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 Lévy 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 Lévy process is an inverse Gaussian process.