Analysis of sound radiation from a vibrating clamped thin rectangular plate without baffle and in the rigid baffle using exact formulas

This study presents an analysis of sound radiation from a vibrating thin clamped rectangular plate using exact formulas. A new analytical approach–referred to here as the theoretical approximate formulas method –is proposed and applied to cases where the plate is either embedded in a rigid infinite baffle or has no baffle at all. The exact eigenfrequencies of the plate are obtained from a system of five coupled characteristic equations, as reported in the literature. The biharmonic equation governing the plate’s vibrations is coupled with the Helmholtz equation on both sides of the plate, thereby incorporating acoustic attenuation into the model. To represent the acoustic pressure and radiated acoustic power, a double Fourier transform is employed. These quantities are expressed as expansion series involving double infinite integrals. The integrals are evaluated exactly using the spectral mapping method, the Dini series, and radial polynomials. The resulting solutions are accurate and rapidly convergent, spanning from frequencies below the plate’s fundamental frequency to those above its critical frequency. Consequently, the proposed method enables effective and precise solutions to both Neumann and Dirichlet boundary value problems, and facilitates detailed analysis of the resulting acoustic fields. The findings can be applied to predict the acoustic behavior of structural casing elements shaped in the form of thin rectangular plates, in industrial environments. Selected numerical examples are also provided to demonstrate the method’s applicability.