## 2 The Design of the Cloak Shell with a Meta-Composite Shell
### Meta-Composite Structural Design for a Three-Dimensional Circle-Type Cloak Shell  
As shown in Figure 3, the specific acoustic impedance values, \(z_{\text{shell}}\), vary in some areas of the cloak shell, which necessitates the use of a cloak shell capable of providing these specific impedance values. Here, we propose a meta-composite structure designed to provide different impedance values in the \(e_{u}\), \(e_{v}\), and \(e_{w}\) directions through the use of specific structural components. As previously described in Figure 2, the anisotropic cloak shell is active by a three-layer meta-composite structure. Figure 6 explains the composited block elements that form the cloak shell by the layer meta-composite structure. The block elements in the layer are designed according to the spatially anisotropic properties of the cloak shell through impedance design. Figure 4: Designed variation in wave phase velocity (relative to background) inside the cloak shell: (**a**) in the \(e_{u}\) direction of the cloak shell region; (**b**) in the \(e_{v}\) direction; (**c**) in the \(e_{w}\) direction. The direction of incoming waves can be guided by altering the wave phase speed using the anisotropic material properties of the cloak shell. Figure 5: Pre-determined path for wave propagation around the cloak shell. The anisotropic material properties of the cloak shell cause the direction of wave propagation to change from its original direction to a pre-determined path. Acoustic structural engineering [26; 27; 28] can be used to develop structural devices, such as tubes and cavities, in order to provide acoustic mass and acoustic compliance, respectively. Thus, the impedance value can be adjusted by altering the configuration of this structural device. As shown in Figure 7a, we assembled block elements with cavities and tubes in three directions, the tubes acting as acoustic mass and the cavity in the center ensuring acoustic compliance. Each block element comprises a single cavity in the center with four tubes in different directions. These block elements provide the acoustic impedance value \(z_{b}\). In the structure of tube, the physical law governing the motion of air inside the tube [26; 28] is \(f(t)=M(\mathrm{d}u(t)/\mathrm{d}t)\), where \(M\) is the mass of volume fluid inside the tube. After dividing the cross-sectional area \(A\) of the tubes, the motion of air leads to \(f(t)/A=(M/A)\cdot\{\mathrm{d}[u(t)A]/\mathrm{d}tA\}\), which can be further expressed as \(p(t)=\left(M/A^{2}\right)\cdot[\mathrm{d}U(t)/\mathrm{d}t]\). Using the above, the relationship can be further explained as follows: \(p(t)=\left(M/A^{2}\right)\cdot[\mathrm{d}U(t)/\mathrm{d}t]=\left[(\rho_{0}l_{e }A)/A^{2}\right]\cdot[\mathrm{d}U(t)/\mathrm{d}t]=M_{A}[dU(t)/dt]=\mathrm{j}wM _{A}U\). In the structure of the cavity, the physical laws governing the compression of air being acted on by a force in a cavity [26; 28] lead to the following equation: \(p(t)=(1/C_{A})\int U(t)\mathrm{d}t=U/\mathrm{j}wC_{A}=-\mathrm{j}U/wC_{A}\). The value \(M_{A}\) is the acoustic mass or acoustic inertance, \(C_{A}\) is the acoustic compliance. Using the above properties, a block element comprising tubes and cavity structures provides the acoustic impedance value \(Z_{b}\) in the respective \(e_{u}\), \(e_{v}\), and \(e_{w}\) directions as follows:  
\[Z_{b}=\mathrm{j}wM_{A}-\frac{1}{wC_{A}}\mathrm{j}=\mathrm{j}w\left(\frac{\rho_ {0}l_{e}}{A}\right)-\frac{1}{w}\left(\frac{\rho_{0}c^{2}}{V}\right)\mathrm{j}. \tag{4}\]  
Figure 6: The three layers of meta-composite structures forming the cloak shell region. Each block element in the layer is designed according to the spatially anisotropic properties of the cloak shell through impedance design. The first part, \(jwM_{A}\) is the acoustic impedance value supplied by the tube structure; the second value, \(j(1/wC_{A})\), is the acoustic impedance value supplied by the cavity structure.