Acoustic scattering by elliptical elastic shells: Exact formalism and physical interpretation

The acoustic scattering by an elastic cylindrical shell of elliptical cross-section with various eccentricities is treated by emphasising its symmetry properties. The elliptical shell remains invariant according to four symmetry transformations of the finite group denoted $C_{2v}$. Four one-dimensional irreducible representations called $A_1$, $A_2$, $B_1$ and $B_2$ are associated with this symmetry group. The incident wave, the unknown scattered pressure and the elastic displacement are expanded over these four representations. The implementation of the boundary conditions leads to solving four matrix equations systems, each associated with an irreducible representation. The unknown scattering coefficients are decoupled owing to symmetry considerations, which considerably simplifies the problem. The resonances characteristic equations are established and provide a physical interpretation in relation to the symmetries of the scatterer. All the physical quantities of interest, such as the scattered pressure, the far-field form function and the elastic displacement, are obtained in exact form. Thus, the problem of scattering by an elastic shell composed of two ellipses of various eccentricities is completely solved. Numerical results are provided for various elliptical shell configurations and physical interpretation is given. Our method could also be generalised to scattering by multilayered shells.