Pulsed beams (PB) are highly collimated wave objects which can simulate actual acoustic fields radiated by time-domain (TD) compact transducers and, by superposition, can analytically synthesize arbitrary pulsed source distributions. While much progress has been made in tracking PB wave packets through nondispersive environments, the TD phenomenologies characterizing PB interaction with dispersive media are as yet inadequately understood. To initiate this study, we review known TD asymptotic techniques for pulsed noncollimated wave objects with real frequency and wave number constituents. The results can be interpreted in terms of space-time ray theory and instantaneous dispersion relations, with interesting graphical representations (in the configuration--spectrum phase space) that highlight the relevant wave physics. Of special interest are inputs which focus the pulsed signal at a specified space-time point inside the medium. The extension to pulsed beams requires consideration of complex spectra, as performed in part II.