Parametric array: Difference between revisions

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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> (currently Professor Emeritus at [[Brown University]]), 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.&nbsp;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 ==