Although homogeneous fluids are impractical to use in the construction of inertial cloaks, recent research has focused on creating fluids which exhibit the prop- erties required. In particular, attention has been focused on acoustic metamaterials, which are materials containing organized microstructures that create extreme effec- tive macroscopic properties that are difficult or impossible to achieve using ordinary materials. Acoustic metamaterials can be thought of as a type of composite struc- ture, although they have the distinct feature of exhibiting effective properties beyond the normal bounds of an ordinary homogenized mixture.
Despite their exotic nature, acoustic metamaterials have been realized using basic physical mechanisms, including the lumped-element behavior of a series of ports and cavities by Zhang et al. [30] and the multiple scattering effects of a lattice of cylindrical scatterers by Torrent and S´anchez-Dehesa [16]. More recent efforts by Popa et al. [31] have used more complex engineered structures, but these fundamentally rely on the same basic principles of lumped-element and scattering effects within each structural unit, in addition to the periodic arrangement of these units, to achieve the desired extreme macroscopic properties.
Figure 2.3(a) shows one type of acoustic metamaterial for 2D acoustic cloak- ing applications proposed by Torrent and S´anchez-Dehesa [15]. In this configuration, each layer is made up of two fluid sublayers (shown in red and blue in the figure),
(a) 1000 100 10 1 0.1
1E−4 1E−2 1 1E2 1E4
ρeff c eff
(b)
Figure 2.3: (a) Schematic view of a cloaked cylinder proposed by Torrent and S´anchez-Dehesa [15]. The cloak consists of alternating fluid sublayers of equal thick- ness. (b) Parametric plot showing the range of material properties needed to achieve cloaking using a configuration like that shown in part (a). The solid lines show the two fluid sublayer properties, and the dashed lines show the design space using a sonic crystal arrangement.
as described by Equations (2.13)–(2.15). To obtain the necessary fluid properties, each sublayer consists of a lattice of elastic scatterers. Utilizing the homogenized properties from this lattice (including multiple scattering effects) creates an acous- tic metamaterial commonly referred to as a sonic crystal [16]. Through the use of sonic crystals, a more realistic design space can considered, as illustrated by the parametric plot presented in Figure 2.3(b). In this figure, the solid lines represent the necessary values for the fluid sublayers based on Equations (2.13)–(2.15), while the dotted lines represent the parameter space which can be used to achieve these values with a sonic crystal.
This noticeable increase in the range of material properties which can be used arises from the variation in the composition and volume fraction of elastic scatterers within each fluid sublayer. Although the choice of using elastic cylinders with a fluid layer is partially responsible for such a broad expansion of the parameter space shown in Figure 2.3(b), the range of achievable densities are extended beyond the
(a) (b) −1 0 1
Figure 2.4: Numerical simulations of the total pressure field for a cloaked rigid cylinder, using an inertial cloak consisting of alternating fluid sublayers containing sonic crystals by Torrent and S´anchez-Dehesa [15], with (a) 50 layers, and (b) 200 layers.
traditional bounds of composite mixtures through the judicious design of the lattice arrangement of the elastic scatterers. Such an effect was verified experimentally by Torrent et al. [32], achieving the effective properties of argon gas using a closely packed lattice of wooden cylinders in air.
Despite the promising characteristics that sonic crystals offer, there are still significant obstacles with regards to implementation in a practical inertial cloak. Figure 2.4(a) and (b) show numerical simulations of the total acoustic pressure field using sonic crystals for 50 layers and 200 layers, respectively. Even with discretiza- tion of the cloak into 50 layers, there is still significant visible disruption of the field, even in the idealized case illustrated which does not include any losses.
