  A Gaussian beam method is presented for the analysis of the energy of the
high frequency solution to the mixed problem of the scalar wave equation in an
open and convex subset, with initial conditions compactly supported in this
set, and Dirichlet or Neumann type boundary condition. The transport of the
microlocal energy density along the broken bicharacteristic flow at the high
frequency limit is proved through the use of Wigner measures. Our approach
consists first in computing explicitly the Wigner measures under an additional
control of the initial data allowing to approach the solution by a
superposition of first order Gaussian beams. The results are then generalized
to standard initial conditions.
