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Parametric array

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">A parametric array, in the field of <a href="Acoustics" title="Acoustics">acoustics</a>, is a nonlinear <a href="Transducer" title="Transducer">transduction</a> mechanism that generates narrow, nearly <a href="Side_lobe" class="mw-redirect" title="Side lobe">side lobe</a>-free beams of low frequency sound, through the mixing and interaction of high frequency <a href="Sound_wave" class="mw-redirect" title="Sound wave">sound waves</a>, effectively overcoming the <a href="Diffraction_limit" class="mw-redirect" title="Diffraction limit">diffraction limit</a> (a kind of spatial 'uncertainty principle') associated with linear acoustics.<a href="#cite_note-1">[1]</a> 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,<a href="#cite_note-nonlinear-underwater-acoustics-book-2">[2]</a> air,<a href="#cite_note-3">[3]</a> and earth materials/rock.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Priority for discovery and explanation of the parametric array owes to <a href="Peter_Westervelt" title="Peter Westervelt">Peter J. Westervelt</a>,<a href="#cite_note-6">[6]</a> winner of the <a href="Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> Medal,<a href="#cite_note-7">[7]</a> although important experimental work was contemporaneously underway in the former Soviet Union.<a href="#cite_note-nonlinear-underwater-acoustics-book-2">[2]</a>

According to Muir<a href="#cite_note-8">[8]</a> and Albers,<a href="#cite_note-:0-9">[9]</a> the concept for the parametric array occurred to Dr. Westervelt while he was stationed at the London, England, branch office of the <a href="Office_of_Naval_Research" title="Office of Naval Research">Office of Naval Research</a> in 1951.

According to Albers,<a href="#cite_note-:0-9">[9]</a> he (Westervelt) there first observed an accidental generation of low frequency sound in air by Captain H.J. Round (British pioneer of the <a href="Superheterodyne_receiver" title="Superheterodyne receiver">superheterodyne receiver</a>) 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 <a href="Acoustical_Society_of_America" title="Acoustical Society of America">Acoustical Society of America</a>. A few years after this, a full paper<a href="#cite_note-10">[10]</a> was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Foundations">Foundations</h2></div>

The foundation for Westervelt's theory of sound generation and scattering in <a href="Nonlinear_acoustics" title="Nonlinear acoustics">nonlinear acoustic</a><a href="#cite_note-14">[14]</a> media owes to an application of <a href="James_Lighthill" title="James Lighthill">Lighthill</a>'s <a href="Aeroacoustics#Lighthill's_equation" title="Aeroacoustics">equation</a> for fluid particle motion.

The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE).<a href="#cite_note-15">[15]</a> Solutions to this equation have been developed using <a href="Green's_functions" class="mw-redirect" title="Green's functions">Green's functions</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a> and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.<a href="#cite_note-18">[18]</a>

An alternate mathematical formalism using <a href="Fourier_operator" title="Fourier operator">Fourier operator</a> methods in <a href="Wavenumber" title="Wavenumber">wavenumber</a> space, was also developed and generalized by Westervelt.<a href="#cite_note-19">[19]</a> 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,<a href="#cite_note-20">[20]</a> but also to other nonlinear acoustic effects, such as the absorption of sound by sound and to the equilibrium distribution of <a href="Sound_intensity" title="Sound intensity">sound intensity</a> spectra in cavities.<a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Practical applications are numerous and include:

<ul><li>underwater sound

<ul><li><a href="Sonar" title="Sonar">sonar</a></li>

<li>depth sounding</li>

<li>sub-bottom profiling</li>

<li>non-destructive testing</li>

<li>and 'see through walls' sensing<a href="#cite_note-22">[22]</a></li>

<li>remote ocean sensing<a href="#cite_note-23">[23]</a></li></ul></li>

<li>medical <a href="Ultrasound" title="Ultrasound">ultrasound</a><a href="#cite_note-24">[24]</a></li>

