## 1 Introduction  
Tapered annular circular plates with different combinations of rectangular and concentric patches has many engineering applications. They are used in many structural components i.e., building, design, diaphragms and deck plates in launch vehicles, diaphragms of turbines, aircraft and missiles, naval structures, nuclear reactors, optical systems, construction of ships, automobiles and other vehicles, the space shuttle etc. These tapering plates with different combinations of rectangular and concentric patches are found to have greater resistance to bending, buckling and vibration in comparison to plates of uniform thickness. It is interesting to know that tapered plates with different thickness variation have drawn the attention of most of the researchers in this field. However, tapered plates with different combination of rectangular and concentric patches can alter the dynamic characteristic of structures with a change in stiffness. Hence, for practical design purposes, the vibration and acoustic characteristics of such tapered plates are equally important. In comparison to the present study, several existing works are presented where the researchers have investigated the vibration response [1; 2; 3; 4; 5; 6; 7; 8; 9] of circular or annular plates of tapered or uniform thickness. But in terms of acoustic behavior, many researchers have contributed most. [PERSON] and [PERSON] [10] used the thin and thick plate theories to determine the sound radiation from out-of-plane modes of a uniform thickness annular circular plate. [PERSON] [11] used the Bouwkamp integral to determine the mutual and self-radiation impedances both for annular and elliptical pistons. [PERSON] and [PERSON] [12] analyzed the sound power generation of a circular plate of uniform thickness using exact integral representation. [PERSON] and [PERSON] [13] determined the acoustic power output of a clamped annular plate using an asymptotic formula. [PERSON] and [PERSON] [14] minimized the acoustic power of circular plates of uniform thickness using the damping layer placement. [PERSON] [15] studied the acoustic radiation from linearly-varying circular plates. [PERSON] and [PERSON] [16] used the flexural and radial modes of a thick annular plate to determine the self and mutual radiation. [PERSON] et al. [17] studied the vibro acoustic behavior of an unbaffled rotating disk. [PERSON] [18] used an isotropic plate with arbitrarily varying thickness to determine its vibro-acoustic behavior using the finite element method (FEM). [PERSON] and [PERSON] [19] studied the forced response of a thin plate of uniform thickness with attached discrete dynamic absorbers. [PERSON] et al. [20] analyzed an isotropic plate with attached discrete patches and point masses with different thickness variation with different taper ratios to determine its vibro acoustic response. [PERSON] and [PERSON] [21] investigated the annular disk acoustic radiation using structural modes through analytical formulations. [PERSON] et al. [22] investigated the sound radiation and sound power of a planar annular membrane for axially-symmetric free vibrations. [PERSON] [23] determined the sound power radiation from clamped annular plates of uniform thickness. [PERSON] et al. [24] investigated the acoustic radiation of a circular plate for a single sound pulse. [PERSON] and [PERSON] [25] determined the acoustic radiation force used on the solid elastic sphere. [PERSON] and [PERSON] [26] used a simplified disk brake rotor to investigate the acoustic radiation through a semi-analytical method. [PERSON] et al. [27; 28] analyzed the modal approach for different boundary conditions to calculate the average radiation efficiency of a rectangular plate. [PERSON] [29] determined the sound radiation from flat finite structures. [PERSON] [30] analyzed the total radiation resistance for ribbed and simple plates using a simplified asymptotic formulation. [PERSON] [31] used the wave number domain and Fourier transform to analyses the acoustic power.