Continua of localized wave solutions via a complex similarity transformation

In the following, we obtain continua of localized wave solutions to the scalar homogeneous wave, damped wave, and Klein-Gordon equations. We do this by utilizing the fact that similar Ans\"atze (all of which involve a free-particle time-dependent Schr\"odinger-like equation) may be used to satisfy all three of these partial differential equations. This Schr\"odinger-like equation is reduced to an ordinary differential equation (ODE) using a dimensionless complex similarity transformation. A general solution to this ODE involving confluent hypergeometric functions is found. For an azimuthal dependence exp(i{ν}{φ}),{ν}{∈}openR, this general solution includes many of the previously determined localized wave solutions as special cases.