Parametric array

<h2 id="history">History</h2>
Priority for discovery and explanation of the parametric array owes to <a href="/facts/Peter_Westervelt/7Sh1cMcD">Peter J. Westervelt</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> winner of the <a href="/facts/Lord_Rayleigh/HcJyYdk5">Lord Rayleigh</a> Medal<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> (currently Professor Emeritus at <a href="/facts/Brown_University/o9MRu4jx">Brown University</a>), although important experimental work was contemporaneously underway in the former Soviet Union.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
According to Muir<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> and Albers,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> 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 <a href="/facts/Acoustical_Society_of_America/0PvCloyB">Acoustical Society of America</a>. A few years after this, a full paper<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h2 id="foundations">Foundations</h2>
The foundation for Westervelt's theory of sound generation and scattering in <a href="/facts/Nonlinear_acoustics/sCN6t1yS">nonlinear acoustic</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> media owes to an application of <a href="/facts/James_Lighthill/dKjIehdI">Lighthill</a>'s <a href="/facts/Aeroacoustics/FJN943nB">equation</a> for fluid particle motion.
The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE).<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> Solutions to this equation have been developed using <a href="/facts/Green%27s_functions/rKxKKPcp">Green's functions</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
An alternate mathematical formalism using <a href="/facts/Fourier_operator/PUU4JepC">Fourier operator</a> methods in <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a> space, was also developed and generalized by Westervelt.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</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 class="footnote-ref" id="fnref:22" href="#fn:22">22</a> 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.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<h2 id="applications">Applications</h2>
Practical applications are numerous and include:

<ul><li>underwater sound
<ul><li><a href="/facts/Sonar/0Rg2G40e">sonar</a></li>
<li>depth sounding</li>
<li>sub-bottom profiling</li>
<li>non-destructive testing</li>
<li>and 'see through walls' sensing<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></li>
<li>remote ocean sensing<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a></li></ul></li>
<li>medical <a href="/facts/Ultrasound/31Pa5IH9">ultrasound</a><a class="footnote-ref" id="fnref:26" href="#fn:26">26</a></li>
<li>and tomography<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a></li>
<li>underground seismic prospecting<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a></li>
<li>active noise control<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a></li>
<li>and directional high-fidelity commercial audio systems (<a href="/facts/Sound_from_ultrasound/Dqv7bUBJ">Sound from ultrasound</a>)<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></li></ul>
Parametric receiving arrays can also be formed for directional reception.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> In 2005, Elwood Norris won the $500,000 <a href="/facts/Lemelson-MIT_Prize/S5AHsHed">MIT-Lemelson Prize</a> for his application of the parametric array to commercial high-fidelity loudspeakers.

