Resonant modal approximation of time-domain elastic scattering from nano-bubbles in elastic materials

Abstract: This paper is devoted to establishing the resonant modal expansion of the low-frequency part of the scattered field for acoustic bubbles embedded in elastic materials in the time domain. Due to the nano-bubble with damping, Minnaert resonance can be induced at certain discrete resonant frequencies, which forms the fundamental basis of effectively constructing elastic metamaterials via the composite material theory. There are two major contributions in this work. First, we ansatz a special form of the density, approximate the incident field with a finite number of modes, and then obtain an expansion with a finite number of modes for the acoustic-elastic wave scattering in the time-harmonic regime. Second, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates. Interestingly, we find that the $0$-th mode is the main contribution to the resonant modal expansion.