Initial-value problem of the one-dimensional wave propagation in a homogeneous random medium

The initial-value problem of the one-dimensional wave propagation in a homogeneous random medium is treated by means of the "Laplace transform," based again on a group-theoretic consideration introduced in the preceding paper. We first define the "Fourier transform" of a random process regarded as a function on the translation group associated with the homogeneity. The inverse "Fourier transform" then gives a general representation of a nonstationary random process generated by a stationary process. Similarly, we define the "Laplace transform" of a random process vanishing on the negative coordinate axis as well as the "Laplace transform" of its derivatives. The one-dimensional wave equation together with the random initial values can be directly treated by means of the "Laplace-transform" technique and is solved approximately in two Gaussian cases where the random media are represented by the well-known O-U (Ornstein-Uhlenbeck) process and by the $Z_0$ process having zero spectrum at the origin. Various statistical parameters associated with the solution can be calculated from the stochastic solution by the averaging procedure. It is shown that the behavior of the average wave is quite different between the two cases and that the result is in agreement with that of the preceding paper. The average of the absolute square of the wave is also calculated using the stochastic solution, and its range of validity is discussed by comparing with the previous results.