Wave propagation in finite periodically ribbed structures with fluid loading

Abstract

The paper studies the problem of wave transmission along a fluid-loaded plane elastic membrane supported by a finite array of equally spaced ribs. One of the ribs is driven by a time-harmonic line force and the rest have infinite impedance, so that fluid loading provides the only mechanism for the transmission of energy. Existing solutions for the infinite analogue exhibit a stop/pass band frequency structure, in which the energy is, alternately, exponentially localized around the driving force and constant along the array. However, at pass band frequencies this is inconsistent with numerical studies of finite arrays, which reveal marked amplitude fluctuations. In this paper an exact solution is given for a general finite configuration. This is used to explain and further explore the response. In particular it is shown that as the array length increases the pass band response becomes increasingly sensitive to frequency, and the solution cannot approach an asymptotic limit. The results give the forces along the array as an interference pattern, which may be thought of as propagating inwards from each end. This solution is obtained by forming a 2 x 2 matrix which relates the forces at any pair of adjacent ribs to those at the next pair. From the action of this matrix the response can be found everywhere, and the detailed properties of the solution are determined by those of the matrix. Special treatment is needed to deal with the band edges, which conform neither to stop nor pass band behaviour.