In this work an analysis of transient wave propagation in forward scattering random media is presented. The analysis is based on evaluation of the two-frequency mutual coherence function, which is an important quantity in itself since it provides a measure of the coherence bandwidth. The coherence function is calculated by using the path integral technique; specifically, by resorting to a cumulant expansion of the path integral. In contrast to the formulas available in the literature, the solution obtained is not limited by the strength of disorder and applies equally well to both dispersive and nondispersive media, with arbitrary spectra of inhomogeneities. For the regime of weak scattering (or relatively short propagation distances) the first cumulant gives an excellent approximation coinciding with the results obtained earlier in a particular case of the Kolmogorov turbulence by solving the corresponding differential equation numerically. In the regime of strong scattering (long distances), which to our knowledge has not been covered previously, our solution demonstrates a different type of scaling dependence. It is shown that, even for power spectra with fractal behavior in a wide range of spatial frequencies, the coherence function is very sensitive to fine details of the spectrum at both small and large spatial scales. Using the cumulant expansion, the temporal moments of the pulsed wave propagating in a random medium are also considered. It is found that the temporal moments of the pulse are determined exactly by accounting for a corresponding number of the cumulants. In particular, the average time delay of the pulse is determined by the first cumulant, and the pulse width is obtained by accounting for the first two cumulants. Although the consideration of the problem is based on the model of a continuous medium, the results are also applicable to wave propagation in media containing discrete particles scattering predominantly in the forward direction.