  Magnetohydrodynamic waves interact with structured plasmas and reveal the
internal magnetic and thermal structures therein, thereby having seismological
applications in the solar atmosphere. We investigate the evolution of fast
magnetoacoustic pulses in randomly structured plasmas, in the context of
large-scale propagating waves in the solar atmosphere. We perform one
dimensional numerical simulations of fast wave pulses propagating perpendicular
to a constant magnetic field in a low-$\beta$ plasma with a random density
profile across the field. Both linear and nonlinear regimes are considered. We
study how the evolution of the pulse amplitude and width depends on their
initial values and the parameters of the random structuring. A randomly
structured plasma acts as a dispersive medium for a fast magnetoacoustic pulse,
causing amplitude attenuation and broadening of the pulse width. After the
passage of the main pulse, secondary propagating and standing fast waves appear
in the plasma. Width evolution of both linear and nonlinear pulses can be well
approximated by linear functions; however, narrow pulses may have zero or
negative broadening. This arises because a narrow pulse is prone to splitting,
while a broad pulse usually deviates less from their initial Gaussian shape and
form ripple structures on top of the main pulse. A linear pulse decays at
almost a constant rate, while a nonlinear pulse decays exponentially. A pulse
interacts most efficiently with a random medium which has a correlation length
of about half of its initial pulse width. The development of a detailed model
of a fast MHD pulse propagating in highly structured medium substantiates the
interpretation of EIT waves as fast magnetoacoustic waves. Evolution of a fast
pulse provides us with a novel method to diagnose the sub-resolution
filamentation of the solar atmosphere.
