  The paper addresses one-dimensional transport in a Goupillaud medium (a
layered medium in which the layer thickness is proportional to the propagation
speed), as a prototypical case of wave propagation in random media. Suitable
stochastic assumptions and limiting procedures lead to characteristic curves
that are L\'evy processes. Solutions corresponding to the discretely layered
medium are shown to converge to limits as the thickness of the layers goes to
zero. The probability distribution of the limiting characteristic curves is
explicitly computed and exemplified when the underlying L\'evy process is an
inverse Gaussian process.
