Witold J. Rdzanek

Theoretical analysis of sound radiation of an elastically supported circular plate

Abstract This study focuses on the analysis of the active and reactive sound power of the axisymmetric modes of free vibrations of elastically supported circular plates embedded in a rigid baffle. Some linear and time-harmonic processes have been considered. It is assumed that the plate radiates some acoustic waves into a hemisphere filled with a lossless gaseous medium. The integral formulations for the active and reactive sound power have been derived and expressed in their Hankel representations. They have been used to derive some elementary formulations in the form of some high-frequency asymptotes valid for frequencies higher than the successive coincidence frequencies of the plate. Therefore, the discussion on some sample numerical results mostly covers the sound power radiated at those frequencies. The asymptotes are easy to express in a computer code and they do not need great processor capacity. They are therefore useful for engineering use. The main benefit of the analysis presented in this paper is that the sound power for all the possible boundary configurations of the boundary stiffnesses, i.e., classical clamped, guided, simply supported or completely free boundaries as well as all the intermediate situations, has been described using the same formulae. This is possible simply by changing the two values of stiffnesses associated with the boundary conditions, whose influence on the radiated sound power has been discussed. The solution of the problem of sound power radiated by a vibrating elastically supported circular plate presented herein is essentially more general than the solutions presented earlier for the classical boundary configurations, such as clamped, simply supported, guided or completely free circular plates.

The modal low frequency noise of an elastically supported circular plate.

The modal low frequency noise generated by a vibrating elastically supported circular plate embedded into a flat infinite baffle has been examined. The main aim of this study is the analysis of the radiation efficiency. Low frequency approximated formulas have been presented. They are valid for all the limiting boundary conditions of the plate with its edge clamped, guided, simply supported or free as well as for all the intermediate axisymmetric boundary configurations. The formulas are expressed in the elementary form, useful for numerical computations. They are a generalization of some earlier published results. First, they are valid for axisymmetric and asymmetric modes since both kinds of modes play an important role in the low frequency range. Second, a single formula for the radiation efficiency, valid for all the axisymmetric boundary configurations, has been proposed. A numerical example for the sound power radiation has been given for some hatchway covers mounted on a ship deck.

ASYMPTOTIC APPROXIMATION OF THE MODAL ACOUSTIC IMPEDANCE OF A CIRCULAR MEMBRANE

The Neumann boundary value problem of the Helmholtz equation of a vibrating circular membrane embedded into a flat rigid baffle is solved. The membrane is excited asymmetrically and radiates acoustic waves into the half-space above the baffle. A set of elementary asymptotic equations for modal radiation self-impedance and mutual impedance is presented. The equations are necessary for numerical computations of the radiated active and reactive acoustic power including the acoustic attenuation. A few equations available in the literature are collected. All the missing equations have been obtained using the methods of analysis of contour integral and stationary phase. The presented equations cover a wide frequency band, with the exception of the lowest frequencies and the frequencies close to coincidence.

Sound Radiation of the Resonator in the Form of a Vibrating Circular Plate Embedded in the Outlet of the Circular Cylindrical Cavity

The Neumann axisymmetric boundary value problem is considered for a vibrating thin clamped circular plate embedded in the flat rigid screen in the outlet of the circular cylindrical cavity. It is assumed that the two pistons, one cylindrical and the other one annular/circular, are vibrating inside the cavity with the same single frequency and different initial phases. The pistons are the only sources of excitation of the fluid. The acoustic pressure difference on both sides of the plate forces its vibrations. The acoustic waves are radiated into the half-space above it. A rigorous theoretical analysis of sound radiation has been performed based on the exact solution of the problem of free vibrations of the plate. The system of three coupled partial differential equations is solved. They are the two Helmholtz equations for the cavity and for the half-space, and the equation of motion of the plate. Consequently, the acoustic pressure distribution in both spaces is presented as well as the acoustic power radiated.