Parametric array: Difference between revisions

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A '''parametric array''', in the field of [[acoustics]], is a nonlinear [[transducer|transduction]] mechanism that generates narrow, nearly [[side lobe]]-free beams of low frequency sound, through the mixing and interaction of high frequency [[sound wave]]s, effectively overcoming the [[diffraction limit]] (a kind of spatial 'uncertainty principle') associated with linear acoustics.<ref>{{cite book| last=Beyer| first=Robert T| title=Nonlinear Acoustics| chapter=Preface to the Original Edition| chapter-url=http://asa.aip.org/books/nonlinear.html#Preface1|archive-date=February 16, 2018 |archive-url=https://web.archive.org/web/20180216214423/https://asa.aip.org/books/nonlinear.html#Preface1 |url-status=dead }}</ref> The main side lobe-free beam of low frequency sound is created as a result of nonlinear mixing of two high frequency sound beams at their difference frequency. Parametric arrays can be formed in water,<ref name=nonlinear-underwater-acoustics-book>{{cite book| last1=Novikov | first1=B. K. | last2=Rudenko | first2=O. V. | last3=Timoshenko | first3=V. I. | translator= Robert T. Beyer| title=Nonlinear Underwater Acoustics| url=http://asa.aip.org/books/nonuw.html |oclc=16240349 |isbn=9780883185223 |publisher=American Institute of Physics |date=1987}}</ref> air,<ref>{{cite journal | doi = 10.1121/1.384959 | volume=68 | issue=4 | title=Experimental study of a saturated parametric array in air | year=1980 | journal=The Journal of the Acoustical Society of America | pages=1214–1216 | last1 = Trenchard | first1 = Stephen E. | last2 = Coppens | first2 = Alan B.| bibcode=1980ASAJ...68.1214T }}</ref> and earth materials/rock.<ref>{{cite journal | doi = 10.1121/1.403453 | volume=91 | issue=4 | title=Finite amplitude wave studies in earth materials | year=1992 | journal=The Journal of the Acoustical Society of America | page=2350 | last1 = Johnson | first1 = P. A. | last2 = Meegan | first2 = G. D. | last3 = McCall | first3 = K. | last4 = Bonner | first4 = B. P. | last5 = Shankland | first5 = T. J.| bibcode=1992ASAJ...91.2350J | doi-access = free }}</ref><ref>[http://www.lanl.gov/orgs/ees/ees11/geophysics/nonlinear/pubs/parabeam.html Parametric Beam Formation in Rock]</ref>

== History ==

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 ==