sinusoidal wave
For an initially sinusoidal, progressive plane wave, an inviscid, weakly nonlinear model predicts the shock-formation distance
\[z_s=\frac{c_0^2}{\beta\omega_0v_0} =\frac{\rho_0c_0^3}{\beta\omega_0p_0},\qquad \beta=1+\frac{B}{2A}.\]Here, \(p_0\) and \(v_0=p_0/(\rho_0c_0)\) are positive pressure and particle-velocity amplitudes at the source, \(\omega_0\) is angular frequency, and \(B/A\) is the equation-of-state nonlinearity parameter. This distance is a lossless reference scale, not a prediction of an infinitely sharp front in a dissipative fluid.
Single-cycle pulses and polarity
A pulse that starts with compression and ends with rarefaction can develop an N-shaped waveform with steep outer fronts. Reversing the polarity changes where the characteristic lines converge and can produce a steep transition near the pulse centre. Pulse duration, amplitude and dissipation all affect the evolution.
Sonic booms
A sonic boom is associated with shock waves generated during supersonic motion. A vehicle produces them continuously while flying supersonically; an observer hears the boom when the shock system reaches their position. It is not a sound emitted only at the instant the vehicle crosses Mach 1.
Whip cracking provides another example of rapid motion and shock formation, although its mechanics differ from plane-wave steepening in a stationary fluid.
References & further reading
- k-Wave: nonlinear plane waves and Mendousse’s solution
- k-Wave: modelling nonlinear wave propagation
- NASA: sonic booms
- Young (2002): Sonic booms of space shuttles approaching Edwards Air Force Base, 1988–1993
- Bernstein, Hall & Trent (1958): On the Dynamics of a Bull Whip
- McMillen & Goriely (2003): Whip waves