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is placed in the large central volume scatters acoustic waves like a much a smaller heterogeneity. The crucial condition is that the new material in the cloak steers the acoustic energy around the “hidden” object, which requires the cloaking material to propagate sound faster in the azimuthal direc- tion than in the exterior fluid while simultaneously slow- ing propagation in the radial direction relative to the back- ground value because of the reduced thickness. As a result, the sound at a given point in the cloak behaves anisotropi- cally, and the speed depends on the propagation direction. This is certainly not the situation in normal fluids. However, acoustic anisotropy is possible in systems comprising lay- ers of different fluids arranged like a sandwich. Significant inertial anisotropy can be achieved if the layer thickness is subwavelength and the fluid densities are quite different. The spherical cloak is an extreme example in that it requires acoustic anisotropy levels that are not reachable at present. The acoustic ground cloak, also known as a carpet cloak, can be realized with relatively small anisotropy. Thus, Zigoneanu et al. (2014) used parallel layers of balsa wood in air to get just the right properties (see http://bit.do/acousticcloak). The recent Acoustics Today article on acoustic cloaking (Norris, 2015) provides a more comprehensive overview of this and related developments.
Although cloaking is the most dramatic manifestation of TA, the underlying concept of coordinate invariance also allows us to view classical effects like focusing in a new light. The convex lens, for instance, is based on the short wavelength approximation of ray acoustics. A superior image is guaran- teed if the lens itself transforms the wave equation according to TA so that the focusing is independent of frequency. TA therefore opens up new possibilities for improved efficiency of passive acoustic devices. As an example, the circle-to- square lens converts a monopole cylindrical source at the lens center into a fourfold plane-wave radiation pattern. Conversely, a plane-wave incident from one of the four di- rections will focus at the center. TA achieves this by map- ping a circular region of acoustic medium into a square using a conformal map. Conformal TA is special in that it maintains isotropy, and, furthermore, the density is unaltered; only the compressibility is modified (one might expect using ray acous- tics that the impedance is constant but that is not the case!).
Figure 4 shows the required distribution of the bulk modu- lus along with the fabricated lens device and comparisons of numerical simulations and experimental measurements (Titovich et al., 2016). The variation in compressibility is ob-
Figure 4. TA, which underlies cloaking effects, also enables efficient and accurate acoustic lenses and radiators. Bottom left: Device is designed to radiate a monopole source preferen- tially in four directions. The square array of 7 × 7 cylindrical tubes uses 9 different materials ranging from brass to polymer selected to approximately match the bulk modulus (top left) defined by transforming a circular region of water to a square. Radiation patterns at four frequencies are also shown, with simulation on the right and underwater measurements on the left (Titovich et al., 2016).
tained to a good approximation using a grid of parallel tubes, each selected to match the density of water and to have the required effective compressibility. Using properties ranging from those of polymers to metals, the lens of Figure 4 was constructed with nine distinct materials. Distributed among the 7 × 7 cells of the lens, these simple elements were suf- ficient to provide a broadband effect as shown in Figure 4.
Emerging Areas in AMMs
Nonlinear Materials
All of the previous AMM examples rely on linear behavior. There are, however, a multitude of acoustical applications where material nonlinearity can be used advantageously. The challenge for AMM researchers is to devise materials whose constitutive curve (pressure versus volume change) includes regions that are highly nonlinear. Most nonlinear AMMs achieve this using structures that can be precon- ditioned, usually with static prestress, such that pressure- volume oscillations about the preconditioned state generate elevated levels of material nonlinearity. The concept is rep- resented in Figure 5 where the linear response is about two different configurations (p = 0 and p = ppre ).
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