A closed form expression for the resonance frequencies of an unbaffled simply-supported rectangular water-loaded plate

Abstract

The sound radiation from a finite, simply-supported rectangular unbaffled plate submerged and vibrating in water is considered. The objective is to obtain a closed form expression for the coupled resonance frequencies of this water-loaded plate. The sound pressure at an arbitrary point in the surrounding fluid medium is expressed as an integral of the product of the pressure jump and the derivative of the Green’s function over the plate surface. Using Euler’s equation in the plane of the plate, a linear system of equations is obtained for the displacement field. The solution procedure involves a certain parameter known as the modal coupling coefficient. This coupling coefficient differs from that of the analogous problem of sound radiation by a baffled plate in that a square root term appears in the numerator. The improper double integral in the coupling coefficient is approximated analytically using the contour integration technique. It is in the nature of panel radiation that due to water loading, several types of modal interactions happen depending upon the frequency of excitation. For underwater applications, keeping 10 kHz as the upper limit of the excitation frequency, approximate analytical expressions for the modal coupling coefficient are derived specifically for the corner–corner and the edge–edge type interactions. Next, a small fluid loading parameter is introduced into the coupled equation of motion for the free vibration problem through the residual contribution of the coupling coefficient. Then, the perturbation method is used to obtain the closed form expression for the coupled resonance frequencies. Using this closed form expression, the coupled natural frequencies are computed for a standard size panel and compared with those obtained from the numerical calculations. A good match is observed between the two results. Along the way, a concerted effort is made to provide bounds on the error in the modal coupling coefficient caused by the various approximations. The closed form natural frequency expression is valid for a range of panel sizes, aspect ratios and thicknesses.