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Figure 2. A carpet cloak. The white region is freed up by squashing the entire white+green+yellow region into the green and yellow segments. The green region is a one dimensional compression, as in Figure 1. The transformation in the yellow region depends on both the vertical and horizontal coordinates, resulting in a compression and an extension in orthogonal directions not aligned with the coordinates axes (Craster and Guenneau, 2013, Ch. 7).
vice requiring a power supply ("They can't have disappeared. No ship that small has a cloaking device." Lorth Needa). Dr. Who’s famous Tardis, which is much larger inside than it ap- pears from the outside, uses “transdimensional engineering” to make the interior and exterior exist in different dimen- sions (!). Yet, this is not too far removed from the actual basis for passive cloaking devices, described next.
Transformation Acoustics
The trick in passive acoustic cloaking is to somehow shrink the object and cloak from the observer’s viewpoint, so that the object appears to be vanishingly small. This geometri- cal metamorphosis of a large virtual region into a smaller physical one is called transformation acoustics (TA). The technical details of TA convert the acoustic wave equation from one coordinate system to another, which can quickly obscure the concepts.
The simple example of acoustic wave reflection in Figure 1 captures the essence of TA. The incident wave reflects from a fixed boundary at the bottom of the uniform fluid (Figure 1(a)). The same response is obtained from a non-uniform acoustic medium (Figure 1(b)), if (i) the time taken for the sound to travel back and forth is unchanged, and (ii) there is no reflection except at the bottom of the medium, assumed to be rigid. These conditions clearly constrain the acoustic speed and impedance in the green section. The no-reflection condition is then met if the green slab has the original acous- tic impedance. Let f be the fractional ratio of the length of the green fluid to the original, the time constraint is then sat- isfied if the green index of refraction is 1/f. The relative den- sity and compressibility are therefore both 1/f >1. The lesson of this simple mirage (Norris, 2009) is that the transformed acoustic parameters depend upon the geometrical quantity f. One can already gain some appreciation for the difficul- ties in the full cloaking problem. In order to achieve a siz- able effect the value of f must be significantly different from unity. A fluid of much greater density and compressibility is
necessary to "squeeze" the original fluid into a much smaller space (f <<1), freeing up a relatively large amount of space. Extreme phenomena require extreme physical properties.
Acoustic anisotropy distinguishes cloaking from everyday acoustics. Consider the mirage in two dimensions, (Figure 1(c)). The index of refraction determined above implies a travel time for the transmitted ray (solid line) different from the original (dashed line). The only resolution is to allow for directional wave speed dependence, also known as anisot- ropy. For a wave incident near glancing the travel time con- dition requires that the green wave speed is the same as the original, hence the horizontal index of refraction is unity as compared to 1/f for normal incidence. A full analysis shows that the index of refraction (i.e. slowness) in any other direc- tion describes an ellipse with major and minor axes corre- sponding to the normal and glancing incidence values.
Figure 3. (a) The cloaked region is flattened by a one dimensional- like mapping. (b) The cloaked region has the scattering cross-section of a thin cylinder. (c) A fully 3D transformation; the cloaked region scatters like a point.
Transformation in more than one dimension (1D) fol- lows the same principle of squeezing a virtual region into a smaller physical volume. The carpet cloak (aka ground-plane cloak) (Figure 2) frees up a finite volume. The anisotropy
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