Using time-reversal to generate generalized transversely localized transient waves (X-waves)

In the traditional approach to X-waves, the X-wave field is synthesized from a superposition of solutions to the homogenous wave equation (in three-dimensions) without regard to boundary conditions. As a consequence the synthesized solution is acausal. Here, it is shown that the solution to the inhomogenous scalar wave equation for the acoustic field from a supersonic source distribution consistent with the radiation condition, i.e., a Mach front, defines a causal X-wave. Using the connection between X-waves and a physical source, it is shown that an X-wave can be generated from a planar aperture using time-reversal. By appealing to the demonstrated self-adaptivity of time-reversal processes, the method should allow for the generation of X-waves in arbitrary (inhomogenous) media. Typically, the generation of approximate acoustic X-waves from a planar aperture is achieved using a complicated annular transducer arrangement. Here, the time-reversal method for the generation of approximate acoustic X-waves is experimentally proven using a line transducer array in two-dimensional geometry in free space.