ACOUSTIC RADIATIONS FOR SUBMERGED ELASTIC STRUCTURES USING NATURAL MODE EXPANSIONS IN CONJUNCTION WITH RADIATION MODES APPROACH

Abstract

This work formulates submerged elastic structures using in-vacuo vibrational mode expansions with which the acoustic impedance loading is derived based on radiation mode theory. The displacement of natural modes on the normal direction is expanded as linear combinations of a set of velocity radiation modes that the expansion coefficients characterize as the radiation characteristics of each vibration mode. This type of expansion allows one to represent the surface pressure by the corresponding set of pressure radiation modes. Thus, a symmetric impedance matrix associated with the natural vibration mode expansions is derived when a variational increment is applied to the virtual work done by the surface pressure against the normal displacement. The equation of the submerged structures is obtained according to Hamilton's principles. By incorporating the description of radiation modes, this equation of natural mode expansions is used to study the coupling among vibration modal amplitudes due to the modal cross-impedances and the convergence of near and farfield solutions. In addition, a slender submerged spheroidal shell vibrating axisymmetrically serves as a numerical example to demonstrate the effectiveness of the analysis procedure. This numerical example reveals that the acoustic impedances decrease with ascending mode numbers, causing the high order vibration modes to react independently. Moreover, the convergence of the surface pressure and normal velocity is examined on the basis of independent reaction of the vibration modes. Accurately predicting farfield solutions depends only on the convergence of the surface quantities whose components pertain to strong radiation modes. The numerical example indicates that the number of vibration modes used in the expansion for predicting farfield solutions is less than the modes required for the surface solution.