The scalar time-dependent equation of radiative transfer is used to develop a theory of bounded beam wave narrow band pulse propagation and scattering in a medium characterized by many random discrete scatters, which scatters energy strongly in the forward scattering direction. Applications include the scattering of highly collimated millimeter waves in vegetation and optical beams in the atmosphere. The specific problem analyzed is that of a periodic sequence of Gaussian shaped pulses normally incident from free space onto the planar boundary surface of a random medium half-space, such as a forest, that possesses a scatter (phase) function consisting of a strong, narrow forward lobe superimposed over an isotropic background. After splitting the specific intensity into the reduced incident and diffused intensities, the solution of the transport equation expressed in cylindrical coordinates in the random medium half-space is obtained by using the Fourier-Bessel transform along with the two-dimensional Gauss quadrature formula and an eigenvalue-eigenvector technique, following the procedure developed by Chang and lshimaru for CW propagation. Curves of received power are obtained for different penetration depths, different incident beamwidths, and different scatter directions. At large penetration depths, as well as for scatter directions different from the incident radiation, the power is shown to attenuate significantly and the pulse widths are shown to broaden, which resulted in considerable pulse distortion. Results for different beam widths are also obtained.