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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> 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>
'''References'''▼
== Foundations ==
[1] H.C. Woodsum and P.J. Westervelt, "A General Theory for the Scattering of Sound by Sound", Journal of Sound and Vibration (1981, 76(2), 179-186.▼
The foundation for Westervelt's theory of sound generation and scattering in [[nonlinear acoustics|nonlinear acoustic]]<ref>{{cite journal | doi = 10.1121/1.380612 | volume=57 | issue=6 | title=The status and future of nonlinear acoustics | year=1975 | journal=The Journal of the Acoustical Society of America | pages=1352–1356 | last1 = Westervelt | first1 = Peter J.| bibcode=1975ASAJ...57.1352W }}</ref> media owes to an application of [[James Lighthill|Lighthill]]'s [[Aeroacoustics#Lighthill's equation|equation]] for fluid particle motion.
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>
[2] Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963.▼
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>
[4] Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", ▼
== Applications ==
[5] Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976 ▼
Practical applications are numerous and include:
* underwater sound
**[[sonar]]
** depth sounding
** sub-bottom profiling
** non-destructive testing
** and 'see through walls' sensing<ref>{{cite journal | doi = 10.1016/S0041-624X(99)00109-2 | pmid=11243456 | volume=37 | issue=8 | title=A non-contact technique for evaluation of elastic structures at large stand-off distances: applications to classification of fluids in steel vessels | year=2000 | journal=Ultrasonics | pages=531–536 | last1 = Kaduchak | first1 = Gregory | last2 = Sinha | first2 = Dipen N. | last3 = Lizon | first3 = David C. | last4 = Kelecher | first4 = Michael J.| url=https://zenodo.org/record/1259727 | doi-access = free }}</ref>
** remote ocean sensing<ref>{{cite journal | doi = 10.1121/1.414208 | volume=98 | issue=5 | title=Remote ocean sensing by parametric array | year=1995 | journal=The Journal of the Acoustical Society of America | page=2915 | last1 = Naugolnykh | first1 = Konstantin A. | last2 = Esipov | first2 = Igor B.| bibcode=1995ASAJ...98.2915N | doi-access=free }}</ref>
* medical [[ultrasound]]<ref>{{cite journal | doi = 10.1088/0031-9155/46/11/314 | pmid=11720358 | volume=46 | issue=11 | title=A focused ultrasound method for simultaneous diagnostic and therapeutic applications—a simulation study | year=2001 | journal=Physics in Medicine and Biology | pages=2967–2984 | last1 = Konofagou | first1 = Elisa | last2 = Thierman | first2 = Jonathan | last3 = Hynynen | first3 = Kullervo| bibcode=2001PMB....46.2967K | s2cid=2036873 | url=https://semanticscholar.org/paper/85b7f120e6568d0a912d79040fd1b0d7810b1053 }}</ref>
* and tomography<ref>{{cite journal |last1=Zhang |first1=Dong |last2=Chen |first2=Xi |last3=Xiu-fen |first3=Gong |year=2001 |title=Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source—Theoretical analysis and computer simulations |journal=The Journal of the Acoustical Society of America |volume=109 |issue=3 |pages=1219–1225 |bibcode=2001ASAJ..109.1219Z |doi=10.1121/1.1344160 |pmid=11303935}}</ref>
* underground seismic prospecting<ref>{{cite journal | doi = 10.1121/1.2022023 | volume=76 | issue=S1 | title=High-resolution seismic profiling with a low-frequency parametric array | year=1984 | journal=The Journal of the Acoustical Society of America | page=S78 | last1 = Muir | first1 = T. G. | last2 = Wyber | first2 = R. J.| bibcode=1984ASAJ...76...78M }}</ref>
* active noise control<ref>{{cite web |url=http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378 |title=Active control of sound using a parametric array |accessdate=2006-12-05 |url-status=dead |archiveurl=https://web.archive.org/web/20070309235859/http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378 |archivedate=2007-03-09 }}</ref>
* and directional high-fidelity commercial audio systems ([[Sound from ultrasound]])<ref>[[n:Elwood Norris receives 2005 Lemelson-MIT Prize for invention.]]</ref>
Parametric ''receiving'' arrays can also be formed for directional reception.<ref>{{Cite book |doi = 10.1109/ICASSP.1979.1170632|chapter = Experiments with a large aperture parametric acoustic receiving array|title = ICASSP '79. IEEE International Conference on Acoustics, Speech, and Signal Processing|year = 1979|last1 = Reeves|first1 = C.|last2 = Goldsberry|first2 = T.|last3 = Rohde|first3 = D.|volume = 4|pages = 616–619}}</ref> In 2005, Elwood Norris won the $500,000 [[Lemelson-MIT Prize|MIT-Lemelson Prize]] for his application of the parametric array to commercial high-fidelity loudspeakers.
