The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.