Aside from the complexity of the structure, there are several other factors that significantly affect the realization of inertial cloaks. From Figure 2.3(b) it is observed that there are two distinct properties exhibited by the two sublayers, one
(a)
(b)
(c)
Figure 2.5: Experimental design by Zhang, Xia and Fang [30] for a cloaked steel cylinder in water, using an inertial cloak with 16 discrete layers of a radially sym- metric lattice of ports and cavities, for (a) the entire cloak, and (b) a close up to show the arrangement of ports and cavities. The measured pressure field passing through the shadow zone of the rigid cylinder are shown in (c) for the cloaked (blue) and uncloaked (red) configurations, relative to a freefield measurement (green).
which is consistently lower in density than the surrounding fluid, and the other which is consistently heavier. Since the required cloak properties are normalized based on the surrounding fluid properties, the relative difficulty in realizing the necessary cloak depends on the particular properties of the surrounding fluid, and can be addressed by varying the filling fraction of each sublayer [33]. However, the most commonly encountered surrounding media in acoustic applications are air and water, which lie on opposite extremes of the range of material densities, potentially limiting the effectiveness of what cloaking layer properties can be achieved, even with an arbitrarily large number of layers.
Another type of acoustic metamaterial which has been developed utilizes transmission-line arrangements of acoustic lumped elements made up of ports and
cavities. Arrangements of acoustic lumped elements have been used for a cylindrical configuration, made up of discs that were stacked together, an example of which is shown in Figure 2.5(a) based on the work of Zhang, Xia and Fang [30]. A close up is shown in Figure 2.5(b), in which the individual ports and cavities can be clearly observed.
To design this cloak, the features illustrated in Figure 2.5(a) and (b) were machined out of aluminum discs, using the surrounding fluid (in this case water) to fill the resulting ports and cavities. To achieve anisotropic inertial effects, different sized ports were used in the radial and tangential directions while using a shared cavity. The cavity sizes were the same for each layer, but were allowed to change size with the radial distance from the inner cylinder.
Using the prescribed properties for ρr, ρθ and κ for an inertial cloak, expres- sions for the acoustical inductance in the radial and tangential directions, Lr and Lθ, and the acoustical compliance C become [30]
Lr= ρw lr Sr , (2.16) Lθ = ρw lθ Sθ r − a r 2 , (2.17) C = V Kw a b − a 2 , (2.18)
where l is the port length, S is the cross-sectional area of the port, V is the volume of the cavity, and ρwand κw are the density and bulk modulus of water, respectively. Note that the sound speed for a transmission-line configuration of acoustic lumped elements is [34]
c = r
1
LC. (2.19)
Using Equations (2.16)–(2.18), the sound speed given by Equation (2.19) can be determined for the radial and tangential directions and equated with the prescribed
values given by Equations (2.11)–(2.12).
Figure 2.5(c) highlights typical data obtained from experimental results us- ing this design. In this plot, the total pressure field was measure in the shadow zone directly behind the target. Note that the axis of symmetry along which the center of the target lies is located at x = 150 mm based on the data illustrated. To interpret the results presented in Figure 2.5(c), a scale in decibels has been added to right-hand side which gives a measure of the sound pressure level (SPL) relative to the freefield levels. Comparing the SPL, it can be seen that the change due to the addition of the cloaking layer is about 3 dB or less over most of the range, with a maximum change of about 12 dB. Even when the cloaking layer is present, there is still a difference about 6 dB in the shadow zone compared with the freefield measurements. Considering that the thickness of the cloak is three times the radius of the target and contains approximately 3000 helmholtz resonators, it is uncertain how much of these observed changes in the data with the cloak present are due to thermoviscous losses.
There are still more significant limitations of designs using this type of acous- tic metamaterial. First, since water (or any ambient fluid) is used as the medium through which sound propagates, this limits how fast the information can propagate within the cloak. Although the high phase speeds needed can be achieved once the acoustic field reaches steady state, this means that the performance with transients will be significantly diminished. Furthermore, one design aspect which appears to have not been considered by Zhang et al. is the end correction of the ports to ac- count for the mass of the entrained fluid [34]. Based on the design presented, thin wide ports are used to achieve the low inertia needed for the high phase speeds near the inner edge of the cloak. However, when the end corrections are properly accounted for, this significantly reduces the achievable phase speeds.