Mathematical analysis of subwavelength resonant acoustic scattering in multi-layered high-contrast structures

TL;DR

The paper addresses subwavelength acoustic resonances in multi-layered high-contrast metamaterials by formulating a boundary-integral problem and applying a Dirichlet-to-Neumann map to reduce the scattering problem to an $N\times N$ generalized capacitance matrix. It presents three resonance characterizations for concentric multi-layer balls and derives sharp asymptotics for the $2N$ subwavelength eigenfrequencies $\omega_i^{\pm}(\delta)$, tied to the positive-definite capacitance matrix via $\lambda_i$ and the weighted eigenvectors $\mathbf{a}_i$. A modal decomposition and a monopole far-field approximation are developed, enabling efficient prediction of scattering and field concentration near resonances. Numerical results confirm the theory and demonstrate significant computational savings, highlighting the framework's potential for analyzing phononic crystals and metamaterials with nested high-contrast layers.

Abstract

Multi-layered structures are widely used in the construction of metamaterial devices to realize various cutting-edge waveguide applications. This paper makes several contributions to the mathematical analysis of subwavelength resonances in a structure of $N$-layer nested resonators. Firstly, based on the Dirichlet-to-Neumann approach, we reduce the solution of the acoustic scattering problem to an $N$-dimensional linear system, and derive the optimal asymptotic characterization of subwavelength resonant frequencies in terms of the eigenvalues of an $N\times N$ tridiagonal matrix, which we refer to as the generalized capacitance matrix. Moreover, we provide a modal decomposition formula for the scattered field, as well as a monopole approximation for the far-field pattern of the acoustic wave scattered by the $N$-layer nested resonators. Finally, some numerical results are presented to corroborate the theoretical findings.

Figures (5)

• Figure 3.1: Schematic illustration of a structure of $N$-layer nested resonators.
• Figure 5.2: The subwavelength resonant frequencies, plotted in the complex plane, of the $24$-layer nested resonators designed by \ref{['str01']} with $\delta = 1/6000$. We compare the values computed using the spherical wave expansion and the values computed using the capacitance matrix. The computations using the spherical wave expansion took 18 seconds while the approximations from the capacitance matrix took just 0.03 seconds, on the same computer.
• Figure 5.3: The acoustic pressure distributions $u_{(1)},u_{(2)},u_{(3)},u_{(4)}$ for the four-layer nested resonators designed by \ref{['str01']}. Each pair of plots corresponds to one of the four eigenfrequencies. The upper plot displays a contour plot of the function $\Re u_k(x_1, x_2,0)$, with the seven-layer concentric ball designed by \ref{['str01']} represented as solid black lines. The lower plot shows the cross section of the upper plot, taken along the line $x_2 = 0$ (passing through the centres of the multi-layered structures). Additionally, red dotted lines represent vertical lines at the coordinates of the radius.
• Figure 5.4: Norm of the acoustic pressure $u$ to equation \ref{['main_equation']} for the four-layer nested resonators designed by \ref{['str01']}.
• Figure 5.5: Norm of the acoustic pressure $u$ to equation \ref{['main_equation']} for the ten-layer nested resonators designed by \ref{['str01']}.

Theorems & Definitions (39)

• Definition 2.1
• Lemma 2.1: see dyatlov2019mathematical
• Theorem 2.2: DKLLZ_MLHC
• Proposition 3.1
• Lemma 3.1
• proof : Proof
• Definition 3.1
• Remark 3.1
• Proposition 3.2
• proof : Proof