###
On the problem of a spherical sound wave diffraction by an elastic inhomogeneous anisotropic ball with an absolutely solid inclusion

Birykov D.R.

#### Abstract:

A mathematical formulation of the problem of diffraction of a spherical sound wave by a linearly elastic radially inhomogeneous transversely isotropic ball with an absolutely solid inclusion is presented. The ball is characterized by density, elastic constants – components of the elasticity tensor – and external and internal radii. The sphere described above is placed in a three-dimensional unlimited space filled with an ideal fluid with certain values of density and speed of sound. The statement describes the input data and some of their limitations. An algorithm for solving the posed diffraction problem is presented. The algorithm is partly analytical, partly numerical. An incident spherical wave, a sound wave scattered by a ball, and elastic waves propagating inside an elastic ball are represented as infinite sums. The definition of the wave scattered by the ball is reduced to the determination of the coefficients of the expansion of the scattered wave field into an infinite sum. To determine these coefficients, a boundary value problem is solved. The differential equations in this boundary value problem are ordinary differential equations describing waves in an elastic ball and obtained from the general equations of motion of a continuous medium. These differential equations are supplemented by boundary conditions on the surfaces of an elastic ball. On the outer surface, the boundary conditions are the continuity of the velocity, normal and shear stresses. On the inner surface – continuity of displacements. The solution of the boundary value problem with the given conditions makes it possible to calculate the displacements inside the ball during the propagation of the wave u. through them, the coefficients of the sound wave scattered by the body. To demonstrate the solution of the problem with the help of software implementation, the results of numerical studies for some particular input data are given.

0