Wave propagation in a cylindrical viscous layer between two elastic shells

Propagation of longitudinal waves in a liquid-filled layer between two thin coaxial shells is investigated. Both liquid viscosity and elasticity of the shells are accounted for. Dynamics of the shells is treated using the Kirchhoff–Love approximation. The elastic deformations of the shells in the sound wave are coupled with the liquid flow in the gap through appropriate dynamic and kinematic boundary conditions. Hydrodynamics of the liquid is described using the quasi-one-dimensional (hydraulic) approach. It is assumed that the external and internal shells are composed of different isotropic elastic materials and have different widths. The dispersion equation for harmonic waves in the system is obtained; it is valid in the low frequency range where the wave length is greater than the external shell radius. In the limiting case for an ideal liquid the dispersion equation yields water hammer speed in the system. The analysis of the dispersion equation has shown strong influence of viscous losses on dispersion and attenuation of pressure signals in the low frequency region. The wave speed and attenuation are highly dependent on the geometrical parameters of the system and elastic properties of the shells.