Unsteady sound scattering by an multilayer elastic cylinder

Abstract

The problem of diffraction of a plane sound pressure pulse by an infinite solid circular multilayer cylinder consisting of an arbitrary number of homogeneous isotropic elastic coaxial cylindrical layers of different thicknesses is considered. The pulse, propagating in an ideal fluid, falls on the cylindrical body parallel to its generatrix. The sound pressure in the wave scattered by the body is determined.
The components of displacement vector and stress tenzor in each homogeneous element of the multilayer body are expressed through the scalar and vector potentials of elastic displacements.
The sought pressure in the fluid, the scalar potential and the only non-zero component of the vector potential of elastic displacements satisfy the wave equations. Their solutions are found for zero initial conditions, conditions of free slip on the surface of the body in contact with the
fluid, conditions of rigid adhesion on the surfaces separating homogeneous elements of the body, the condition of attenuation of the scattered sound wave and the condition of boundedness of
the wave field in the cylinder.
The integral Laplace transform with respect to time is used to solve the problem. In the image space, the sought pressure and potentials are represented as expansions in series in cylindrical basis solutions of the Helmholtz equation, taking into account the conditions of radiation
at infinity and boundedness. The unknown coefficients included in the series are determined from a system of linear algebraic equations written for each summation index and obtained by substituting the images of the solutions into the images of the boundary conditions. The transition to the space of originals is carried out numerically. Using the previously obtained solution by the authors to the problem of scattering a plane acoustic pressure pulse by a homogeneous elastic cylinder with a continuously inhomogeneous elastic coating, the possibility of mathematical modeling of such a coating by a multilayer coating in a non-stationary problem of sound diffraction is shown.