Sound radiation of a vibrating elastically supported circular plate embedded into a flat screen revisited using the Zernike circle polynomials

Abstract

This study deals with the problem of sound radiation by an elastically supported thin circular plate. The plate is excited asymmetrically. In such cases, usually different approximate methods are used to calculate the acoustic power. These methods are time-consuming, and some of them are applicable only to axisymmetric problems. They work only for the lowest and the highest frequencies. Consequently, their applications are limited. Another shortcoming of such methods is that they work only for either clamped circular plates or simply supported plates. In practical applications, the boundary conditions of circular plates are often required. Therefore, the model of an elastically supported plate is of a great interest in this regard. Finally, the methods presented in the literature are time-consuming and not highly accurate. They often incorporate numerical integration to calculate the acoustic pressure and the acoustic power. The use of the radial polynomials leads to much more accurate and efficient results. The acoustic power is expressed in terms of the modal impedance coefficients. The coefficients are calculated without numerical integration and with arbitrary precision in a wide low-frequency band. They are expressed in terms of a rapidly converging expansion series. The formulas presented are useful to solve the problem of vibration of a plate including arbitrary excitation, material damping, and fluid-structure interactions. This results in a system of three coupled differential equations. The first one is the Helmholtz equation governing the fluid vibrations in the upper half-space, the second one is also the Helmholtz equation governing the fluid vibrations in the lower half-space, and the third one deals with the motion of the plate. Consequently, it leads to a system of algebraic equations. Finally, the effects of the plate's boundary conditions on the acoustic power, the far field, and the near field are analyzed numerically.