Keywords: eigenpairs, rectangular cavity, coupling, evanescent modes, standing wave modes, modal orthogonality.

Noise inside an acoustic-structural coupled cavity is a common problem in numerous practical situations including those involving aircrafts, automobiles, and trains. The term "coupling" indicates interference between a structural and an acoustic field of a cavity, resulting in the alternation of the eigenpairs of uncoupled system dynamics. Depending on the degree of coupling, cavity systems can be classified into two categories: a modally coupled cavity system and a coupled cavity system.

A modally coupled cavity system often introduced in sound transmission control problems is based on the modal coupling theorem established under the assumption that the fluid medium is non-dense and the cavity walls not "thin". The characteristic of this system is that the eigenfunctions of a coupled system remain the same as those of an acoustically rigid walled cavity, while only the eigenfrequencies of the cavity change.

When cavity walls become thin and the cavity gap shallow, the assumption of a modal coupling is no longer valid; thus, such a case falls into the second category, i.e. a coupled cavity system. Considerable efforts have been made in the literature to derive the exact solution of a coupled rectangular cavity system that comprises five rigid walls and a flexible panel. The main focus of these studies was, however, to understand sound transmission through a cavity-backed panel, and thus their concern was directed towards deriving the exact solution of a forced vibration of the cavity-backed panel subject to external sound pressure. As such, the exact eigenpairs of the coupled cavity, which are intrinsic parameters governing the system dynamics and independent of any extraneous forces, have yet to be found.

The purpose of this paper is to derive explicitly the eigenpairs of a coupled rectangular cavity comprising five rigid walls and a flexible panel. First, the coupling orthogonality conditions the eigenpairs need to satisfy are derived, thereby verifying the validity of the eigenpairs newly found or already existent. Using the coupling orthogonality conditions, the modal equation of the coupled cavity system is then obtained, permitting one to deal with a forced response of the coupled cavity. Furthermore, eigenfunctions governing the dynamics of both the sound field and the vibration field are expressed as the infinite sum of cluster eigenfunctions that possess the same attribute in common. The characteristic matrix equation is then derived, enabling one to specify the eigenpairs of the coupled cavity. In order to investigate the fundamental properties of the eigenpairs derived, a numerical analysis is conducted, revealing the presence of evanescent modes together with the conventional standing wave modes. It is shown that the evanescent modes emerge when the eigen wavenumber of the cluster eigenfunction becomes pure imaginary and the associated coefficient large.

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