## 2 The Design of the Cloak Shell with a Meta-Composite Shell
### A Numerical Simulation of the Cloak Effect in Practice Working Conditions
The cloaked field exhibited field patterns with less alteration in the overall field than in the uncloaked object case, the results having a similarity to a free field. These results could be attributed to the wave guiding effect of the cloak shell, and these findings verified the effectiveness of the meta-composite structure in acting as a cloak shell. The wave cloak spatially varied the impedance values of the cloak shell, successfully producing the  
Figure 11: The 3D geometric models and boundary conditions in numerical analysis. The object and cloak shell are enclosed inside a rectangular acoustic domain (5000 mm \(\times\) 2500 mm \(\times\) 100 mm) for a background medium (air). Surfaces A and B are assumed to be radiation boundaries and the other surfaces of acoustic domain are assumed to be the free surface. The 3D geometric model is meshed using a fluid structure interaction (FSI) interface to represent a fluid–structure interaction. cloak effect by changing the wave phase velocity of \(c_{eu}\) and \(c_{ev}\) along the horizontal \(e_{u}\)-\(e_{v}\) plane and in the vertical \(e_{w}\) direction, as described in Figure 4.  
As with the case at 800 Hz, the results at 1250 Hz in Figure 14 show no apparent cloak effect. The cloaked object case, cloak shell concentrated the wave scattering energy and forming strong wave beam in both forward and backward directions. Those strong wave beams break the cloak effect. The field patterns, the wave propagating paths, and the resulting scattering fields of the cloaked and uncloaked cases of the incident wave at 1000 Hz are explained in Figure 15. In the uncloaked case, the incident wave at the center path interacted head on with the cylinder object and produced a typical scattering field in both forward and backward directions. However, the waves in the outer path propagated straight forward unhindered. The scattering field, combining the waves of both paths, was composed of two rectangular high-pressure regions (Region A), which were attributed to scattered energy and wave diffraction. The object in the space shadows the incident wave and forms the Low-Pressure Shadow Region B. This makes obvious pressure disturbance in the sound field and states the uncloaked condition. In the cloaked case, the incoming wave on the center path was guided by the cloak shell and was smoothly restored after passing through the object. These mechanisms avoid the strong scattering effect of the object. Within the unaffected waves in the outer path, the total scattering field of the cloaked case showed flat wave fronts. The pressure field patterns clearly showed more uniform field patterns in both the forward and backward scattering directions in Region C, evidence that the cloak shell guided the waves and created the cloak effect. These results matched the theoretical findings presented in [9]. Our simulation results agreed with the original design concept of transformation acoustics [8] and served to verify the effectiveness of the proposed metamaterial structure of the cloak shell design. Figure 12: Wave field of uncloaked object (upper plot), cloaked object (middle plot), and free field at 800 Hz. Plane waves with a specific frequency propagate from right to left. The dotted lines indicate the geometry of the cloaked object and cloak shell. Figure 14: The wave field of the uncloaked object (upper plot), the cloaked object (middle plot), and free field at 1250 Hz. Plane waves with a specific frequency propagate from right to left. The dotted lines indicate the geometry of the cloaked object and cloak shell. Figure 10.0 Wave field of uncloaked object (upper plot), cloaked object (middle plot), and free field at 1000 Hz. Plane waves with a specific frequency propagate from right to left. The dotted lines indicate the geometry of the cloaked object and cloak shell. In the investigation of the scattering field, the pressure values with and without the cloak were measured at 11 section lines along the wave fronts in the scattering field.