Propagation of longitudinal waves in a random binary rod

The one-dimensional wave equation describes many phenomena in physics [1]. In a periodic medium, the solution of the wave equation has a Floquet–Bloch structure. Very long waves propagate without attenuation and the medium can be viewed as homogeneous. However, for higher frequencies there are certain intervals (so-called gaps) where waves cannot propagate even though the system is perfectly conservative. Due to multiple scattering, destructive interference occurs, so that the wave amplitude decreases exponentially in the medium. The rate of decay of the solution is called the Lyapunov exponent (the inverse of the localization length). The Lyapunov exponent is strictly positive inside the gaps and equals zero in the spectral bands. The Floquet–Bloch structure of the solution is completely destroyed if the medium is not perfectly periodic because of random independent variations of geometric or material parameters [2]. Now as the wave propagates, its amplitude decreases exponentially with probability one and the medium can be considered homogeneous for long waves only on a finite time interval. This phenomenon is known as localization. Calculation of the Lyapunov exponent for disordered systems usually employs the transfer matrix formalism, asymptotic analysis, and numerical modelling [3–6]. Our approach is based on the phase-amplitude representation of the solution and stochastic methods. This allows us to find exactly the joint probability density distribution for the phase of the solution that leads to an exact formula for the Lyapunov exponent.