  We use random matrix theory (RMT) to study the first two moments of the wave
power transmitted in time reversal invariant systems having ergodic motion.
Dissipation is modeled by a number of loss channels of variable coupling
strength. To make a connection with ultrasonic experiments on ergodic
elastodynamic billiards, the channels injecting and collecting the waves are
assumed to be negligibly coupled to the medium, and to contribute essentially
no dissipation. Within the RMT model we calculate the quantities of interest
exactly, employing the supersymmetry technique. This approach is found to be
more accurate than another method based on simplifying naive assumptions for
the statistics of the eigenfrequencies and the eigenfunctions. The results of
the supersymmetric method are confirmed by Monte Carlo numerical simulation and
are used to reveal a possible source of the disagreement between the
predictions of the naive theory and ultrasonic measurements.