<li>and tomography<a href="#cite_note-25">[25]</a></li>

<li>underground seismic prospecting<a href="#cite_note-26">[26]</a></li>

<li>active noise control<a href="#cite_note-27">[27]</a></li>

<li>and directional high-fidelity commercial audio systems (<a href="Sound_from_ultrasound" title="Sound from ultrasound">Sound from ultrasound</a>)<a href="#cite_note-28">[28]</a></li></ul>

Parametric receiving arrays can also be formed for directional reception.<a href="#cite_note-29">[29]</a> In 2005, Elwood Norris won the $500,000 <a href="Lemelson-MIT_Prize" class="mw-redirect" title="Lemelson-MIT Prize">MIT-Lemelson Prize</a> for his application of the parametric array to commercial high-fidelity loudspeakers.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBeyer" class="citation book cs1">Beyer, Robert T. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180216214423/https://asa.aip.org/books/nonlinear.html#Preface1">"Preface to the Original Edition"</a>. Nonlinear Acoustics. Archived from <a rel="nofollow" class="external text" href="http://asa.aip.org/books/nonlinear.html#Preface1">the original</a> on February 16, 2018.</cite>

</li>

<li id="cite_note-nonlinear-underwater-acoustics-book-2">^ <a href="#cite_ref-nonlinear-underwater-acoustics-book_2-0">a</a> <a href="#cite_ref-nonlinear-underwater-acoustics-book_2-1">b</a> <cite id="CITEREFNovikovRudenkoTimoshenko1987" class="citation book cs1">Novikov, B. K.; Rudenko, O. V.; Timoshenko, V. I. (1987). <a rel="nofollow" class="external text" href="http://asa.aip.org/books/nonuw.html">Nonlinear Underwater Acoustics</a>. Translated by Robert T. Beyer. American Institute of Physics. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780883185223</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/16240349">16240349</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFTrenchardCoppens1980" class="citation journal cs1">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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980ASAJ...68.1214T">1980ASAJ...68.1214T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.384959">10.1121/1.384959</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFJohnsonMeeganMcCallBonner1992" class="citation journal cs1">Johnson, P. A.; Meegan, G. D.; McCall, K.; Bonner, B. P.; Shankland, T. J. (1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.403453">"Finite amplitude wave studies in earth materials"</a>. The Journal of the Acoustical Society of America. 91 (4): 2350. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ASAJ...91.2350J">1992ASAJ...91.2350J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.403453">10.1121/1.403453</a>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a rel="nofollow" class="external text" href="http://www.lanl.gov/orgs/ees/ees11/geophysics/nonlinear/pubs/parabeam.html">Parametric Beam Formation in Rock</a>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a rel="nofollow" class="external text" href="http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000005003231000004&idtype=cvips&gifs=yes">Professor Peter Westervelt and the parametric array</a>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a rel="nofollow" class="external text" href="http://www.ioa.org.uk/medals-and-awards/">Institute of Acoustics - Medals & Awards Programme</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090628181721/http://www.ioa.org.uk/medals-and-awards/">Archived</a> 2009-06-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFMuir1976">Muir 1976</a>, p. 554.

</li>

<li id="cite_note-:0-9">^ <a href="#cite_ref-:0_9-0">a</a> <a href="#cite_ref-:0_9-1">b</a> <a href="#CITEREFAlbers1972">Albers 1972</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFWestervelt1963">Westervelt 1963</a>

