Directional Decomposition in Anisotropic Heterogeneous Media for Acoustic and Electromagnetic Fields

Abstract: Directional wave-field decomposition for heterogeneousanisotropic media with in-stantaneous response is establishedfor both the acoustic and the electromagnetic equations.We derive a sufficient condition for ellipticity of thesystem's matrix in the Laplace domain and show that theconstruction of the splitting matrix via a Dunford-Taylorintegral over the resolvent of the non-compact, non-normalsystem's matrix is well de ned. The splitting matrix also hasproperties that make it possible to construct the decompositionwith a generalized eigenvector procedure. The classical way ofobtaining the decomposition is equivalent to solving analgebraic Riccati operator equation. Hence the proceduredescribed above also provides a solution to the algebraicRiccati operator equation.The solution to the wave-field decomposition for theisotropic wave equation is expressed in terms of theDirichlet-to-Neumann map for a plane. The equivalence of thisDirichlet-to-Neumann map is the acoustic admittance, i.e. themapping between the pressure and the particle velocity. Theacoustic admittance, as well as the related impedance aresolutions to algebraic Riccati operator equations and are keyelements in the decomposition. In the electromagnetic case thecorresponding impedance and admittance mappings solve therespective algebraic Riccati operator equations and henceprovide solutions to the decomposition problem.The present research shows that it is advantageous toutilize the freedom implied by the generalized eigenvectorprocedure to obtain the solution to the decomposition problemin more general terms than the admittance/impedancemappings.The time-reversal approach to steer an acoustic wave eld inthe cavity and half space geometries are analyzed from aboundary control perspective. For the cavity it is shown thatwe can steer the field to a desired final configuration, withthe assumption of local energy decay. It is also shown that thetime-reversal algorithm minimizes a least square error forfinite times when the data are obtained by measurements. Forthe half space geometry, the boundary condition is expressedwith help of the wave-field decomposition. In the homogeneousmaterial case, the response of the time-reversal algorithm iscalculated analytically. This procedure uses the one-wayequations together with the decomposition operator.