  The present study focuses on the interaction of gravity waves in the
atmosphere with the tropopause. As the vertical extent of the latter is small
compared to the density scale height, wave propagation is described by the
Taylor-Goldstein equation as derived from the linearised Boussinesq
approximation. Of particular interest in the construction of gravity wave
parameterisations for the upper atmosphere are the transmission and reflection
properties of the tropopause as these determine the upward fluxes of energy and
momentum carried by internal waves.
  A method is presented that decomposes internal waves explicitly into upward
and downward propagating contributions, thus giving direct access to
transmission and reflection coefficients of finite regions of non-uniform
stratification in a stationary atmosphere. The scheme utilizes a piecewise
constant approximation for the background stratification and matches up- and
downward propagating plane wave solutions in each layer through physically
meaningful coupling conditions. As a result, transmission and reflection
coefficients follow immediately.
