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- Trapped modes in a fluid with three layers topped by a rigid lidPublication . Cal, Filipe; Dias, Gonçalo A. S.; Pereira, Bruno M. M.We consider trapping of linear water waves by a submerged horizontal cylinder in a three-layer fluid topped by a rigid lid. Trapped modes correspond to time harmonic oscillations with finite energy of the fluid surrounding a submerged structure and can be found as eigenfunctions of a certain spectral boundary-value problem. Our main result is a geometric condition relating the cross sections of the submerged parts of the obstacles and the line integrals along the parts of the interfaces pierced by the obstacles and guaranteeing the existence of trapped modes: This follows from variational techniques applied to a suitable operator formulation of the problem. Several examples of structures (piercing or not the interfaces between the fluid layers) satisfying the condition and supporting trapped modes are given.
- Trapped modes along periodic structures submerged in a two-layer fluid with free surface and a background steady flowPublication . Dias, Gonçalo; Pereira, BrunoThis study examines the trapping of linear water waves by an endless structure of stationary, three-dimensional periodic obstacles within a two-layer fluid system. The setup features a lower layer of either limited or unlimited depth, overlaid by an upper layer of finite thickness bounded by a free surface, with each layer exhibiting its own constant background speed relative to the fixed reference frame. For real roots to emerge in the dispersion relation, an additional stability condition on the layer velocities is necessary. By selecting adequate choices for the background flow, a non-linear eigenvalue problem is derived from the variational formulation; its reasonable approximation yields a geometric criterion that guarantees the presence of trapped modes (subject to the aforementioned stability bounds). The selection of the eigenvalue is influenced by velocity owing to the presence of an interface and free surface. Due to inherent symmetries, the overall analysis can be confined to the positive quadrant of the velocity domain. Illustrations are provided for various obstacle setups that produce trapped modes in diverse ways.