We derive from first principles the dynamical equations that govern the interaction of small-amplitude water waves with freely floating obstacles in a stratified multilayer fluid. Focusing on two-layer fluids, we present the equations in an easily manageable matrix form, write down conditions for the stability of equilibrium and, by limiting ourselves to time-harmonic motions, recast the problem as a spectral boundary-value problem composed of a differential equation and an algebraic system, coupled through boundary conditions. Proceeding with a suitable variational and operator formulation, we present an elimination scheme that simplifies the system to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we derive a sufficient condition guaranteeing the existence of trapped modes in a two-layer fluid channel.
