Page 36 - 2016Fall

Acoustic Metamaterials
The above model of membrane masses and Helmholtz resonators obviously allows for this possibility.
Negative index AMMs also rely on the
negative relationship between phase and
group velocity to achieve what appears to
be a radical violation of normal physics.
Recall that a plane wave incident at a flat
interface between two materials produces
reflected and transmitted waves such that
the components of the directions of propa-
gation relative to the interface are equal for
the incident and reflected waves and are
of the same sign for the transmitted wave,
with the magnitude given by the Snell-
Descartes law of refraction. If one blindly
uses the standard formula with a negative
index (inverse of speed) for the transmit-
ted wave, then the sign of the transmitted
component becomes opposite to that of
the incident wave. As a result, the trans-
mitted wave appears to reverse its direction relative to the incident wave, hence negative index material (NIM). Unlike the one-dimensional LWA, the negative index phenomenon is two- or three-dimensional because it requires the trans- mitted wave to have a negative group velocity for all incident directions.
The concept of negative refraction is difficult to appreciate, and the authors of this article are the first to admit to having trouble understanding it. One way to think about negative refraction is that at an interface, it is the phase of the wave that is matched, phase rules so to speak. Hence, if the group velocity is parallel to the phase velocity on one side and anti- parallel on the other, the energy flow along the interface must switch direction as the wave crosses into the NIM. If this is too far-fetched to comprehend, you do not have to believe us but can watch the movie instead (see http://bit.do/negref).
One of the primary motivations for interest in NIMs is their ability to achieve higher resolution in imaging compared with classical lenses. Pendry (2000) showed that a rectan- gular slab of NIM acts, in principle, as a perfect lens, which provided a major stimulus in the development of metama- terials. This interactive program (see http://bit.do/NIMlens) allows you to get a feel for the NIM lens. Phononic crys- tals (PCs) are periodic structures designed to display spe- cific dispersion properties and are at present the method of
34 | Acoustics Today | Fall 2016
Figure 3. A rectangular slab of a negative index material can act as a near- perfect lens because of the negative refraction and near-field effects (Pendry, 2000). The simulated response of a point source below a negative index material (NIM) lens in water (left) shows the expected separation between the source and the focal point equal to twice the lens thickness. Top right: Measured pressure field in the source and the focal regions (Hladky-Hennion et al., 2014). Bottom right: Close-up view of the lens, an aluminum lattice designed to have effective static density and compressibility matched to water, hence called metal water.
choice for realizing NIMs. However, it is a delicate balancing act to obtain approximately constant phase and group ve- locities for all wave numbers at a given frequency (equifre- quency contours) while also ensuring uniform transmission. In practice, to act as the perfect lens, the slab should be of infinite extent, the equifrequency contours must be circular and the transmission amplitude independent of wave num- ber. The PC water-based flat lens device of Hladky-Hennion et al. (2014) in Figure 3 achieves the first two constraints well but not the third. Despite this limitation, the lateral fo- cus achieved is superior to a normal acoustic lens, making this arguably the most effective acoustic NIM lens to date.
Transformation Acoustics
The second concept driving the development of AMMs is transformation acoustics (TA) that provides the basis for acoustic cloaking and other wave steering systems. The fun- damental idea behind TA is that (1) a change of coordinates, the transformation, modifies the acoustic equations, while (2) the coefficients in the new equations are the physical pa- rameters in the transformed region. For instance, a spheri- cal acoustic cloak uses the transformation of a thick annular region of fluid into a thinner one so that the original central volume is enlarged. If the thinner annulus, the cloak, faith- fully mimics the equations of acoustics in the original an- nulus, then, to an outside observer, the cloak plus whatever