CONTINUA OF LOCALIZED WAVE SOLUTIONS VIA A COMPLEX SIMILARITY TRANSFORMATION

In the following, we obtain continua of localized wave solutions to th e scalar homogeneous wave, damped wave, and Klein-Gordon equations. We do this by utilizing the fact that similar Ansatze (all of which invo lve a free-particle time-dependent Schrodinger-like equation) may be u sed to satisfy all three of these partial differential equations. This Schrodinger-like equation is reduced to an ordinary differential equa tion (ODE) using a dimensionless complex similarity transformation. A general solution to this ODE involving confluent hypergeometric functi ons is found. For an azimuthal dependence exp(inuphi), nu is-an-elemen t-of R, this general solution includes many of the previously determin ed localized wave solutions as special cases.