Continuous gradient and discretized layered designs for control of stress wave scattering

The scattering of stress waves in graded media is discussed in this paper using a conversion of the heterogeneous wave equation to a potential form. In this form, the scattering and inverse scattering methods of mathematical physics can be used with proper enforcement of the boundary conditions on the mechanical quantities, i.e. displacements and tractions. Two approaches for design of non-reflective media are analyzed, one for all frequencies with infinite thickness, and one with finite thickness for a target frequency range. In the latter approach, the transformed potential is set to zero, and the wave speed can have almost any desired profile, from which the profile of impedance is derived. The dispersive behavior of the finite thickness design is studied. Finally, an example is discussed in which a continuous non-reflective profile is discretized to a piece-wise constant profile and the effect of such unavoidable practical constraints on the scattering of the designed layer is shown to be minimal.