[6] Ronald A. Roy and Junru Wu, "An Experimental Investigation of the Interaction of Two Non-Collinear Beams of Sound", Proceedings of the 13th International Symposium on Nonlinear Acoustics, H. Hobaek, Editor, Elsevier Science Ltd., London (1993).▼
{{Reflist}}
== Further reading ==
[7] Harvey C. Woodsum, "Analytical and Numerical Solutions to the 'General Theory for the Scattering of Sound by Sound”, J. Acoust. Soc. Am. Vol. 95, No. 5, Part 2 (2PA14), June, 1994 (Program of the 134th Meeting of the Acoustical Society of America, Cambridge Massachusetts)▼
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* {{Wikicite|reference=H.C. Woodsum, Proc. 17th International Conference on Nonlinear Acoustics, AIP Press (NY), 2006; " Comparison of Nonlinear Acoustic Experiments with a Formal Theory for the Scattering of Sound by Sound", paper TuAM201.|ref=CITEREFWoodsum2006}}
* {{Wikicite|reference=T.G. Muir, Office of Naval Research Special Report - "Science, Technology and the Modern Navy, Thirtieth Anniversary (1946-1976), Paper ONR-37, "Nonlinear Acoustics: A new Dimension in Underwater Sound", published by the Department of the Navy (1976)|ref=CITEREFMuir1976}}
* {{Wikicite|reference=V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)|ref=CITEREFAlbers1972}}
* {{Wikicite|reference=M. Cabot and Seth Putterman, "Renormalized Classical Non-linear Hydrodynamics, Quantum Mode Coupling and Quantum Theory of Interacting Phonons", Physics Letters Vol. 83A, No. 3, 18 May 1981, pp. 91–94 (North Holland Publishing Company-Amsterdam)|ref=CITEREFCabotPutterman1981}}
* Nonlinear Parameter Imaging Computed Tomography by Parametric Acoustic Array Y. Nakagawa; M. Nakagawa; M. Yoneyama; M. Kikuchi. IEEE 1984 Ultrasonics Symposium. Volume, Issue, 1984 Page(s):673–676
* Active Nonlinear Acoustic Sensing of an Object with Sum or Difference Frequency Fields. Zhang, W.; Liu, Y.; Ratilal, P.; Cho, B.; Makris, N.C.; Remote Sens. 2017, 9, 954. https://doi.org/10.3390/rs9090954
{{DEFAULTSORT:Parametric Array}}
▲[8] Robert T. Beyer , Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.
[[Category:Sound]]
[[Category:Acoustics]]
▲[9] H.O. Berktay and D.J. Leahy, Journal of the Acoustical Society of Amercia, 55,p. 539 (1974)
[[Category:Nonlinear systems]]
▲[10] M.J. Lighthill, “ On Sound Generated Aerodynamically”, Proc. R. Soc. Lond.
▲[11] M. J. Lighhill, “ On Sound Generated Aerodynamically”, Proc. R. Soc. Lond.,
▲[12] J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32,
▲[13] M. J. Lighthill, Math.Revs. 19, 915 (1958)]
▲[14] H.C. Woodsum, Bull. Of Am. Phys. Soc., Fall 1980; “A Boundary Condition Operator for Nonlinear Acoustics”
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