Wave Field Decomposition in Anisotropic Fluids: A Spectral Theory Approach

An extension of directional wave field decomposition in acoustics from heterogenous isotropic media to generic heterogenous anisotropic media is established. We make a connection between the Dirichlet-to-Neumann map for a level plane, the solution to an algebraic Riccati operator equation, and a projector defined via a Dunford–Taylor type integral over the resolvent of a nonnormal, noncompact matrix operator with continuous spectrum.In the course of the analysis, the spectrum of the Laplace transformed acoustic system's matrix is analyzed and shown to separate into two nontrivial parts. The existence of a projector is established and using a generalized eigenvector procedure, we find the solution to the associated algebraic Riccati operator equation. The solution generates the decomposition of the wave field and is expressed in terms of the elements of a Dunford–Taylor type integral over the resolvent.