Trapped modes along periodic structures submerged in a two-layer fluid with background steady flow

The trapping of linear water waves by infinite arrays of three-dimensional fixed periodic structures in a two-layer fluid, where the layers are considered semi-infinite in depth, have a common interface and move each with an independent uniform velocity with respect to the ground, is studied. The existence of real solutions to the dispersion relation demands a further stability condition on the layer velocities. From the variational formulation, after certain choices of background steady flow, results a nonlinear spectral problem, which upon a sensible linearization gives a geometric condition ensuring the existence of trapped modes (within the limits set by the stability condition). Symmetries reduce the global problem to the first quadrant of the velocity space. Examples are shown of configurations of obstacles that are both independent of the layer velocities and dependent only on their difference. Future developments are suggested.