On Asymptotic Behaviors of Acoustic Waves Due to High-Contrast Material Inclusions

This paper investigates the asymptotic behaviors of time-harmonic acoustic waves generated by an incident wave illuminating inhomogeneous medium inclusions with high-contrast material parameters. We derive sharp asymptotic estimates and obtain several effective acoustic obstacle scattering models when the material parameters take extreme values. The results clarify the connection between inhomogeneous medium scattering and obstacle scattering for acoustic waves, providing a clear criterion for identifying the boundary conditions of acoustic obstacles in practice. The contributions of this paper are twofold. First, we provide a rigorous mathematical characterization of the classical sound-hard and sound-soft obstacle scattering models. We demonstrate that a sound-hard obstacle can be viewed as an inhomogeneous medium inclusion with infinite mass density, while a sound-soft obstacle corresponds to an inclusion with zero mass density and zero bulk modulus. Second, we introduce two novel acoustic obstacle scattering models when the mass density of the inclusion degenerates to zero. These new models offer a fresh perspective on considering inhomogeneous medium inclusions with high-contrast material parameters.