We formulate a coupled vibration problem between a structure and an acoustic field in a mathematically rigorous fashion. A typical example of the structure is a car body that can be modeled by a cluster of thin plates. The problem leads to a nonstandard eigenvalue problem in a function space. Furthermore, we introduce a coupling strength parameter e as a multiplier applied to the nondiagonal coupling terms. A natural interpretation of this parameter is given. We represent an eigenpair for the coupled system by a perturbation series with respect to e which enable us to express the eigenpair for the coupled case by those for the decoupled case. We prove that the series consists only of even order terms of e. In some practical applications, by using this perturbation series, it would become unnecessary to perform time consuming computations to get coupled eigenvalues, and hence the present results obtained by the perturbation analysis might have considerable engineering importance. We confirm the adequacy of this perturbation analysis by investigating some numerical examples. The results are given for a two-dimensional coupled problem. Without loss of generality, the same operator theoretical approach applies to the eigenvalue problems of the coupled vibration problem in a more general three-dimensional acoustic region with a plate for a part of its boundary.