Nondiffracting X Waves-Exact Solutions to Free-Space Scalar Wave Equation and Their Finite Aperture Realizations

Jian Yu Lu, James F. Greenleaf

Research output: Contribution to journalArticlepeer-review

440 Scopus citations


Novel families of generalized nondiffracting waves have been discovered. They are exact nondiffracting solutions of the isotropic/homogenous scalar wave equation and are a generalization of some of the previously known nondiffracting waves such as the plane wave, Durnin's beams, and the nondiffracting portion of the Axicon beam equation in addition to an infinity of new beams. One subset of the new nondiffracting waves have X-like shapes that are termed “X waves.” These nondiffracting X waves can be almost exactly realized over a finite depth of field with finite apertures and by either broad band or bandlimited radiators. With a 25 mm diameter planar radiator, a zeroth-order broadband X wave will have about 2.5 mm lateral and 0.17 mm axial -6-dB beam widths with a -6-dB depth of field of about 171 mm. The phase of the X waves changes smoothly with time across the aperture of the radiator, therefore, X waves can be realized with physical devices. A zeroth-order bandlimited X wave was produced and measured in water by our 10 element, 50 mm diameter, 2.5 MHz PZT ceramic/polymer composite J0 Bessel nondiffracting annular array transducer with -6-dB lateral and axial beam widths of about 4.7 mm and 0.65 mm, respectively, over a -6-dB depth of field of about 358 mm. Possible applications of X waves in acoustic imaging and electromagnetic energy transmission are discussed.

Original languageEnglish (US)
Pages (from-to)19-31
Number of pages13
JournalIEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control
Issue number1
StatePublished - Jan 1992

ASJC Scopus subject areas

  • Instrumentation
  • Acoustics and Ultrasonics
  • Electrical and Electronic Engineering


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