We consider propagation and time reversal of wave pulses in a random environment.
The focus of our analysis is the development of an expression for the two frequency
mutual coherence function for the harmonic wave field. This quantity plays
a crucial role in the analysis of many wave propagation phenomena and we
illustrate by explicitly considering time reversal in the context of
time pulses with a high carrier frequency.
In a time-reversal experiment the wave received by an active
transducer or antenna (receiver-emitter) array,
is recorded in a finite time window and then re-emitted into the medium
time reversed, that is, the tails of the recorded signals
are sent first.
The re-emitted wave pulse will focus approximately on the original
We use explicit expressions for the mutual coherence
functions and their asymptotic approximations in the regime of
long or short propagation distance and a high carrier frequency to analyze
the refocusing of the wave pulse in the time reversal experiment.
A novel aspect of our analysis is that we are able to characterize
precisely the decoherence length in temporal frequency.
This allows us to analyze for instance the time reversal experiment
when the mirror has a finite aperture in time.