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Acoustic Cloaking
Figure 7. Cloaking of flexural waves. The black cloaked region of di-ameter 3 cm is clamped, the dashed lines depict the outer bound- ary of the cloak. Reprinted from (Stenger et al., 2012) with permis- sion. This movie shows experimental results at 200 Hz. (To view
movie please visit http://wp.me/p4zu0b-MY )
Cloaking of seismic waves has obvious potential but is es- sentially a pipe dream, the wavelengths are so long that any structure providing significant cloaking effect would be huge. Nevertheless, several ideas have been floated for cloak- like effects. Kim and Das (2013) proposed attenuating seis- mic energy using large buried "meta-boxes" with resonance
frequencies of the seismic waves. Simulations with boxes of volume ≈ 100 m2 show significant attenuation at 10 Hz. Brûlé et al. (2014) proposed a phononic crystal design to reflect geophysical surface noise such as pile drivers. Large scale tests were made on a periodic 2D mesh of cylindrical empty boreholes of 0.32 m diameter drilled 5 m into the top soil layer, with a lattice constant of 1.73 m. The configuration was designed to have a bandgap at the operating frequency of 50 Hz. Measurements showed greater than 50% decrease in surface wave amplitude for transmitted waves.
Cloaking by Active Cancellation
The methods discussed so far are passive, reliant on material properties for cloaking. Active cloaking, on the other hand, uses sound sources to cancel the incident wave. It is closely related to active noise control and anti-sound which creates a zone of silence, although unlike cloaking, the sound is gen- erally not required to be non-radiating. Recent theoretical work on active acoustic cloaking provides insight into the older problem of active sound control.
Miller (2006) proposed cloaking a region by sensing sound on a closed surface while simultaneously exciting sources with amplitudes defined by the measurements. The method relies on the Kirchhoff-Helmholtz integral (Nelson and El- liott, 1992) whereby a continuous distribution of monopoles and dipoles completely suppresses sound. The difficulty with this approach is realizing acoustically transparent sen- sor and actuator surfaces, replacing the surfaces by a finite number of discrete sensors and sources is preferable. A solu- tion to this problem was provided by Guevara Vasquez et al. (2011) who showed, remarkably, that cloaking requires as few as three active sources in 2D and four in 3D. Norris et al. (2012) subsequently found explicit formulas for the source amplitudes. The catch with this approach is that the sources are multipoles, and full cloaking requires multipoles of all order. Point multipoles are not possible in practice, neither are monopoles or dipoles for that matter. Furthermore, the multipole expansion is divergent, a point noted earlier in an- ti-sound research (Nelson and Elliott, 1992, pp. 262-4). This motivates truncating the series, limiting accuracy, although numerical simulations indicate that only a small number of multipoles may be required. Despite these difficulties, Gue- vara Vasquez et al. (2011) provide a rigorous basis for sub- sequent approximation. Other approaches to active cloaking have been suggested; for instance, Bobrovnitskii (2010) pro- posed acoustic cloaking using a non-local impedance coat- ing extended reaction.
44 | Acoustics Today | Winter 2015