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. 
Priority for discovery and explanation of the Parametric Array  owes to Peter J. Westervelt, 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 , 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 , 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 1950's, was later explained theoretically in 1960, at a meeting of the Acoustical Society of America. A few years after this, a full paper  was published as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound, as described in [8,6,12].
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  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 , 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 .
Practical applications are numerous and include:
- underwater sound
- medical ultrasound 
- and tomography ,
- underground sesimic prospecting 
- active noise control 
- and directional high-fidelity commercial audio systems (Sound from ultrasound
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.
 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. Lighhill, “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 Departmetn 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