# Parametric array

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Nonlinear transduction mechanism

A **parametric array**, in the field of [acoustics](/source/Acoustics), is a nonlinear [transduction](/source/Transducer) mechanism that generates narrow, nearly [side lobe](/source/Side_lobe)-free beams of low frequency sound, through the mixing and interaction of high frequency [sound waves](/source/Sound_wave), effectively overcoming the [diffraction limit](/source/Diffraction_limit) (a kind of spatial '[uncertainty principle](/source/Uncertainty_principle)') associated with linear acoustics.[1] 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,[2] air,[3] and earth materials/rock.[4][5]

## History

Priority for discovery and explanation of the parametric array owes to [Peter J. Westervelt](/source/Peter_Westervelt),[6] winner of the [Lord Rayleigh](/source/Lord_Rayleigh) Medal,[7] although important experimental work was contemporaneously underway in the former Soviet Union.[2]

According to Muir[8] and Albers,[9] 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](/source/Office_of_Naval_Research) in 1951.

According to Albers,[9] 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](/source/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](/source/Acoustical_Society_of_America). A few years after this, a full paper[10] was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.[11][12][13]

## Foundations

The foundation for Westervelt's theory of sound generation and scattering in [nonlinear acoustic](/source/Nonlinear_acoustics)[14] media owes to an application of [Lighthill](/source/James_Lighthill)'s [equation](/source/Aeroacoustics#Lighthill's_equation) for fluid particle motion.

The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE).[15] Solutions to this equation have been developed using [Green's functions](/source/Green's_functions)[16][17] and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.[18]

An alternate mathematical formalism using [Fourier operator](/source/Fourier_operator) methods in [wavenumber](/source/Wavenumber) space, was also developed and generalized by Westervelt.[19] 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,[20] but also to other nonlinear acoustic effects, such as the absorption of sound by sound and to the equilibrium distribution of [sound intensity](/source/Sound_intensity) spectra in cavities.[21]

## Applications

Practical applications are numerous and include:

- underwater sound - [sonar](/source/Sonar) - [depth sounding](/source/Depth_sounding) - sub-bottom profiling - non-destructive testing - and 'see through walls' sensing[22] - remote ocean sensing[23]

- medical [ultrasound](/source/Ultrasound)[24]

- and tomography[25]

- underground seismic prospecting[26]

- [active noise control](/source/Active_noise_control)[27]

- and directional high-fidelity commercial audio systems ([Sound from ultrasound](/source/Sound_from_ultrasound))[28]

Parametric *receiving* arrays can also be formed for directional reception.[29] In 2005, Elwood Norris won the $500,000 [MIT-Lemelson Prize](/source/Lemelson-MIT_Prize) for his application of the parametric array to commercial high-fidelity loudspeakers.

## References

1. **[^](#cite_ref-1)** Beyer, Robert T. ["Preface to the Original Edition"](https://web.archive.org/web/20180216214423/https://asa.aip.org/books/nonlinear.html#Preface1). *Nonlinear Acoustics*. Archived from [the original](http://asa.aip.org/books/nonlinear.html#Preface1) on February 16, 2018.

1. ^ [***a***](#cite_ref-nonlinear-underwater-acoustics-book_2-0) [***b***](#cite_ref-nonlinear-underwater-acoustics-book_2-1) Novikov, B. K.; Rudenko, O. V.; Timoshenko, V. I. (1987). [*Nonlinear Underwater Acoustics*](http://asa.aip.org/books/nonuw.html). Translated by Robert T. Beyer. American Institute of Physics. [ISBN](/source/ISBN_(identifier)) [9780883185223](https://en.wikipedia.org/wiki/Special:BookSources/9780883185223). [OCLC](/source/OCLC_(identifier)) [16240349](https://search.worldcat.org/oclc/16240349).

1. **[^](#cite_ref-3)** Trenchard, Stephen E.; Coppens, Alan B. (1980). "Experimental study of a saturated parametric array in air". *The Journal of the Acoustical Society of America*. **68** (4): 1214–1216. [Bibcode](/source/Bibcode_(identifier)):[1980ASAJ...68.1214T](https://ui.adsabs.harvard.edu/abs/1980ASAJ...68.1214T). [doi](/source/Doi_(identifier)):[10.1121/1.384959](https://doi.org/10.1121%2F1.384959).

