(2+1)-dimensional X-shaped localized waves

A hybrid spectral superposition method is presented that allows a smooth transition between two seemingly distinct classes of localized wave solutions to the homogeneous scalar wave equation in free space; specifically, luminal or focus wave modes, and superluminal or X waves. This representation, which is based on superpositions of products of forward plane waves moving at a fixed speed v>c and backward plane waves moving at the speed c, is used to construct a large class of finite-energy superluminal-type X-shaped localized waves. The latter are characterized by arbitrarily high-frequency bands and are suitable for applications in the microwave and optical regime. In the limiting case v-->c, one recaptures the well-known focus wave mode-type localized wave solutions. A modified hybrid spectral representation, based on superpositions of products of forward plane waves moving at a fixed speed c and backward plane waves moving at the speed v>c, allows in the limit v-->c a smooth transition from superluminal localized waves to paraxial luminal pulsed beams. Although the proposed methods are applicable to a (n+1)-dimensional, n> or =2, scalar wave equation, the discussion will be limited to the case n=2 for simplicity; also, so that comparisons can be made to related recent results in the literature.