Unsteady sound scattering by an elastic cylinder with a continuously non-uniform coating

Incidence of a plane non-stationary sound wave on a homogeneous elastic cylinder in ideal liquid with a coating in the form of an elastic cylindrical layer with a density and elastic moduli continuously changing in thickness is considered. It is assumed that the front of the incident wave is parallel to the axis of rotation of the cylinder. The pressure field in the sound wave scattered by the body is found.
Mathematical model of the diffraction process under consideration is constructed, based on the linearized model of hydrodynamics of an ideal compressible fluid and a model of the linear theory of elasticity. The acoustic pressure in the fluid, equal to the sum of the pressures in the incident and scattered fields, is a solution to the wave equation. The propagation of elastic waves in a homogeneous cylinder is described by two wave equations with respect to the scalar and vector potentials of elastic displacements. In this case, due to the formulation of the problem,
the vector equation is reduced to a scalar equation. The wave process in an inhomogeneous elastic coating is described by the general equations of motion of a continuous medium and Hooke’s law. In addition to the above equations, the model includes: zero initial conditions, free slip conditions on the outer surface of the coating, rigid adhesion conditions on the inner surface of the coating, attenuation at infinity condition for the scattered acoustic field, and a boundedness condition for wave fields in the body.
Integral Laplace transform with respect to time and the method of separating variables by radial and angular coordinates are applied to the equations of the constructed model. In the image space, the sought pressure and potentials are represented as expansions in series in modified cylindrical Bessel functions taking into account the radiation and boundedness conditions. Images of the components of the displacement vector, normal and tangential stresses in the coating are sought in the form of Fourier series with unknown coefficients depending
on the radial coordinate. To determine them, a boundary value problem is constructed for a
system of linear ordinary differential equations of the first order. The boundary value problem is reduced to problems with initial conditions. The transition to the space of originals is carried out numerically. The results of pressure calculations in the acoustic field scattered by the body are presented.

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