In this paper we re-interpret a recently introduced method for obtaining non-separable, localized solutions of homogeneous partial differential equations. This re-interpretation is in the form of a geometrical consideration of the algebraic constraint that the Fourier transforms of such solutions must satisfy in the transform domain (phase space). With this approach we link two classes of localized, non-separable solutions of the homogeneous wave equation, and examine the transform domain characteristic that determines the space-time localization properties of these classes. This characterization allows us to design classes of solutions with better localization properties. In particular, we design and discuss the properties of several novel subluminal and superluminal solutions of the homogeneous wave equation. We also design families of non separable, localized, subluminal and superluminal solutions of the Klein-Gordon equation by using the same technique.