**3. Sound radiation from finite periodic ABH shells**

### **3.1 Radiation theory for cylindrical shells**

Once the radial displacement *w x*<sup>0</sup> , *θ*<sup>0</sup> ð Þ (here *θ*<sup>0</sup> ¼ *y*<sup>0</sup> *=R*) is obtained, the normal velocity to the surface can be further expressed as *vw x*<sup>0</sup> , *θ*<sup>0</sup> ð Þ¼ j*ωw x*<sup>0</sup> , *θ*<sup>0</sup> ð Þ. For a baffled cylindrical shell, the radiated sound pressure of an arbitrary external point ð Þ *r*, *θ*, *x* can be analytically obtained by [29, 30].

$$p(\mathbf{x}, \theta, r) = \frac{\mathbf{j}\rho\_a a}{4\pi^2} \Big|\_{-\pi}^{\pi} \int\_{-a}^{a} \nu\_w(\mathbf{x}', \theta') \sum\_{n = -\infty}^{+\infty} \cos\left[n(\theta - \theta')\right] \times \tag{17}$$

$$\int\_{-\infty}^{+\infty} \frac{\exp\left[\mathbf{j}k\_x(\mathbf{x} - \mathbf{x}')\right]}{k\_\mathcal{\mathcal{R}}R} \frac{\mathbf{H}\_n^{(1)}(k\_\mathcal{\mathcal{R}}r)}{\mathbf{H}\_n^{(1)'}(k\_\mathcal{\mathcal{R}}R)} \mathrm{d}k\_\mathcal{\mathcal{R}}\mathbf{d}\mathbf{x}' \mathrm{d}\mathbf{d}\theta',$$

where *<sup>ρ</sup><sup>a</sup>* represents the density of air, *kx* and *ky* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> � *<sup>k</sup>*<sup>2</sup> *x* � �<sup>0</sup>*:*<sup>5</sup> symbolize the wavenumber components in the *x* and the *y* direction, respectively, and *k* stands for the total sound wavenumber. Here *θ* ¼ *y=R* stands for the circumferential angle. Hð Þ<sup>1</sup> *<sup>n</sup>* indicates the *n*-th Hankel function of the first kind, and Hð Þ0 <sup>1</sup> *<sup>n</sup>* denotes its first derivative with respect to the argument *kyR*.

For numerical estimation, the cylindrical shell can be segmented into *N* elementary radiators, with each surface area Δ*S*. The surface velocity can be assembled as a vector **v***w*, which can be further used to calculate the sound pressure vector on the cylindrical surface

$$\mathbf{p}\_{N \times 1} = \mathbf{Z}\_{N \times N} \mathbf{v}\_{N \times 1},\tag{18}$$

where *Z* represents the acoustic impedance matrix, with entries

$$\mathbf{Z}\_{\vec{\eta}} = \frac{\mathbf{j}\rho\_a a \Delta \mathbf{S}}{2\pi^2} \sum\_{n=0}^{+\infty} \varepsilon\_n \cos\left[n\left(\theta\_i - \theta\_j\right)\right] \int\_0^{+\infty} \frac{\cos\left[k\_\mathbf{x}\left(\mathbf{x}\_i - \mathbf{x}\_j\right)\right]}{k\_\mathbf{y}a} \frac{\mathbf{H}\_n^{(1)}\left(k\_\mathbf{y}r\right)}{\mathbf{H}\_n^{(1)'}\left(k\_\mathbf{y}R\right)} \mathrm{d}k\_\mathbf{x}, \quad \text{(19)}$$

here *ε<sup>n</sup>* is a normalized coefficient, and it is given by

$$\varepsilon\_n = \begin{cases} \mathbf{1}, & n = \mathbf{0}, \\ \mathbf{2}, & n > \mathbf{0}. \end{cases} \tag{20}$$

Next, we can write the sound power as

$$\mathbf{W}\_s = \boldsymbol{\nu}^\mathrm{H} \mathbf{R} \boldsymbol{\nu},\tag{21}$$

where the superscript H stands for the Hermite transpose and *<sup>R</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>S</sup>* <sup>2</sup> Re *Z*j *r*¼*R* � � is the radiation resistance matrix (real symmetric and positive-definite). The sound radiation efficiency can be further obtained by

$$\sigma = \frac{\mathcal{W}\_s}{\rho\_a c\_a N \Delta \mathcal{S} \langle v\_w^2 \rangle\_{overall}},\tag{22}$$

where *v*<sup>2</sup> *w* � � *overall* represents the mean square velocity (MSV) over the whole surface of the ABH cylindrical shell.

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