Editing Shock wave (section)

== Shock waves due to nonlinear steepening ==
Shock waves can form due to steepening of ordinary waves. The best-known example of this phenomenon is [[ocean wave]]s that form [[breaking wave|breakers]] on the shore. In shallow water, the speed of surface waves is dependent on the depth of the water. An incoming ocean wave has a slightly higher wave speed near the crest of each wave than near the troughs between waves, because the wave height is not infinitesimal compared to the depth of the water. The crests overtake the troughs until the leading edge of the wave forms a vertical face and spills over to form a turbulent shock (a breaker) that dissipates the wave's energy as sound and heat.

Similar phenomena affect strong [[sound wave]]s in gas or plasma, due to the dependence of the sound speed on temperature and pressure. Strong waves heat the medium near each pressure front, due to adiabatic compression of the air itself, so that high pressure fronts outrun the corresponding pressure troughs. There is a theory that the sound pressure levels in brass instruments such as the trombone become high enough for steepening to occur, forming an essential part of the bright timbre of the instruments.{{refn|{{Citation|last1=Hirschberg|first1=A.|last2=Gilbert|first2=J.|last3=Msallam|first3=R.|last4=Wijnands|first4=A. P. J.|title=Shock Waves in Trombones|journal=Journal of the Acoustical Society of America|volume=99|issue=3|pages=1754–1758|date=March 1996|url=http://www.physics.mcgill.ca/~guymoore/ph225/shock.pdf|bibcode=1996ASAJ...99.1754H|doi=10.1121/1.414698|access-date=2017-04-17|archive-date=2019-12-10|archive-url=https://web.archive.org/web/20191210004134/http://www.physics.mcgill.ca/%7Eguymoore/ph225/shock.pdf|url-status=dead}}}} While shock formation by this process does not normally happen to unenclosed sound waves in Earth's atmosphere, it is thought to be one mechanism by which the [[Sun|solar]] [[chromosphere]] and [[solar corona|corona]] are heated, via waves that propagate up from the solar interior.

Similar nonlinear steepening and shock formation have been observed in supersonic jet flows, where pressure fluctuations amplify rapidly and convert to shock fronts, contributing to the characteristic jet noise. This principle is also applied in shock tube experiments, where controlled wave steepening helps simulate blast wave behavior for aerospace and defense research.

The mathematical modeling of such nonlinear wave behavior often involves the Burgers’ equation, which includes a viscous damping term and a nonlinear convective term to describe wave steepening and eventual shock development. These models are crucial in predicting real-world phenomena like sonic booms, atmospheric reentry heating, and even medical applications such as focused shock wave therapy.