Well-posedness for the biharmonic scattering problem for a penetrable obstacle

Abstract

We address the direct scattering problem for a penetrable obstacle in an infinite elastic two--dimensional Kirchhoff--Love plate. Under the assumption that the plate's thickness is small relative to the wavelength of the incident wave, the propagation of perturbations on the plate is governed by the two-dimensional biharmonic wave equation, which we study in the frequency domain. With the help of an operator factorization, the scattering problem is analyzed from the perspective of a coupled boundary value problem involving the Helmholtz and modified Helmholtz equations. Well-posedness and reciprocity relations for the problem are established. Numerical examples for some special cases are provided to validate the theoretical findings.

Figures (1)

• Figure 1: Left: Boundary of the kite-shaped scatterer, $\partial D$, and measurement curve, $\Gamma$. Center and Right: An incident fundamental wave located at $z=(-3,0)$ propagates through a medium with $\tau_+= 15$ and impinges upon a scatterer with $\tau_-=5$. The center panel shows the real part of the total wavefield (blue is negative, red is positive), while the right panel shows the magnitude of the real part.

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Lemma 4.1
• Lemma 4.2
• Theorem 4.2
• proof
• Theorem 5.1
• proof