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Diffraction of the cylindrical sound wave on the continuous inhomogeneous thermoelastic spherical shell

Larin N.V.

#### Abstract:

The diffraction of a cylindrical harmonic sound wave emitted by an infinitely long linear source is considered on an isotropic thermoelastic spherical shell of random thickness. The physical and mechanical characteristics of the shell material are described by continuous functions of the radial coordinate. It is assumed that the shell surfaces are bordered by non-viscous heat-conducting liquids, which are mostly different. The desired velocity potentials of acoustic and thermal waves outside and inside the shell cavity are the solutions of the Helmholtz equations, which satisfy the radiation conditions at infinity and the boundedness condition. The displacement of particles and temperature change in the thermoelastic shell are described by the system of equations of the linear coupled dynamic problem of the inhomogeneous isotropic body thermoelasticity. To simplify this system of equations, two new unknown functions are introduced. The functions are related by certain relations with the angular components of the displacement vector. The radial component of the displacement vector, the two new introduced functions and the change of the body temperature are in the form of expansions into series in spherical harmonics with unknown coefficients, which depend on the radial coordinate. Taking these expansions into account, the system of equations for describing thermomechanical perturbations in the shell reduces to a system of linear ordinary differential equations of the second order. Conditions for an ideal thermomechanical contact are satisfied on the outer and inner surfaces of the shell. Expressions for the coefficients of potential functions and boundary conditions for a system of differential equations are found from the boundary conditions. The obtained boundary value problem is solved by a spline-collocation method with the help of the cubic B -splines device. Analytic expressions describing the wave fields outside and inside the cavity of the shell are obtained. The results of the frequency and angular dependences calculations of the scattered sound field amplitude in the far zone are presented. We showed the noticeable difference in the characteristics of the sound scattering, caused by both different laws of the inhomogeneity of the shell material and the thermoelasticity of its material.

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