Scattering of mechanical waves from the perspective of open systems

In this paper, we consider the problem of mechanical wave scattering from a spatially finite system into an infinite surrounding environment. The goal is to illuminate why the scattering spectrum undergoes peaks and dips (resonances) at specific locations and how these locations connect to the vibrational properties of the scatterer. The resonance locations are connected to the eigenvalues of a finite dimensional effective operator, $H_{eff}$, corresponding to the scatterer. The developments are presented from the perspective of open systems, which seeks to convert the infinite dimensional scattering problem (scatterer+environment) into a finite dimensional effective problem involving only the finite scatterer. This is achieved through a projection operator formalism which allows us to formally calculate $H_{eff}$. An interesting corollary of our analysis is the deep connection between resonance locations in the scattering spectrum and the eigenfrequencies of the scatterer under Neumann boundary condition. We bring out this point further by considering 3D scattering from an elastic shell, connecting our results to classical results in acousto-elastic scattering theory.