The parametric array is a nonlinear transduction mechanism that allows the generation of narrow, nearly sidelobe free beams of low frequency sound, through the mixing of higher frequency sound wave beams, effectively overcoming the diffraction limit (a kind of spatial 'uncertainty principle') associated with linear means of sound generation and scattering. Applications include underwater sound,medical ultrasound, underground sesimic prospecting, and directional high-fidelity commercial audio systems.
Priority for discovery and explanation of the Parametric Array owes to Westervelt, although experimental work was contemporaneously underway in the former Soviet Union. The phenomenon of the parametric array was seen first experimentally by Westervelt in the 1950's and theoretically presented first in 1960 at a meeting of the Acoustical Society of America, as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.
The foundation for Westervelt's theory of sound generation and scattering in nonlinear acoustic media owes to the equation of Lighthill [see http://en.wikipedia.org/wiki/James_Lighthill]. The application of Lighthill’s theory in the nonlinear acousic realm yields the Westervelt-Lighthill (nonlinear acoustic wave) Equation.Solutions to this equation have been developed using Green's functions and Parabolic Equation Methods, most notably the 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.
References
[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 Amercia, 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”