An analytical partial solution for the exploding reflector model for scattering of waves from a boundary

Abstract

The exploding reflector model has been suggested as a way to simplify calculation of wave propagation. Previous work has produced numerical solutions to a model where two media are in contact and a signal is generated at the surface of the upper layer. Scatter from the interface was modelled as an exploding reflector. The strength and shape of the signal that arrives at the base of the lower layer were calculated following the scatter from the interface. These numerical solutions gave results that were comparable to finite difference solutions of the wave equation. The physical meaning and an extension of the exploding reflector model are discussed here. Exact analytical solutions to a two layer model separated by an incoherent scattering layer are derived by two distinct methods, one using the mathematical computer package Maple and then by simple geometrical arguments. Exact analytical solutions could be found for the case where the layers had identical thickness and sound velocities and the interface scattered the waves incoherently. Two different laws for the reduction in intensity of the signal as a function of distance were examined being the inverse unity power law and inverse square law. Two different laws for scattering from the interface were also examined being an isotropic scatter for which scattering intensity is independent of scattering angle, and a Lambertian scatter which obeys Lambert's Cosine Law. Analytical solutions to the model with unequal thicknesses or unequal acoustic velocities were investigated. Analytical solutions could be found in some cases but could not usually be reduced to simple expressions that could be presented here. The general solution is possible but required the solution to the general quartic equation. Solution expressions are very long and can only reasonably be given as process flows in a mathematical computing package such as Maple. The models described here and the solutions found previously may find a use in describing the incoherent scatter from interfaces between layers.