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for which a time domain approach is more suitable. In Figure 1, the peaks of the acoustic waveforms propagate faster than the troughs, and there are two reasons for this. First, when the medium is compressed, the local sound speed increases, and when rarified, it decreases. Second, the acoustic wave sets the particles in motion, and the local particle veloc- ity is highest at the peaks and lowest at the troughs. Thus, for nonlinear acoustic waves, the regions of compression (“wave crests”) in the waveform propagate faster and the regions of rarefaction (“wave troughs”) propagate slower. Even if changes in the local wave speed are small, over long-enough propagation distances, they manifest as sig- nificant distortion.
If the wave is sufficiently strong, substantial nonlinear distortion can occur before the wave is attenuated due to absorption, scattering, or divergence. However, the large gradients associated with nonlinear steepening result in increased importance of loss mechanisms such as viscosity, which competes against nonlinearity. When the nonlin- earity and loss mechanisms balance, a stable shock front is formed, and if the shock is sufficiently thin, it can be approximated as a discontinuous jump. Across a shock, all physical properties such as pressure, particle velocity, and density undergo abrupt jumps in their values. Shock waves have unusual properties in terms of how they propagate and the effects they can introduce in the medium through which they propagate, ranging from heating to structural damage.
The distortion of a wave profile and its evolution into a shock wave are observed in any elastic medium (air, water, or a solid). The shock waves can be generated by explo- sions, lightning, electric spark, gunshots, or other pulsed sources that cause sudden pressure changes. Even sources of moderate amplitude radiating smooth waveforms, such as piezoelectric sources of ultrasound, can achieve shock formation, especially when focused.
Some examples of measured shock wave profiles are shown in Figure 4. Regardless of the propagation medium and the type of source, they are all seen to possess a universal form in which the shocks are connected by smooth transitions. The characteristic durations of the waveforms can vary over a wide range, from microseconds to fractions of seconds, but the duration of the shock itself (referred to as shock rise time) can be extremely short, in fact, so short that often it cannot be resolved by a microphone or hydrophone. Shock waves
Figure 5. Supersonic aircraft produce a shock wave that is heard on the ground as a sonic boom. Top: NASA photo (created by NASA and in the public domain in the United States [PD-USGov-NASA]) showing a schlieren image (based on refraction of light by density gradients in the medium) of shock waves produced by two US Air Force T-38 aircraft flying at supersonic speeds. Bottom: cartoon showing the Mach cone (yellow) that is created in the wake of a supersonic aircraft. The waveforms inside the cone show the evolution of the sonic boom, due to nonlinear acoustic distortion, from an initial irregular waveform near the aircraft to an N wave at the ground. A sonic boom sounds similar to an explosion or a thunderclap to the human ear and may even cause damage to some structures.
do not propagate in the same way as ordinary weak acous- tic waves. Compression shock waves (e.g., all of the shocks at time = 0 in Figure 4) propagate faster than sound waves of infinitesimal amplitude, and their speed increases with increasing amplitude. The amplitude of the shock wave also
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