<h2 id="further-reading">Further reading</h2>
<ul><li>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.</li>
<li>Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963</li>
<li>Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977</li>
<li>Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976</li>
<li>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)</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>Robert T. Beyer, Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.</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>J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960)</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>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.</li>
<li>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)</li>
<li>V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)</li>
<li>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)</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 href="https://doi.org/10.3390/rs9090954">https://doi.org/10.3390/rs9090954</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Beyer, Robert T. "Preface to the Original Edition". Nonlinear Acoustics. <a href="http://asa.aip.org/books/nonlinear.html#Preface1" target="_blank">http://asa.aip.org/books/nonlinear.html#Preface1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Novikov, B. K.; Rudenko, O. V.; Timoshenko, V. I. (1987). Nonlinear Underwater Acoustics. Translated by Robert T. Beyer. American Institute of Physics. ISBN 9780883185223. OCLC 16240349. <a href="9780883185223" target="_blank">9780883185223</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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:1980ASAJ...68.1214T. doi:10.1121/1.384959. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Johnson, P. A.; Meegan, G. D.; McCall, K.; Bonner, B. P.; Shankland, T. J. (1992). "Finite amplitude wave studies in earth materials". The Journal of the Acoustical Society of America. 91 (4): 2350. Bibcode:1992ASAJ...91.2350J. doi:10.1121/1.403453. <a href="https://doi.org/10.1121%2F1.403453" target="_blank">https://doi.org/10.1121%2F1.403453</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Parametric Beam Formation in Rock <a href="http://www.lanl.gov/orgs/ees/ees11/geophysics/nonlinear/pubs/parabeam.html" target="_blank">http://www.lanl.gov/orgs/ees/ees11/geophysics/nonlinear/pubs/parabeam.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Professor Peter Westervelt and the parametric array <a href="http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000005003231000004&idtype=cvips&gifs=yes" target="_blank">http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000119000005003231000004&idtype=cvips&gifs=yes</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Institute of Acoustics - Medals & Awards Programme Archived 2009-06-28 at the Wayback Machine <a href="http://www.ioa.org.uk/medals-and-awards/" target="_blank">http://www.ioa.org.uk/medals-and-awards/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Novikov, B. K.; Rudenko, O. V.; Timoshenko, V. I. (1987). Nonlinear Underwater Acoustics. Translated by Robert T. Beyer. American Institute of Physics. ISBN 9780883185223. OCLC 16240349. <a href="9780883185223" target="_blank">9780883185223</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Muir 1976, p. 554. - 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) <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Albers 1972 - V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972) <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Albers 1972 - V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972) <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Westervelt 1963 - Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963 <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Roy & Wu 1993 - 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) <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Beyer 1974 - Robert T. Beyer, Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Bellin & Beyer 1960 - J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960) <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Westervelt, Peter J. (1975). "The status and future of nonlinear acoustics". The Journal of the Acoustical Society of America. 57 (6): 1352–1356. Bibcode:1975ASAJ...57.1352W. doi:10.1121/1.380612. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Sources of Difference Frequency Sound in a Dual-Frequency Imaging System with Implications for Monitoring Thermal Surgery[permanent dead link‍] <a href="https://dspace.mit.edu/bitstream/1721.1/28762/1/59823423.pdf" target="_blank">https://dspace.mit.edu/bitstream/1721.1/28762/1/59823423.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Moffett & Mellen 1977 - Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977 <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Moffett & Mellen 1976 - Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">"Texas KZK Time Domain Code". <a href="http://people.bu.edu/robinc/kzk/" target="_blank">http://people.bu.edu/robinc/kzk/</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Woodsum & Westervelt 1981 - 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. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Woodsum 2006 - 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. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Cabot & Putterman 1981 - 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) <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">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". Ultrasonics. 37 (8): 531–536. doi:10.1016/S0041-624X(99)00109-2. PMID 11243456. <a href="https://zenodo.org/record/1259727" target="_blank">https://zenodo.org/record/1259727</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Naugolnykh, Konstantin A.; Esipov, Igor B. (1995). "Remote ocean sensing by parametric array". The Journal of the Acoustical Society of America. 98 (5): 2915. Bibcode:1995ASAJ...98.2915N. doi:10.1121/1.414208. <a href="https://doi.org/10.1121%2F1.414208" target="_blank">https://doi.org/10.1121%2F1.414208</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Konofagou, Elisa; Thierman, Jonathan; Hynynen, Kullervo (2001). "A focused ultrasound method for simultaneous diagnostic and therapeutic applications—a simulation study". Physics in Medicine and Biology. 46 (11): 2967–2984. Bibcode:2001PMB....46.2967K. doi:10.1088/0031-9155/46/11/314. PMID 11720358. S2CID 2036873. <a href="https://semanticscholar.org/paper/85b7f120e6568d0a912d79040fd1b0d7810b1053" target="_blank">https://semanticscholar.org/paper/85b7f120e6568d0a912d79040fd1b0d7810b1053</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">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:2001ASAJ..109.1219Z. doi:10.1121/1.1344160. PMID 11303935. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">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:1984ASAJ...76...78M. doi:10.1121/1.2022023. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">"Active control of sound using a parametric array". Archived from the original on 2007-03-09. Retrieved 2006-12-05. <a href="https://web.archive.org/web/20070309235859/http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378" target="_blank">https://web.archive.org/web/20070309235859/http://www.mecheng.adelaide.edu.au/anvc/abstract.php?abstract=378</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">n:Elwood Norris receives 2005 Lemelson-MIT Prize for invention. <a href="https://en.wikinews.org/wiki/Elwood_Norris_receives_2005_Lemelson-MIT_Prize_for_invention." target="_blank">https://en.wikinews.org/wiki/Elwood_Norris_receives_2005_Lemelson-MIT_Prize_for_invention.</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">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:10.1109/ICASSP.1979.1170632. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
</ol>

Parametric array open-in-new

Parametric array