## 3 Experimental Analysis of the Acoustic Cloak  
### Fabrication and Experimental Analysis of Meta-Composite Cloak Shell  
Experiments were conducted to evaluate the cloak shell design. As in the previous design, the 840 mm diameter of the cloak shell was impracticable for acoustic experiments in a free field; therefore, we adopted the techniques of the acoustic model measurement [35]. The test specimen was scaled down by defining the scaling factor \(\sigma\) and denoting quantities referring to the model by a prime, for a geometric relationship between the real structure and the scale model of \(l^{\prime}=l/\sigma\). The time relationship in the scale model is as follows:  
\[t^{\prime}=l^{\prime}/c^{\prime}=l/c^{\prime}\sigma=ct/c^{\prime}\sigma. \tag{7}\]  
Using the same scaling factor \(\sigma\) for the sound frequencies, the wavelength and the frequency relationships between the original and scaled models can be written, respectively, as \(\lambda^{\prime}=\lambda/\sigma\) and \(f^{\prime}=c^{\prime}/\lambda^{\prime}=\sigma c^{\prime}/c\). The experiment was performed in air; therefore, the speed of the acoustic waves of the experimental specimen and the original cloak shell were equivalent. According to the relationship \(c=c^{\prime}\), the equations above can be rewritten as  
\[l^{\prime} =l/\sigma \tag{8}\] \[t^{\prime} =l/\sigma\] (9) \[\lambda^{\prime} =\lambda/\sigma\] (10) \[f^{\prime} =\sigma f. \tag{11}\]  
If the difference in sound level between the two cases is not subject to scaling, from the equivalence of the decrease in energy in the decaying sound field [35], we obtain the following relationship:  
\[\mathrm{e}^{\{-[m^{\prime}/c-n^{\prime}\ln(1-\alpha^{\prime})]t^{\prime}\}}= \mathrm{e}^{\{-[m^{\prime}/c-n^{\prime}\ln(1-\alpha^{\prime})]t^{\prime}\}} \tag{12}\]  
where the primed and unprimed systems represent the scale and original propagation condition of the acoustic waves, respectively. Parameter \(m\) represents the attenuation constant of air and \(n\) represents the average number of wall reflections associated with a single sound ray each second. Value \(\alpha\) is the absorption coefficient of air. In this study, the experiments were performed within a semi-anechoic room under free-field conditions. Reflected waves could be disregarded and \(n^{\prime}=n=0\). To fulfill the experiments with regard to the equivalence of measured sound levels in the scaled specimen and the original cloak shell as written in Equation (12), we used the following field relations between the original and the scale model experimental measurements: \(t^{\prime}=t/\sigma\), \(\alpha^{\prime}=\alpha\), and \(m^{\prime}=\sigma m\). Air was the background medium in both cases, which satisfied the requirement of \(\alpha^{\prime}=\alpha\). The design frequency of the original cloak shell structure was 1000 Hz. To reduce the size of the sample object and shell, we increased the frequency of the incident sound source from 1000 to 2000 Hz. According to the attenuation constant of acoustic waves at different frequencies [35], the attenuation relationship between the frequencies in the experiment and those in the original case was \(m^{\prime}=0.38m\), for a scaling factor of the model of \(\sigma=0.38\). We scaled down the cloak shell and object according to the scale factor \(\sigma\), which resulted in a scaled down cloak shell with a diameter of just 319 mm.  
In accordance with the design parameters in Table 1 and scale factor \(\sigma\), single block elements were simply duplicated, resulting in a total of 144 block elements. The block elements were scaled down using 3 ds Max software prior to fabrication using a 3D printer, as shown in Figure 23a. The real product of the block element components is shown in Figure 23b. Figure 23c presents the assembled block elements of the cloak shell for the experiment.