Uniaxial waves in randomly heterogeneous elastic media

Propagation of uniaxial sound waves in heterogeneous linear elastic one-dimensional media is considered. The scalar wave equation is transformed by the Liouville substitution and the Dyson integral equation is applied for a statistically homogeneous field of heterogeneities that results in an integral spatial representation for the mean wave field. The mean field is analysed in detail for the following three correlation functions: (i) an exponential one; (ii) a mean-square differentiable correlation function with a hidden periodicity; and (iii) a non-differentiable correlation function with a hidden periodicity. Despite the variety of the stochastic properties of the media, the equation for the mean field takes formally the same form for the three cases. For this reason, the general case of the random elastic medium with an arbitrary heterogeneity of small scale is considered and simple closed form expressions for the mean field and attenuation are derived. Applicability of the modelling of extended complex engineering structures by one-dimensional random media is discussed. The overall mechanical parameters of the primary structure and the secondary systems determine average rigidity, average mass density and average wave speed, while the secondary systems determine the attenuation. The latter is shown to depend upon the size and range of the secondary systems.