Root-finding absorbing boundary conditions for scalar and elastic waves in infinite media

Abstract

Root-Finding Absorbing Boundary Conditions (RFABCs) for scalar and elastic waves in infinite media are developed. First, the existing RFABC formulation for scalar-wave propagation problems is refined. Specifically, a Fourier series expansion or an eigenfunction expansion, as in Sturm–Liouville problems, is applied to general scalar-wave fields and a consistent nodal flux is derived for eventual combination with the discrete representation of the problem under consideration. This newly-proposed approach for scalar waves is extended to elastic waves, governed by two scalar-wave equations for pressure and shear waves. Stability of the refined RFABCs can be proven at both the continuous and discrete levels. The newly-developed RFABCs for scalar and elastic waves are applied to various wave-propagation problems. It is verified that they produce accurate and stable results. Perfectly matched layers and high-order absorbing boundary conditions are the most popular of the available tools for wave-propagation problems. But, their stability properties have not been established in applications to problems of elastic waves in waveguides. It is expected that existing high-order absorbing boundaries will lead to stable results for elastic waves in waveguides, when implemented using the technique proposed in the present study.