An exact eigenfrequency equation for the oscillations of a viscous fluid contained in an open and rectangular vessel with a flexible wall

Our concern is with the problem of small amplitude motion of free surface viscous liquid interacting with the vibration of an elastic side wall of a rectangular container. This note extends the study of Chai et al. (1996) by including the viscous effects. The note also extends the work of Bauer and Komatsu (1998) who considered a similar problem, that they solved approximately after abandoning no-slip conditions on rigid walls. Here, an analytical description, which takes into account all the boundary conditions, is given for the fluid flow by performing a suitable decomposition of the stream function. This leads to an exact equation for complex frequencies.