Ultrasonic wave propagation in randomly layered heterogeneous media

This article considers the propagation of high frequency elastic waves in a layered material. Each layer is locally anisotropic and the layer thicknesses and slowness surface orientations are modelled by a (Markovian) process. This work is important in deepening our understanding of the ultrasonic non-destructive testing of carbon fibre reinforced polymer (CFRP) composites and polycrystalline materials. The paper focuses on monochromatic shear waves propagating in two-dimensional ( plane) heterogeneous media. The displacement is in the  direction and the model focuses on the reflection and transmission of the wave at layer interfaces. The rotation of the slowness surface in each layer lies in the  plane and varies with the wave propagation direction ( ) only. Expressions for the local and global coefficients for the reflected and transmitted wave amplitudes are derived and shown to satisfy energy conservation. The resulting stochastic differential equations lead to a self-adjoint infinitesimal generator which can be used to produce a Fokker–Planck equation to study the probability distribution of the transmission coefficient. Explicit expressions for the moments of the probability distributions of the power transmission and reflection coefficients are then derived. The dependency of the mean and standard deviation of the power transmission coefficient on the depth of wave penetration, the localisation length, and the direction of wave propagation is then reported.