Parametric array
Encyclopedia
The parametric array is a nonlinear transduction
mechanism that generates narrow, nearly sidelobe free beams of low frequency sound, through the mixing and interaction of high frequency sound waves, effectively overcoming the diffraction limit (a kind of spatial 'uncertainty principle') associated with linear acoustics. Parametric arrays can be formed in water, air, and earth materials/rock.
, winner of the Lord Rayleigh Medal (currently Professor Emeritus at Brown University
), although important experimental work was contemporaneously underway in the former Soviet Union.
According to Muir [16, p. 554] and Albers [17], 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 [17], 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 [2] was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound, as described in [8,6,12].
media owes to an application of Lighthill
's equation (see Aeroacoustics) for fluid particle motion.
The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE). Solutions to this equation have been developed using Green's functions [4,5] and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.
An alternate mathematical formalism using Fourier operator
methods in wavenumber
space, was also developed by Westervelt, and generalized in [1] for solving the WLE in a most general manner. 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 [15], 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 [18].
Parametric receiving arrays can also be formed for directional reception. In 2005, Elwood Norris won the $500,000 MIT-Lemelson Prize
for his application of the parametric array to commercial high-fidelity loudspeakers.
[2] Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963
[4] Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977
[5] Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976
[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)
[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)
[8] Robert T. Beyer , Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.
[9] H.O. Berktay and D.J. Leahy, Journal of the Acoustical Society of America, 55, p. 539 (1974)
[10] M.J. Lighthill, "On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A211, 564-687 (1952)
[11] M.J. Lighhill, “On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A222, 1-32 (1954)
[12] J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960)
[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”
[15] 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.
[16] 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 Departmetn of the Navy (1976)
[17] V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)
[18] 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)
[19] 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
Transducer
A transducer is a device that converts one type of energy to another. Energy types include electrical, mechanical, electromagnetic , chemical, acoustic or thermal energy. While the term transducer commonly implies the use of a sensor/detector, any device which converts energy can be considered a...
mechanism that generates narrow, nearly sidelobe free beams of low frequency sound, through the mixing and interaction of high frequency sound waves, effectively overcoming the diffraction limit (a kind of spatial 'uncertainty principle') associated with linear acoustics. Parametric arrays can be formed in water, air, and earth materials/rock.
History
Priority for discovery and explanation of the parametric array owes to Peter J. WesterveltPeter Westervelt
Peter Westervelt is an American physicist and Professor Emeritus of Physics at Brown University. Westervelt is known for his work in nonlinear acoustics...
, winner of the Lord Rayleigh Medal (currently Professor Emeritus at Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
), although important experimental work was contemporaneously underway in the former Soviet Union.
According to Muir [16, p. 554] and Albers [17], 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 [17], 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
Acoustical Society of America
The Acoustical Society of America is an international scientific society dedicated to increasing and diffusing the knowledge of acoustics and its practical applications.-History:...
. A few years after this, a full paper [2] was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound, as described in [8,6,12].
Foundations
The foundation for Westervelt's theory of sound generation and scattering in nonlinear acousticNonlinear acoustics
Non-linear acoustics is a branch of physics dealing with sound waves being distorted as they travel.-Introduction:A sound wave propagates through a material as a localized pressure change...
media owes to an application of Lighthill
James Lighthill
Sir Michael James Lighthill, FRS was a British applied mathematician, known for his pioneering work in the field of aeroacoustics.-Biography:...
's equation (see Aeroacoustics) for fluid particle motion.
The application of Lighthill’s theory to the nonlinear acoustic realm yields the Westervelt–Lighthill Equation (WLE). Solutions to this equation have been developed using Green's functions [4,5] and Parabolic Equation (PE) Methods, most notably via the Kokhlov–Zablotskaya–Kuznetzov (KZK) equation.
An alternate mathematical formalism using Fourier operator
Fourier operator
The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform.It may be thought of as a limiting case for when the size of the discrete Fourier transform increases without bound while its spatial resolution also increases without bound,...
methods in wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
space, was also developed by Westervelt, and generalized in [1] for solving the WLE in a most general manner. 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 [15], 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 [18].
Applications
Practical applications are numerous and include:- underwater sound
- sonarSonarSonar is a technique that uses sound propagation to navigate, communicate with or detect other vessels...
- depth sounding
- sub-bottom profiling
- non-destructive testing
- and 'see through walls' sensing
- remote ocean sensing
- sonar
- medical ultrasoundUltrasoundUltrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...
- and tomography http://dx.doi.org/10.1121/1.1344160
- underground seismic prospecting
- active noise control
- and directional high-fidelity commercial audio systems (Sound from ultrasoundSound from ultrasoundSound from ultrasound is the name given here to situations when modulated ultrasound can make its carried signal audible without needing a receiver set...
)
Parametric receiving arrays can also be formed for directional reception. In 2005, Elwood Norris won the $500,000 MIT-Lemelson Prize
Lemelson-MIT Prize
The Lemelson Foundation awards several prizes yearly to inventors in United States. The largest is the Lemelson-MIT Prize which was endowed in 1994 by Jerome H. Lemelson, and is administered through the Massachusetts Institute of Technology...
for his application of the parametric array to commercial high-fidelity loudspeakers.
Further reading
[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.[2] Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963
[4] Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources", J. Acoust. Soc. Am. Vol. 61, No. 2, Feb. 1977
[5] Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976
[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)
[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)
[8] Robert T. Beyer , Nonlinear Acoustics, 1st Edition (1974),. Published by the Naval Sea Systems Command.
[9] H.O. Berktay and D.J. Leahy, Journal of the Acoustical Society of America, 55, p. 539 (1974)
[10] M.J. Lighthill, "On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A211, 564-687 (1952)
[11] M.J. Lighhill, “On Sound Generated Aerodynamically”, Proc. R. Soc. Lond. A222, 1-32 (1954)
[12] J.S. Bellin and R. T. Beyer, “Scattering of Sound by Sound”, J. Acoust. Soc. Am. 32, 339-341 (1960)
[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”
[15] 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.
[16] 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 Departmetn of the Navy (1976)
[17] V.M. Albers,"Underwater Sound, Benchmark Papers in Acoustics, p.415; Dowden, Hutchinson and Ross, Inc., Stroudsburg, PA (1972)
[18] 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)
[19] 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