Transient transport equations for high-frequency power flow in heterogeneous cylindrical shells

Abstract

We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic partial differential equations to derive exact transient power flow equations for vibrations of randomly heterogeneous cylindrical shells. The theory shows that the angularly resolved energy densities of an heterogeneous, elastic medium satisfy transient transport equations at higher frequencies. The behaviour of solutions of such equations short of their diffusion limit—if any—is fundamentally different from that of the solution of a diffusion equation, although the latter one is often invoked in the analyses of high-frequency vibrations of elastic structures. A condition by which diffusion equations can be obtained from transport equations is the presence of reflectors or heterogeneities such that scattering mean free paths are short with respect to the characteristic dimensions of the structure. The diffusion limit is reached in this study taking account of scattering by random heterogeneities of the background medium at the scale of the wavelength. This approach fills the gap between transport theory and the diffusion approach in structural dynamics, and clarifies the range of validity of the latter. Our results can be extended to fully coupled dynamic equations for compression, shear and bending of Timoshenko beams or Mindlin plates.