**3.2 Sound radiation from unstiffened ABH shells**

Now we investigate the sound radiation from a finite unstiffened shell containing five ABH cells. Based on the reconstructed GEM presented in Section 2.1, we cancel the axial periodic conditions, namely Eqs. (12) and (13), such that the vibration field of the finite ABH shell can be characterized. It is well-known that there is a cut-on frequency for an ABH, which is mainly determined by its size. Only beyond this frequency (wavelength smaller than the ABH size, 2*rabh*), the wave can be trapped and the ABH effect can be triggered. According to **Table 1**, the radius cut-on frequency *fr* <sup>¼</sup> *<sup>π</sup>huni* 4*r*<sup>2</sup> *abh* ffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>E</sup>* <sup>3</sup>*<sup>ρ</sup>* <sup>1</sup>�*ν*<sup>2</sup> ð Þ <sup>q</sup> ¼ 282 Hz.

As plotted in **Figure 6a**, the radial mean square velocity (MSV) on the surface of the ABH shell is compared to that of the uniform (UNI) shell having the same damping layer configuration. From the figure, it can be seen that in the BGs, the vibration of the ABH shell is very low. In the passbands, the vibration beyond *fr* ¼ 282 Hz is stronger but never exceeds the MSV of the reference shell. Particularly, the average reduction reaches Δ*MSV* ¼ 10 dB. Looking at **Figure 6b**, the sound power level (SWL) is also very small in the BGs. Even for the passbands, the SWL is effectively suppressed, with an average reduction approaching Δ*SWL* ¼ 15 dB. Then, we examine the radiation efficiency of the ABH shell. As shown in **Figure 6c**, the radiation efficiency of the uniform shell grows to the maxima in the vicinity of the critical frequency *f <sup>c</sup>* ¼ 390 Hz. After embedding the ABHs, however, the radiation efficiency is significantly impaired, almost in the whole frequency band of

#### **Figure 6.**

*Comparison of the (a) mean square velocity (MSV), (b) sound power level (SWL), and (c) radiation efficiency, between the ABH and uniform shells. The shaded areas stand for the BGs.*

interest. Careful readers may notice that the radiation efficiency is very small in the 2-nd BG (the 1-st BG is not shown here because it is too narrow), but in the latter BGs, it becomes larger. This is due to the weakening of the local resonance effect at higher frequencies where the uniform portions start to activate (see **Figure 4**).

To clearly manifest the characteristics of the ABH shell, we have further calculated the vibration field at each frequency. As illustrated in **Figure 7**, compared to the uniform shell whose distribution of vibration nodes is very regular to location and frequency (over the ring frequency 173 Hz), the vibration in the BGs is obviously isolated as propagating in the axial direction. For frequencies outside of the BGs, the amplitude of the displacement is also clearly reduced to the right direction. Specifically, we choose 287 Hz and 340 Hz as two representative frequencies in the BG and the passband, respectively, for demonstrating the effectiveness of the ABH shell. In **Figure 8a**, it is clearly seen that the local resonance effect in the ABHs is very strong, such that vibration can be substantially stopped when compared to the uniform shell. While in **Figure 8b** we can see that the wave can be transmitted to the whole shell, but the amplitude is very small because of the highly efficient damping effect by the ABH + damping layer configuration.

**Figure 7.** *Displacements (wref* ¼ 1 *m) of (a) the ABH shell and (b) the uniform shell, changing with frequency.*

#### **Figure 8.**

*Forced vibration shapes for the finite shell having five cells at (a) 258 Hz (at the center of the second bandgap), (b) 340 Hz (in the passband), compared to a uniform cylindrical shell having the same damping layer configuration. The shaded areas represent the ABH portions.*

*Periodic Acoustic Black Holes to Mitigate Sound Radiation from Cylindrical Structures DOI: http://dx.doi.org/10.5772/intechopen.101959*
