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Priority for discovery and explanation of the parametric array owes to [[Peter Westervelt|Peter J. Westervelt]],<ref>[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000005003231000004&idtype=cvips&gifs=yes Professor Peter Westervelt and the parametric array]</ref> winner of the [[Lord Rayleigh]] Medal,<ref>[http://www.ioa.org.uk/medals-and-awards/ Institute of Acoustics - Medals & Awards Programme] {{webarchive|url=https://web.archive.org/web/20090628181721/http://www.ioa.org.uk/medals-and-awards/ |date=2009-06-28 }}</ref> although important experimental work was contemporaneously underway in the former Soviet Union.<ref name=nonlinear-underwater-acoustics-book />
According to Muir<ref>{{Harvard citation no brackets|Muir|1976}}, p. 554.</ref> and Albers,<ref name=":0">{{Harvnb|Albers|1972}}</ref> the concept for the parametric array occurred to Dr. Westervelt while he was stationed at the London, England, branch office of the [[Office of Naval Research]] in 1951.
According to Albers,<ref name=":0" /> he (Westervelt) there first observed an accidental generation of low frequency sound ''in air'' by Captain H.J. Round (British pioneer of the [[superheterodyne receiver]]) via the parametric array mechanism.
The phenomenon of the parametric array, seen first experimentally by Westervelt in the 1950s, was later explained theoretically in 1960, at a meeting of the [[Acoustical Society of America]]. A few years after this, a full paper<ref>{{Harvard citation no brackets|Westervelt|1963}}</ref> was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.<ref>{{Harvard citation no brackets|Roy|Wu|1993}}</ref><ref>{{Harvnb|Beyer|1974}}</ref><ref>{{Harvnb|Bellin|Beyer|1960}}</ref>
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The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE).<ref>[https://dspace.mit.edu/bitstream/1721.1/28762/1/59823423.pdf Sources of Difference Frequency Sound in a Dual-Frequency Imaging System with Implications for Monitoring Thermal Surgery]{{dead link|date=January 2018 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> Solutions to this equation have been developed using [[Green's functions]]<ref>{{Harvnb|Moffett|Mellen|1977}}</ref><ref>{{Harvnb|Moffett|Mellen|1976}}</ref> and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.<ref>{{Cite web|url=http://people.bu.edu/robinc/kzk/|title = Texas KZK Time Domain Code}}</ref>
An alternate mathematical formalism using [[Fourier operator]] methods in [[wavenumber]] space, was also developed and generalized by Westervelt.<ref>{{Harvnb|Woodsum|Westervelt|1981}}</ref> The solution method is formulated in Fourier (wavenumber) space in a representation related to the beam patterns of the primary fields generated by linear sources in the medium. This formalism has been applied not only to parametric arrays,<ref>{{Harvnb|Woodsum|2006}}</ref> but also to other nonlinear acoustic effects, such as the absorption of sound by sound and to the equilibrium distribution of [[sound intensity]] spectra in cavities.<ref>{{Harvnb|Cabot|Putterman|1981}}</ref>
== Applications ==
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