Acoustic double negativity induced by position correlations within a disordered set of monopolar resonators

Using a multiple scattering theory algorithm, we investigate numerically the transmission of ultrasonic waves through a disordered locally resonant metamaterial containing only monopolar resonators. By comparing the cases of a perfectly random medium with its pair correlated counterpart, we show that the introduction of short range correlation can substantially impact the effective parameters of the sample. We report, notably, the opening of an acoustic transparency window in the region of the hybridization band gap. Interestingly, the transparency window is found to be associated with negative values of both effective compressibility and density. Despite this feature being unexpected for a disordered medium of monopolar resonators, we show that it can be fully described analytically and that it gives rise to negative refraction of waves.

Figure 1

Schematic illustration of the numerical setup together with the pressure field spectra obtained for two different configurations: (a) a spatially random distribution of noninterpenetrating scatterers and (b) a spatially random distribution of pairs of scatterers separated by a distance $d=250\mu \mathrm{m}$ and aligned parallel to the $z$ axis. In both cases, $a=100\mu \mathrm{m}$ and $n=2.38{\text{mm}}^{-3}$.

Figure 2

Real (a) and imaginary (b) parts of the reduced wave vector projected along the propagation (Oz) axis for a spatially random sample of $a=100\mu \mathrm{m}$ scatterers (blue open symbols) and for a pair correlated disordered sample of the same scatterers (red symbols). The concentration is $n=2.38{\text{mm}}^{-3}$. Solid lines refer to analytical predictions obtained by applying Eqs. (1) and (2) in the case of an isotropic scatterer (blue dashed line) and in the case of a pair of isotropic scatterers (red solid line). The horizontal dotted lines indicate positions of the single scatterer resonance frequency (${\omega }_0$), symmetrical mode frequency (${\omega }_1$), and antisymmetrical mode frequency (${\omega }_2$). (c) Real part of the reduced effective density obtained by numerical computation in the correlated scenario for three different concentrations: $n=0.10{\text{mm}}^{-3}$, $n=0.14{\text{mm}}^{-3}$, and $n=0.18{\text{mm}}^{-3}$ (respectively light, medium, and dark green symbols).

Figure 3

Refraction patterns for an obliquely incident Gaussian beam ($\theta =51.5^{\circ }$). Left: Perfectly random sample. Right: pair-correlated sample. Two frequencies are investigated: ${\omega }_1$ (top) and ${\omega }_2$ (bottom). Arrows indicate the propagation directions. The edges of each quadrant are $6\mathrm{cm}$ long and the sample thickness is 5 mm. Parameters: $a=100\mu \mathrm{m}$, $d=250\mu \mathrm{m}$, $n=2.38{\text{mm}}^{-3}$.

Figure 4

(a) Dispersion curves along the $z$ direction for a random (blue open symbols) and for a pair-correlated disordered sample (red solid symbols) of air bubbles in a yield stress fluid. Dashed and solid lines refer to analytical predictions. The horizontal dotted lines indicate frequencies ${\omega }_0$, ${\omega }_1$, and ${\omega }_2$. Parameters: $a=100\mu \mathrm{m}$, $d=250\mu \mathrm{m}$, $n=2.38{\text{mm}}^{-3}$. (b) Numerical (symbols) and analytical (solid line) evolution of the real part of the reduced wave vector with scatterer concentration at frequency ${\omega }_2$. The horizontal dashed line represents the prediction of Eq. (6).