## 2 The Design of the Cloak Shell with a Meta-Composite Shell
### Meta-Composite Structural Design for a Three-Dimensional Circle-Type Cloak Shell
In Equation (4), \(j=\sqrt{-1}\), \(w\) is the circular frequency, \(\rho_{0}\) is the density of the background medium (air), and \(c\) is the wave speed in air. Values \(A\) represent the cross section of the tubes and value \(V\) is the volume of the cavity. Because tubes connected the cavity, forming as the flanged end in the cavity side, the effective length \(l_{e}\) is the original tube length plus the correction value of the flanged end [27] as shown in Figure 7b. Equation (4) provides impedance values of the cloak shell in \(e_{u}\), \(e_{v}\), and \(e_{w}\) directions. The frequency of the meta-composite structure is set as 1000 Hz, an undistorted frequency for human hearing. Using the relationship \(|Z_{b}|\cdot A_{i}=Z_{\text{shell}}\) in conjunction with the specific cavity dimensions, the cloak shell can be represented using a three-layer metamaterial comprising a structural block element,  
Figure 7: Structure of the meta-composite material used to make up the cloak shell: (**a**) the block elements include cavities and tubes arranged as circular elements in three layers. The inset on the upper right shows the details of single block elements with rectangular cavities and tubes. The bottom right inset exhibits both the cloak shell layered structures and cylinder object inside the cloaked region; (**b**) the part slices of the meta-composite material layer exhibit the interior cavity and tube in directions. The coefficient \(a\) explains the cross section dimension of and individual element, and \(l_{e}\) represents the effective lengths of the tube connected with the cavity. as shown in Figure 7. Note that \(A_{i}=a^{2}\) is the cross-sectional area of the individual element forming the cloak shell layers, where the element size \(a\) of a specific layer in the study is presented in Figure 7b. The design parameters were described in Table 1. Because the metamaterial structure provides acoustic mass and acoustic compliance rather than acoustic resistance, the shell is able to store acoustic energy rather than consume energy and it has a low-loss nature. Figure 8 explains the interior details of the meta-composite material structure, showing the interior details of block elements with rectangular cavities and tubes sliced by a horizontal surface in the \(e_{u}\)-\(e_{v}\) plane and by a vertical surface in the \(e_{u}\)-\(e_{w}\) plane, respectively. Based on the principle of Equation (4), the combination of cavity and tubes of the block element provides the acoustic impedance value \(Z_{b}\) in the \(e_{u}\), \(e_{v}\), and \(e_{w}\) directions, respectively. The tubes connected the cavity without extending inside the interior of the cavity. In the design, the cloak shell structure element unit is composed of cavity and a pair of tubes series connected in the respective direction. They give the impedance performance as the Helmholtz resonator [29]. Table 1 lists the design parameters of the structure. A total of 144 block elements were used in the cloak shell. The volume of the cavities was set as 27,000 mm\({}^{3}\) for each layer. The largest size of the cavities was 30 mm, and the wavelength of the design frequency was 343 mm (1000 Hz, \(c\) = 343 m/s); therefore, the size of the elements did not exceed 1/10 of the wavelength. The cloak shell structure elements were designed based on the principles of the long wavelength limit [30] and the structure of the cloak shell could then be treated as an effective medium for incident waves.