An X wave transform

Limited diffraction beams such as X waves can propagate to an infinite dist ance without spreading if they are produced with an infinite aperture and e nergy. In practice, when the aperture and energy are finite, these beams ha ve a large depth of field with only limited diffraction. Because of this pr operty, limited diffraction beams could have applications in medical imagin g, tissue characterization, blood flow velocity vector imaging, nondestruct ive evaluation (NDE) of materials, communications, and other areas such as optics and electromagnetics. In this paper, a new transform, called X wave transform, is developed. In the transform, any well behaved solutions to th e isotropic-homogeneous wave equation or limited diffraction beams can be e xpanded using X waves as basis functions. The coefficients of the expansion s can be calculated with the properties that X waves are orthogonal. Exampl es are given to demonstrate the efficacy of the X wave transform. The X wav e transform reveals all intrinsic relationship between any well behaved. so lutions to the wave equation and X waves, including limited diffraction bea ms. This provides a theoretical foundation to develop new limited diffracti on beams or solutions to the wave Equation that may have practical usefulne ss.