A uniqueness theorem for the scattering of harmonic waves in a fluid- solid medium.

*(English)*Zbl 0688.76059In recent years considerable attention has been addressed to both direct and inverse problems for acoustic waves. In a typical inverse problem an obstacle, whose nature and shape are unknown, is striken by an incident wave propagating through a medium which surrounds the obstacle. The scattered waves should provide information on the nature and the shape of the obstacle. Such problems appear in diverse areas of applications, as non destructive testing, medical imaging, radar and sonar. This fact is at the origin of the interest in the various numerical techniques for the solution of scattering problems. In particular considerable effort has been concentrated on the analysis of rather simple models, e.g. the case of an impenetrable obstacle which is immersed in an inviscid fluid, because of the intrinsic nonlinearity and improper posedness of the problem.

This work is meant as a step toward the analysis of more realistic situations. In practice this view is followed by modelling the obstacle as general as possible, namely an inhomogeneous, anisotroic solid. The surrounding medium, instead, is regarded as a homogeneous isotropic fluid. To get a model realistic enough we have to account for dissipation; the solid is modelled as viscoelastic, the fluid as viscous.

Particular attention is devoted to the thermodynamic restrictions. So we are able to write strict inequalities for the imaginary part of the material parameters which account for the constitutive properties when harmonic waves occur. Moreover, we have recourse to an integral relation which involves the Green function for the problem. As a result in this note we prove the uniqueness of the scattered wave produced by an incident wave and a body force.

This work is meant as a step toward the analysis of more realistic situations. In practice this view is followed by modelling the obstacle as general as possible, namely an inhomogeneous, anisotroic solid. The surrounding medium, instead, is regarded as a homogeneous isotropic fluid. To get a model realistic enough we have to account for dissipation; the solid is modelled as viscoelastic, the fluid as viscous.

Particular attention is devoted to the thermodynamic restrictions. So we are able to write strict inequalities for the imaginary part of the material parameters which account for the constitutive properties when harmonic waves occur. Moreover, we have recourse to an integral relation which involves the Green function for the problem. As a result in this note we prove the uniqueness of the scattered wave produced by an incident wave and a body force.

##### MSC:

76Q05 | Hydro- and aero-acoustics |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |