## 2 The Design of the Cloak Shell with a Meta-Composite Shell
### The Design of a Circular Cloak Shell by Three-Dimensional Coordinate Transformation
In accordance with the methods outlined in [10], and using the wave equation equivalence between the coordinate transformed domain and the outside field, the cloak shell material parameters of both radial \(e_{u}\) and vertical \(e_{w}\) directions in the transformed domain can be written as (see the Supplementary Materials):  
\[\rho=\left[ \begin{array}{ccc}\frac{e_{u}}{f}\frac{\mathrm{d}f}{\mathrm{d}e_{ u}}&0&0\\ 0&\frac{f}{e_{u}}\left(\frac{\mathrm{d}f}{\mathrm{d}e_{u}}\right)^{-1}&0\\ 0&0&(\delta)^{2}\frac{e_{u}}{f}\left(\frac{\mathrm{d}f}{\mathrm{d}e_{u}}\right) ^{-1}\\ \end{array} \right] \tag{2a}\] \[\lambda=\frac{e_{u}}{f}\left(\frac{\mathrm{d}f}{\mathrm{d}e_{u}}\right)^{-1}. \tag{2b}\]  
Following the above method, we designed the 3D acoustic circle cloak shell working in air. Firstly, in the horizontal plane, we selected the transform function \(f\) of the cloak shell in the horizontal \(e_{u}-e_{v}\) plane as \(e_{u}^{\prime}=R_{1}+e_{u}(R_{2}-R_{1})/R_{2}\), where the radial positions are \(R_{1}=3\) and \(R_{2}=6\), and the cloak shell region along the radial direction is \(3<e_{u}<6\). The transform function \(f\) leads to \(e_{u}^{\prime}=f(e_{u})=3+0.5e_{u}\), \(4.5<e_{u}^{\prime}<6\), \(e_{v}^{\prime}=e_{v}\), and \(e_{w}^{\prime}=e_{w}\). The above coordinate transformation procedure in the radial direction compresses the region in the real domain within \(e_{u}<3\) in the horizontal \(e_{u}-e_{v}\) plane into region \(e_{u}^{\prime}>3\) in the transformed domain. This process makes region \(e_{u}<3\) in the real domain non-existent in the transformed domain and transforms region \(e_{u}<3\) into a cloaked region. Objects that one wishes to cloak can be placed inside the cloaked region to protect them from being hit by incoming sound waves. Secondly, in the vertical \(e_{w}\) direction, we randomly selected \(\delta=0.4\) to obtain the coordinate transformation function in the vertical direction,\(e_{w}^{\prime}=g(e_{w})=0.4e_{w}\). Substituting the transformation function into Equations (2a) and (2b) provides the material parameters of the 3D transformed cloak shell as follows:  
\[\rho=\left[ \begin{array}{ccc}0.33\,\sim 0.5&0&0\\ 0&2\,\sim 3&0\\ 0&0&0.21\,\sim 0.32\\ \end{array} \right]\rho_{0} \tag{3a}\] \[\lambda=1.33\lambda_{0}\,\sim 2\lambda_{0}. \tag{3b}\]  
According to the above results, the material properties of the cloak shell along \(e_{u}\), \(e_{v}\), and \(e_{w}\) are spatially varying in three directions. The circular acoustic cloak shell is composed of an anisotropic inhomogeneous material. Since the acoustic impedance of the material is the primary factor determining the propagation behavior of waves through the medium; therefore, we use the guiding principle in the design of the cloak shell material. The structural design of the cloak shell is divided into three layers along the radial direction as plotted in Figure 2. The three layers (\(e_{u}=3\), 4.5, and 6; \(e_{v}=0\)-\(2\pi\)) construct the cloak shell region; therefore, they will be physically realized by proposing meta-composite structures. Substituting the spatially varied cloak shell material parameters \(\boldsymbol{\rho}\) and \(\lambda\) in Equations (3a) and (3b) into relationship \(z=\rho\cdot c=\rho\sqrt{\lambda/\rho}=\sqrt{\lambda\rho}\) (the definition of specific acoustic impedance), we can obtain the required ratio of specific acoustic impedance of the cloak shell, \(z_{\text{shell}}\), in \(e_{u}\), \(e_{v}\) and \(e_{w}\), as shown in Figure 3a-c. The diagram in Figure 3 is plotted according to the relationships between the Cartesian coordinates and the cylindrical coordinate system on the horizontal plane as \(Sx=e_{u}\cos(e_{v})\) and \(Sy=e_{u}\sin(e_{v})\), respectively. As shown in Figure 3, the ratio of the specific acoustic impedance values of the cloak shell varies spatially in three different directions.