</li>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFBeyer1974">Beyer 1974</a>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFBellinBeyer1960">Bellin & Beyer 1960</a>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFWestervelt1975" class="citation journal cs1">Westervelt, Peter J. (1975). "The status and future of nonlinear acoustics". The Journal of the Acoustical Society of America. 57 (6): 1352–1356. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975ASAJ...57.1352W">1975ASAJ...57.1352W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.380612">10.1121/1.380612</a>.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a rel="nofollow" class="external text" href="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</a>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFMoffettMellen1977">Moffett & Mellen 1977</a>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFMoffettMellen1976">Moffett & Mellen 1976</a>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://people.bu.edu/robinc/kzk/">"Texas KZK Time Domain Code"</a>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFWoodsumWestervelt1981">Woodsum & Westervelt 1981</a>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a href="#CITEREFWoodsum2006">Woodsum 2006</a>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFCabotPutterman1981">Cabot & Putterman 1981</a>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFKaduchakSinhaLizonKelecher2000" class="citation journal cs1">Kaduchak, Gregory; Sinha, Dipen N.; Lizon, David C.; Kelecher, Michael J. (2000). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1259727">"A non-contact technique for evaluation of elastic structures at large stand-off distances: applications to classification of fluids in steel vessels"</a>. Ultrasonics. 37 (8): 531–536. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0041-624X%2899%2900109-2">10.1016/S0041-624X(99)00109-2</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11243456">11243456</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFNaugolnykhEsipov1995" class="citation journal cs1">Naugolnykh, Konstantin A.; Esipov, Igor B. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.414208">"Remote ocean sensing by parametric array"</a>. The Journal of the Acoustical Society of America. 98 (5): 2915. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995ASAJ...98.2915N">1995ASAJ...98.2915N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.414208">10.1121/1.414208</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFKonofagouThiermanHynynen2001" class="citation journal cs1">Konofagou, Elisa; Thierman, Jonathan; Hynynen, Kullervo (2001). <a rel="nofollow" class="external text" href="https://semanticscholar.org/paper/85b7f120e6568d0a912d79040fd1b0d7810b1053">"A focused ultrasound method for simultaneous diagnostic and therapeutic applications—a simulation study"</a>. Physics in Medicine and Biology. 46 (11): 2967–2984. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001PMB....46.2967K">2001PMB....46.2967K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9155%2F46%2F11%2F314">10.1088/0031-9155/46/11/314</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11720358">11720358</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2036873">2036873</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFZhangChenXiu-fen2001" class="citation journal cs1">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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001ASAJ..109.1219Z">2001ASAJ..109.1219Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.1344160">10.1121/1.1344160</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11303935">11303935</a>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFMuirWyber1984" class="citation journal cs1">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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984ASAJ...76...78M">1984ASAJ...76...78M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1121%2F1.2022023">10.1121/1.2022023</a>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070309235859/http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378">"Active control of sound using a parametric array"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378">the original</a> on 2007-03-09. Retrieved 2006-12-05.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="https://en.wikinews.org/wiki/Elwood_Norris_receives_2005_Lemelson-MIT_Prize_for_invention." class="extiw external" title="n:Elwood Norris receives 2005 Lemelson-MIT Prize for invention.">n:Elwood Norris receives 2005 Lemelson-MIT Prize for invention.</a>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFReevesGoldsberryRohde1979" class="citation book cs1">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. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FICASSP.1979.1170632">10.1109/ICASSP.1979.1170632</a>.</cite>

</li>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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</style><cite class="citation wikicite" id="CITEREFWoodsumWestervelt1981">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.</cite></li>

<li><cite class="citation wikicite" id="CITEREFWestervelt1963">Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963</cite></li>

<li><cite class="citation wikicite" id="CITEREFMoffettMellen1977">Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977</cite></li>

<li><cite class="citation wikicite" id="CITEREFMoffettMellen1976">Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976</cite></li>

<li><cite class="citation wikicite" id="CITEREFRoyWu1993">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)</cite></li>

<li>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)</li>

<li><cite class="citation wikicite" id="CITEREFBeyer1974">Robert T. Beyer, Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.</cite></li>

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

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

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

<li><cite class="citation wikicite" id="CITEREFBellinBeyer1960">J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960)</cite></li>

<li>M.J. Lighthill, Math. Revs. 19, 915 (1958)</li>

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

<li><cite class="citation wikicite" id="CITEREFWoodsum2006">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.</cite></li>

<li><cite class="citation wikicite" id="CITEREFMuir1976">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)</cite></li>

<li><cite class="citation wikicite" id="CITEREFAlbers1972">V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)</cite></li>

<li><cite class="citation wikicite" id="CITEREFCabotPutterman1981">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)</cite></li>

<li>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</li>

<li>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. <a rel="nofollow" class="external free" href="https://doi.org/10.3390/rs9090954">https://doi.org/10.3390/rs9090954</a></li></ul></div><div><div class="zim-footer">

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