Diffraction of long elastic waves by an isolated inclusion in a homogeneous medium

We consider the one-particle problem for self-consistent methods. The solution of this problem is constructed for incident waves longer than characteristic sizes of the inclusion (long-wave approximation). Explicit equations for the elastic field inside an ellipsoidal inclusion and its limiting forms (oblate and prolate spheroids) are obtained in Sections 5.1–5.3. Thin soft and hard inclusions, and hard axisymmetric fibers are considered in Section 5.4. The final Section 5.5 is devoted to the proof of the optical theorem for diffraction of elastic waves, and the calculation of the total scattering cross-sections of inclusions of various forms.