Predictive modeling of ultrasonic pulse propagation in elastic solids is usually formulated in the frequency domain. Tractable solutions can then be obtained by using, for example, the powerful technique of geometrical elastodynamics and ray theory for wavefront propagation [1]. Recent advances [2,3] allow us to incorporate the finite pulse width by means of Gaussian profiles. However, a more realistic model should also include the fact that the pulse is of limited duration and therefore spatially localized in all directions. This paper outlines a theory for pulses in the form of a localized disturbance with a Gaussian envelope. The theory is valid if the associated carrier wavelength is short in comparison with typical length scales encountered in the solid. The method provides results explicitly in the time domain without the necessity of intermediate FFTs required by frequency domain methods. Applications to pulse propagation in smoothly varying inhomogeneous media, interface scattering and edge diffraction are discussed. The present theory contains an extra degree of freedom not explicitly considered before, i. e., the temporal width or duration of the pulse. An extensive treatment of the related problem for the scalar wave equation can be found in reference 4.