**1. Introduction**

An acoustic black hole (ABH) is usually realized by reducing structural thickness following a power law *h x*ð Þ¼ *<sup>ε</sup>xm* ð Þ *<sup>m</sup>* <sup>≥</sup> <sup>2</sup> , as illustrated in **Figure 1**. When an incident flexural wave impinges at the edge of the ABH, its wavelength and wave speed get gradually decreased. Meanwhile, the wavenumber is however increased and the amplitude is intensified. In the ideal case, the thickness at the ABH tip decays to zero, where the wave velocity vanishes as well, such that the traveling time to its center becomes infinite. In other words, the wave will never reach the tip. In analogy with cosmology, the termination behaves like a "Black Hole" in which nothing can escape from it. This is the story of how the term "Acoustic Black Hole" was coined [1]. Howbeit, in real applications, generally, an ABH is imperfect. Namely, there exists a truncation near the ABH tip, which results in obvious reflection because of the residual thickness [2]. Fortunately, attaching a thin viscoelastic

### **Figure 1.**

*Illustration of ABH effect: the incident wave is localized in the ABH tip as it propagates toward the ABH.*

layer at the ABH tip, where the energy is highly concentrated, can alleviate this problem [3]. Recently, constrained viscoelastic layers have been suggested to enhance the damping effects, by changing the normal tensile and compressive deformation of damping material into the shear one [4].

It has been shown that the ABHs are very efficient to reduce vibration from straight beams [5, 6] and flat plates [7, 8]. The shapes and lengths of the damping layers have been extensively investigated [4, 9]. Also, different ABH designs have been proposed for the purpose of enhancing energy consumption [7, 10–12]. Thanks to the vibration reduction because of highly efficient damping, the sound radiation from ABH structures is accordingly reduced [13]. Not only that, recent studies have shown that the ABHs can also impair the sound radiation efficiency because of the thickness reduction [14, 15]. Particularly, for cavity noise, the ABH profile can destroy the coupling strength between structural and acoustic modes, which is the third underlying mechanism of the ABHs for reducing room noise [16, 17]. Furthermore, periodic and gradient-index ABHs are investigated for steering waves [18–20]. It is also worthwhile mentioning that ABHs can also enhance energy harvesting due to wave focalization [21], using piezoelectric layers rather than viscoelastic ones.

The state-of-the-art reviewed above are mainly centered on flat structures. However, in aeronautics, astronautics, and underwater vehicles, cylindrical shells are very common. There, the vibroacoustic problems are very critical to determine their comfortability and safety, thus it is very demanding to apply ABH features on them. Our previous efforts have been focused on the vibration of cylindrical beams [22] and shells [23, 24], together with the sound radiation from a finite cylinder [25]. In this chapter, we continue this topic but analyze the sound radiation from periodic ABH shells.

As shown in **Figure 2a**, an infinite periodic ABH shell is considered, with each unit cell having radius *R* and length *Lcell*. An ABH plus a thin viscoelastic layer (see the green layer) is laid in the center of the cell. The geometries of the ABH and the damping layer are detailed in **Figure 2b**. Here, the profile of the ABH is defined by *h x*ð Þ¼ *<sup>ε</sup>*j j *<sup>x</sup> <sup>m</sup>* <sup>þ</sup> *hc*, where *<sup>ε</sup>* <sup>¼</sup> ð Þ *huni* � *hc <sup>r</sup>*�*<sup>m</sup> abh* stands for the ABH slope and, *rabh*, *hc*, and *m* respectively are the ABH radius, residual thickness, and order. We will characterize the band gaps (BGs) for infinite periodic ABH shells and their dependence on the ABH geometry. Next, a finite periodic ABH shell containing five cells will be characterized, under a ring excitation acting at *x <sup>f</sup>* (see **Figure 2c**). The translational springs *ki*, *i* ¼ 1, 2 and rotational ones *pi* , *i* ¼ 1, 2 are intended for boundary conditions (distributed circumferentially). As one could expect, the appearance of ABHs weakens the stiffness of the whole structure. This may deteriorate the structural problems. To partially solve this, we can introduce *N* stiffeners for each cell (see **Figure 2d**), with each width being *W*. The effects of the stiffeners will also be investigated at the end of this chapter.

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

**Figure 2.**

*Illustration of the periodic ABH cylindrical shell. (a) A unit ABH cell having a thin damping layer (green). (b) The geometrical detail of the ABH profile. (c) A supported finite ABH cell having five cells under a ring excitation. (d) Illustration of the stiffener for enhancing the structure (with N stiffeners in the circumferential direction).*