1. **[^](#cite_ref-4)** Johnson, P. A.; Meegan, G. D.; McCall, K.; Bonner, B. P.; Shankland, T. J. (1992). ["Finite amplitude wave studies in earth materials"](https://doi.org/10.1121%2F1.403453). *The Journal of the Acoustical Society of America*. **91** (4): 2350. [Bibcode](/source/Bibcode_(identifier)):[1992ASAJ...91.2350J](https://ui.adsabs.harvard.edu/abs/1992ASAJ...91.2350J). [doi](/source/Doi_(identifier)):[10.1121/1.403453](https://doi.org/10.1121%2F1.403453).

1. **[^](#cite_ref-5)** [Parametric Beam Formation in Rock](http://www.lanl.gov/orgs/ees/ees11/geophysics/nonlinear/pubs/parabeam.html)

1. **[^](#cite_ref-6)** [Professor Peter Westervelt and the parametric array](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000005003231000004&idtype=cvips&gifs=yes)

1. **[^](#cite_ref-7)** [Institute of Acoustics - Medals & Awards Programme](http://www.ioa.org.uk/medals-and-awards/) [Archived](https://web.archive.org/web/20090628181721/http://www.ioa.org.uk/medals-and-awards/) 2009-06-28 at the [Wayback Machine](/source/Wayback_Machine)

1. **[^](#cite_ref-8)** [Muir 1976](#CITEREFMuir1976), p. 554.

1. ^ [***a***](#cite_ref-:0_9-0) [***b***](#cite_ref-:0_9-1) [Albers 1972](#CITEREFAlbers1972)

1. **[^](#cite_ref-10)** [Westervelt 1963](#CITEREFWestervelt1963)

1. **[^](#cite_ref-11)** [Roy & Wu 1993](#CITEREFRoyWu1993)

1. **[^](#cite_ref-12)** [Beyer 1974](#CITEREFBeyer1974)

1. **[^](#cite_ref-13)** [Bellin & Beyer 1960](#CITEREFBellinBeyer1960)

1. **[^](#cite_ref-14)** Westervelt, Peter J. (1975). "The status and future of nonlinear acoustics". *The Journal of the Acoustical Society of America*. **57** (6): 1352–1356. [Bibcode](/source/Bibcode_(identifier)):[1975ASAJ...57.1352W](https://ui.adsabs.harvard.edu/abs/1975ASAJ...57.1352W). [doi](/source/Doi_(identifier)):[10.1121/1.380612](https://doi.org/10.1121%2F1.380612).

1. **[^](#cite_ref-15)** [Sources of Difference Frequency Sound in a Dual-Frequency Imaging System with Implications for Monitoring Thermal Surgery](https://dspace.mit.edu/bitstream/1721.1/28762/1/59823423.pdf)[*[permanent dead link](https://en.wikipedia.org/wiki/Wikipedia:Link_rot)*]

1. **[^](#cite_ref-16)** [Moffett & Mellen 1977](#CITEREFMoffettMellen1977)

1. **[^](#cite_ref-17)** [Moffett & Mellen 1976](#CITEREFMoffettMellen1976)

1. **[^](#cite_ref-18)** ["Texas KZK Time Domain Code"](http://people.bu.edu/robinc/kzk/).

1. **[^](#cite_ref-19)** [Woodsum & Westervelt 1981](#CITEREFWoodsumWestervelt1981)

1. **[^](#cite_ref-20)** [Woodsum 2006](#CITEREFWoodsum2006)

1. **[^](#cite_ref-21)** [Cabot & Putterman 1981](#CITEREFCabotPutterman1981)

1. **[^](#cite_ref-22)** Kaduchak, Gregory; Sinha, Dipen N.; Lizon, David C.; Kelecher, Michael J. (2000). ["A non-contact technique for evaluation of elastic structures at large stand-off distances: applications to classification of fluids in steel vessels"](https://zenodo.org/record/1259727). *Ultrasonics*. **37** (8): 531–536. [doi](/source/Doi_(identifier)):[10.1016/S0041-624X(99)00109-2](https://doi.org/10.1016%2FS0041-624X%2899%2900109-2). [PMID](/source/PMID_(identifier)) [11243456](https://pubmed.ncbi.nlm.nih.gov/11243456).

1. **[^](#cite_ref-23)** Naugolnykh, Konstantin A.; Esipov, Igor B. (1995). ["Remote ocean sensing by parametric array"](https://doi.org/10.1121%2F1.414208). *The Journal of the Acoustical Society of America*. **98** (5): 2915. [Bibcode](/source/Bibcode_(identifier)):[1995ASAJ...98.2915N](https://ui.adsabs.harvard.edu/abs/1995ASAJ...98.2915N). [doi](/source/Doi_(identifier)):[10.1121/1.414208](https://doi.org/10.1121%2F1.414208).

