Transformational acoustic metamaterials based on pressure gradients

We apply a homogenization process to the acoustic velocity potential wave equation. The study of various examples shows that the resulting effective properties are different from those of the homogenized pressure wave equation for the same underlying acoustic parameters. A careful analysis reveals that a given set of inhomogeneous parameters represents an entirely different physical system depending on the considered equation. Our findings unveil a different way of tailoring acoustic properties through gradients of the static pressure. In contrast to standard metafluids based on isobaric composites, this alternative kind of metafluid is suitable for the implementation of transformational devices designed via the velocity potential equation. This includes acoustic systems in a moving background or arising from general space-time transformations. As an example, we design a device able to cloak the acoustic velocity potential.

Figure 1

Effective parameters obtained for periodic arrays (square lattice with periodicity $l)$ of wood inclusions embedded in air as a function of the wood filling fraction $f$ [$f=\pi r^2/l^2$ for infinite cylinders and $f=4\pi r^3/\left(3l^3\right)$ for spheres, $r$ being the radius]. (a) Effective density and (b) speed of sound of a 2D array of cylinders. The speed of sound is calculated using two different methods (homogenization and simulation of a finite array). (c) Effective density and (d) speed of sound of a 3D array of spheres.

Figure 2

Simulation of the acoustic velocity potential in a finite array of wood cylinders embedded in air. A plane wave impinges onto the array from the left. A perfectly matched layer (PML) absorbs the output wave on the right. Periodic conditions are applied at the top and bottom simulation boundaries. Note that a stairlike behavior appears at the cylinder-background interfaces due to the discontinuity in the acoustic parameters.

Figure 3

Same set of acoustic parameters arising from two different physical systems. (a) One material at different static pressures (this kind of system is modeled through the $\varphi$ equation). (b) Two different materials at the same static pressure (this kind of system is modeled through the $p$ equation).

Figure 4

Gaussian beam (a) propagating through air, (b) impinging onto a high-density cylinder, and (c) onto the cloak-surrounded cylinder. The cloak consists of 50 different layers with $b=2a=1.7\lambda$, and $d\approx \lambda /100$, where $\lambda$ is the acoustic wavelength. In all simulations, the left boundary is defined as the source of the Gaussian beam. The rest of the computational domain is terminated by PMLs.