  In this paper we analyze a time-reversal experiment in a random underwater
acoustic channel. In this kind of waveguide with semi-infinite cross section a
propagating field can be decomposed over three kinds of modes: the propagating
modes, the radiating modes and the evanescent modes. Using an asymptotic
analysis based on a separation of scales technique we derive the asymptotic
form of the the coupled mode power equation for the propagating modes. This
approximation is used to compute the transverse profile of the refocused field
and show that random inhomogeneities inside the waveguide deteriorate the
spatial refocusing. This result, in an underwater acoustic channel context, is
in contradiction with the classical results about time-reversal experiment in
other configurations, for which randomness in the propagation medium enhances
the refocusing.
