  We study cumulative scattering effects on wave front propagation in time
dependent randomly layered media. It is well known that the wave front has a
deterministic characterization in time independent media, aside from a small
random shift in the travel time. That is, the pulse shape is predictable, but
faded and smeared as described mathematically by a convolution kernel
determined by the autocorrelation of the random fluctuations of the wave speed.
The main result of this paper is the extension of the pulse stabilization
results to time dependent randomly layered media. When the media change slowly,
on time scales that are longer than the pulse width and the time it takes the
waves to traverse a correlation length, the pulse is not affected by the time
fluctuations. In rapidly changing media, where these time scales are similar,
both the pulse shape and the random component of the arrival time are affected
by the statistics of the time fluctuations of the wave speed. We obtain an
integral equation for the wave front, that is more complicated than in time
independent media, and cannot be solved analytically, in general. We also give
examples of media where the equation simplifies, and the wave front can be
analyzed explicitly. We illustrate with these examples how the time
fluctuations feed energy into the pulse.