1. **[^](#cite_ref-24)** Konofagou, Elisa; Thierman, Jonathan; Hynynen, Kullervo (2001). ["A focused ultrasound method for simultaneous diagnostic and therapeutic applications—a simulation study"](https://semanticscholar.org/paper/85b7f120e6568d0a912d79040fd1b0d7810b1053). *Physics in Medicine and Biology*. **46** (11): 2967–2984. [Bibcode](/source/Bibcode_(identifier)):[2001PMB....46.2967K](https://ui.adsabs.harvard.edu/abs/2001PMB....46.2967K). [doi](/source/Doi_(identifier)):[10.1088/0031-9155/46/11/314](https://doi.org/10.1088%2F0031-9155%2F46%2F11%2F314). [PMID](/source/PMID_(identifier)) [11720358](https://pubmed.ncbi.nlm.nih.gov/11720358). [S2CID](/source/S2CID_(identifier)) [2036873](https://api.semanticscholar.org/CorpusID:2036873).

1. **[^](#cite_ref-25)** Zhang, Dong; Chen, Xi; Xiu-fen, Gong (2001). "Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source—Theoretical analysis and computer simulations". *The Journal of the Acoustical Society of America*. **109** (3): 1219–1225. [Bibcode](/source/Bibcode_(identifier)):[2001ASAJ..109.1219Z](https://ui.adsabs.harvard.edu/abs/2001ASAJ..109.1219Z). [doi](/source/Doi_(identifier)):[10.1121/1.1344160](https://doi.org/10.1121%2F1.1344160). [PMID](/source/PMID_(identifier)) [11303935](https://pubmed.ncbi.nlm.nih.gov/11303935).

1. **[^](#cite_ref-26)** Muir, T. G.; Wyber, R. J. (1984). "High-resolution seismic profiling with a low-frequency parametric array". *The Journal of the Acoustical Society of America*. **76** (S1): S78. [Bibcode](/source/Bibcode_(identifier)):[1984ASAJ...76...78M](https://ui.adsabs.harvard.edu/abs/1984ASAJ...76...78M). [doi](/source/Doi_(identifier)):[10.1121/1.2022023](https://doi.org/10.1121%2F1.2022023).

1. **[^](#cite_ref-27)** ["Active control of sound using a parametric array"](https://web.archive.org/web/20070309235859/http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378). Archived from [the original](http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378) on 2007-03-09. Retrieved 2006-12-05.

1. **[^](#cite_ref-28)** [n:Elwood Norris receives 2005 Lemelson-MIT Prize for invention.](https://en.wikinews.org/wiki/Elwood_Norris_receives_2005_Lemelson-MIT_Prize_for_invention.)

1. **[^](#cite_ref-29)** Reeves, C.; Goldsberry, T.; Rohde, D. (1979). "Experiments with a large aperture parametric acoustic receiving array". *ICASSP '79. IEEE International Conference on Acoustics, Speech, and Signal Processing*. Vol. 4. pp. 616–619. [doi](/source/Doi_(identifier)):[10.1109/ICASSP.1979.1170632](https://doi.org/10.1109%2FICASSP.1979.1170632).

## Further reading

- 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.

- Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963

- Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977

- Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976

- 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)

- 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)

- Robert T. Beyer, Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.

- H.O. Berktay and D.J. Leahy, Journal of the Acoustical Society of America, 55, p. 539 (1974)

- M.J. Lighthill, "On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A211, 564-687 (1952)

- M.J. Lighthill, “On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A222, 1-32 (1954)

- J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960)

- M.J. Lighthill, Math. Revs. 19, 915 (1958)

- H.C. Woodsum, Bull. Of Am. Phys. Soc., Fall 1980; “A Boundary Condition Operator for Nonlinear Acoustics”

- 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.

- 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)

- V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)

- 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)

- 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.](/source/Purnima_Ratilal); Cho, B.; Makris, N.C.; Remote Sens. 2017, 9, 954. [https://doi.org/10.3390/rs9090954](https://doi.org/10.3390/rs9090954)

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Adapted from the Wikipedia article [Parametric array](https://en.wikipedia.org/wiki/Parametric_array) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Parametric_array?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.