## 2 Mathematical Modeling and Analysis
### Analytical and Numerical Formulation for Acoustic Radiation from Tapered Annular Circular Plate  
It is considered that an annular circular plate of inner radius 'b' and outer radius 'a' in flexural vibration is set on flat rigid baffle having infinite extent as reported in Figure 1. Acoustic scattering of the edges of a vibrating structure is neglected in this study. Let P be the sound pressure amplitude, S\({}_{\text{s}}\) be the surface of the sound source, \(q\) be the Green methods function in free field, \(l_{\text{s}}\) and \(l_{p}\) be the position vectors of source and receiver and the surface normal vector at \(l_{\text{s}}\) be \(f\); then structure sound radiation can be obtained by the Rayleigh integral [10] as given by Equation (3):  
\[\text{P}\big{(}l_{\text{P}}\big{)}=\int_{S_{\text{s}}}\left(\frac{\partial q} {\partial f}\big{(}l_{p},l_{s}\big{)}P\big{(}l_{\text{P}}\big{)}-\frac{ \partial P}{\partial f}(l_{s})q\big{(}l_{p},l_{s}\big{)}\right)ds(l_{s}) \tag{3}\]  
Figure 1: Sound radiation investigated for thick annular circular plate in Z direction enclosed in a sphere. The sound pressure, radiated from non-planar source in far and free field environment based on plane wave approximation can be expressed by Equation (4):  
\[\mathrm{P}\big{(}l_{p}\big{)}=\frac{\rho_{0}c_{0}B}{4\pi}\int_{S_{s}}\frac{e^{iB| l_{p}-l_{s}|Ul}\big{(}l_{s}\big{)}}{\big{|}l_{p}-l_{s}\big{|}}(1+\cos\eta)dS \tag{4}\]  
Let \(\rho_{0}\) be the mass density of air, \(c_{0}\) be the speed of sound in air, B be the corresponding acoustic wave number, and \(\mathrm{U}\) and \(\mathrm{\dot{u}}\) be the corresponding vibratory velocity amplitude and spatial dependent vibratory velocity amplitude in the \(z\) direction at \(l_{s}\), then from a normal plane [10], the modal sound pressure \(\mathrm{P}_{mn}\) for an annular plate with (m, n)th mode is obtained from simplifying Equation (4) with Hankel transform and is expressed by Equations (5) and (6):  
\[\mathrm{P}_{mn}(R,\alpha,\beta)=\frac{\rho_{0}c_{0}Be^{iB_{mn}R_{d}}}{2R_{d}} \cos n\beta(-i)^{n+1}A_{n}\bigg{[}u(l)\bigg{]}(1+\cos\eta) \tag{5}\]  
\[A_{f}\bigg{[}u(l)\bigg{]}=\int\limits_{0}^{\infty}u(l)J_{n}(B_{l}l)ldl,\ Bl=B \sin\theta;\ R_{d}=\big{|}l_{p}-l_{s}\big{|} \tag{6}\]  
where \(\mathrm{X}_{n}\) is Bessel function of order n, (\(\alpha\), \(\beta\)) are the cone and azimuthal angles of the observation positions, respectively, \(\eta\) is the angle between the surface normal vector and the vector from source position to receiver position, and A is the Hankel transform. According to the far field condition, \(\mathrm{R}_{d}\) in the denominator is approximated by R where \(\mathrm{R}=\big{|}l_{p}\big{|}\) is considered to be radius of the sphere. Consider that on a sphere \(\mathrm{S}_{v}\) the observation positions are represented by some points having equal angular increments (\(\Delta\varphi,\ \Delta\alpha\)). If '\(\Delta\varphi\)' represents the small increment in the circumferential direction of the plate, at all of the observation positions, the sound pressures is given by Equations (4)-(6). The modal sound power \(\mathrm{S}_{mn}\) for the (m, n)th mode [10; 16] from the far-field is given by Equation (7):  
\[S_{mn}=\big{(}D_{mn}S_{v}\big{)}_{s}=\frac{1}{2}\!\!\int\limits_{0}^{2\pi}\int \limits_{0}^{\pi}\frac{p_{mn}^{2}}{\rho_{0}c_{0}}R^{2}\sin\alpha d\alpha d\beta \tag{7}\]  
where the acoustic intensity is represented by \(\mathrm{D}_{mn}\) and area of the control surface is represented by \(\mathrm{S}_{v}\). The radiation efficiency \(\sigma_{mn}\) of the plate [10] is given by Equation (8):  
\[\sigma_{mn}=\frac{S_{mn}}{\big{|}u_{mn}\big{|}ts}\bigg{|}\overset{.}{u}_{mn} ^{2}\bigg{|}ts=\frac{1}{2\pi(a^{2}-b^{2})}\int_{b}^{a}\int\limits_{0}^{2\pi}U ^{2}d\varphi dl \tag{8}\]  
where, for the two normal surfaces of the plate the spatially average r.m.s velocity is represented as \(\big{|}\overset{.}{u}_{mn}^{2}\big{|}ts\).