In previous publications [L.B. Felsen and E. Heyman, Proc. SPIE Vol 873 pp. 320-328 1988; E. Heyman, Wave Motion, in press], it has been shown how highly focused pulsed fields in vacuum can be generated analytically by assigning complex values to the space-time source coordinates of the conventional transient free-space Green's function. These new wave objects have been called complex source pulsed beams (CSPB). Their utility can be extended to generating new solutions for pulsed beam propagation and diffraction in a perturbed environment by making the space-time source coordinates in the corresponding Green's function complex. The analytic extension required in this process is performed systematically via the spectral theory of transients (STT) [E. Heyman and L.B. Felsen, IEEE Trans. Antennas Propagat. AP-35 (1987), 80-86, 574-580]. A canonical test case for reflection, including critical angle and lateral (head) wave eflEaTis provided by a dielectric half space. The exact solution for CSPB scattering is derived in spectral integral form, evaluated in terms of the time-dependent and time-independent spatial spectrum singularities in the complex plane, and interpreted physically. Numerical evaluation reveals the detailed space-time behavior of these physical constituents and their role in establishing the total scattered field.