LYAPUNOV'S INDEX OF ONE-DIMENSIONAL WAVE EQUATION WITH RANDOM COEFFICIENTS.

One-dimensional wave equation omega **2 rho (x)u plus ( sigma ** minus **1(x)u//x)//x equals 0, is considered in which rho (x) and sigma (x) are random functions. Lyapunov's index of this equation is studied as a function of the frequency omega for certain random processes. Qualitative curves of this relationship are plotted and various asymptotics are found. The typical situation is one where Lyapunov's index grows monotonically with frequency. An example where no monotonicity is observed is constructed.