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Figure 6. Acoustic signals usually behave in a reciprocal man- ner, meaning that the response is unchanged if the source and receiver are swapped. The schematic shows how nonrecipro- cal wave propagation can be induced by transmission through medium II with fluid flow. An acoustic beam (psig ) incident from the left and sensed at A is refracted as it transits me- dium II. It exits with a different propagation direction than it entered with and is detected at point B. If the acoustic sig- nal is time reversed at B (double arrow), it will be refracted and detected at point C on the other side of medium II. Thus tAB ≠ tBA and transmission between any two points on opposite sides of II (A and B or B and C) breaks reciprocity.
propagation. The challenge of creating media and devices displaying nonreciprocal behavior is thus to violate micro- scopic reversibility. Fleury et al. (2014) achieved this using fluid flow to induce momentum field bias coupled with a resonant cavity; Figure 6 shows an illustrative schematic of this approach without the cavity.
Another means to break reciprocity is to violate some as- sumptions of the Onsager-Casimir principle. Nonreciprocal transmission can thus be induced by breaking linearity or time invariance. For instance, Liang et al. (2009) employed narrowband acoustic filters made from periodic plates and a nonlinear medium consisting of contrast agents in water to produce one-way propagation.
Active and Reconfigurable Materials
Active AMMs are heterogeneous media with some constitu- ent components controlled by an external stimulus. A simple example would replace the diaphragms in Figure 2 used to generate negative dynamic density with piezoelectric mem- branes. It then becomes possible to control the resonance, and thus the dispersive nature, of the AMMs. The first ex-
ample of this in the context of AMMs was an active trans- mission line (Baz, 2010) that injected energy into the system using electromechanically coupled elements to produce tun- able negative dynamic density. Another application of active AMMs employs nonlinear electric circuits to control non- linear acoustic propagation (Popa and Cummer, 2014).
The earlier discussion of engineered nonlinear acoustic me- dia opens the door to the concept of a reconfigurable AMM. Reconfigurable here implies that one can somehow alter the medium (actively or passively) in a way that fundamentally changes its dispersive properties. Analogues are deployable structures used in aerospace applications or the everyday ex- ample of a deployable structure, the umbrella. An umbrella is highly compliant and dense (i.e., contained in a small vol- ume) before expanded into its “usable” configuration. When deployed, however, the structure is considerably stiffer and less dense. A reconfigurable metamaterial is one that em- ploys similar structural concepts on the subwavelength scale to produce a material that can control acoustic waves in dra- matically different ways depending on the configuration.
To date, most reconfigurable AMMs have employed per- forated buckling elastomeric structures (Shim et al., 2015). This area of research is very new, however, so it is highly probable that different methods for producing reconfigu- rable acoustic media will be found.
Finally, it is important to highlight that many nonlinear AMMs represent the flipside of the reconfigurable AMM coin. The interest in reconfigurable AMMs is to support wave propagation in two or more regimes with large linear regions and then to vary the linear properties by switching between configurations. Nonlinear AMM design, on the other hand, seeks to optimize configurations whose consti- tutive behavior does not require large-amplitude acoustic pressures to generate nonlinear behavior. For this reason, it is possible that the same structure can be used to create both nonlinear and reconfigurable AMMs.
Conclusions
Although many of the topics of AMM research sound exot- ic, it is important to recognize that acousticians have a long history of designing subwavelength structures to control acoustic waves. Examples include the design of bass traps to absorb low-frequency sound in performance spaces, Helm- holtz resonators to limit acoustic wave propagation in ducts, and contrast agents to improve ultrasonic imaging capa- bilities. Nature provides exquisite prototypes; the subwave-
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