Wave Equations and Solutions of In Vacuo and Fluid-Filled Elliptical Cylindrical Shells

Abstract

A perturbation-based approach is used to formulate the governing equations for wave propagation at high frequencies in infinite in vacuo and fluid-filled elliptical cylindrical shells. The in vacuo equations thus formulated have the form of a perturbation over the corresponding equations for the circular cylindrical shell. Here, eccentricity of the cross-section is assumed to be small and used as the perturbation parameter. Next, the coupled equations for the fluid-filled elliptical shell are obtained as a perturbation over the in vacuo shell equations (by using a single fluid-loading parameter μ). Asymptotic arguments are used to neglect various terms in the derivation, and a compact form of the non-dimensional governing equations is found. Presenting the governing equations in this form is the main contribution of this work. Due to the eccentricity, all spatial quantities need to be represented in terms of a harmonic series instead of a single harmonic term. Using symmetry and asymptotic arguments, the nature of the harmonic series is obtained. Using the harmonic series expansion, the dispersion relation is formulated for both the in vacuo and the fluid-filled cases. The in vacuo dispersion equation is solved using the regular perturbation method, while the transcendental coupled dispersion equation is solved numerically.