David Natroshvili, Guram Sadunishvili, Irine Sigua

Some Remarks Concerning Jones Eigenfrequencies and Jones Modes

Three-dimensional fluid-solid interaction problems with regard for thermal stresses are considered. An elastic structure is assumed to be a bounded homogeneous isotropic body occupying a domain $\Omega^+\subset\mathbb{R}^3$, where the thermoelastic four dimensional field is defined, while in the unbounded exterior domain $\Omega^-=\mathbb{R}^3\setminus\ov{\Omega^+}$ there is defined the scalar (acoustic pressure) field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface $\partial\Omega^{\pm}$. We show that uniqueness of solutions strongly depends on the geometry of the boundary $\pa\Omega^{\pm}$. In particular, we prove that for the corresponding homogeneous transmission problem for a ball there exist infinitely many exceptional values of the oscillation parameter (Jones eigenfrequencies). The corresponding eigenvectors (Jones modes) are written explicitly. On the other hand, we show that if the boundary surface $\partial\Omega^+$ contains two flat, non-parallel sub-manifolds then there are no Jones eigenfrequencies for such domains.

Elasticity, thermoelasticity, fluid-solid interaction, Jones eigenfrequencies, Jones modes.

MSC 2000: 74F10, 74F05