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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.
- Velocity and energy of periodic travelling interfacial waves between two bounded fluidsPublication . Cal, Filipe; Dias, Gonçalo A. S.For a periodic travelling irrotational wave propagating at the interface between two homogeneous, incompressible and inviscid fluids bounded by horizontal planes, we generalise the Stokes definitions for the velocity of the wave propagation. Under certain conditions imposed on the horizontal velocity of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we prove that the mean horizontal velocity of propagation of the wave is greater than the generalised mean horizontal velocity of the mass of the fluid. We show that, for interfacial waves of small amplitude, the excess kinetic and potential energy of the fluid have the same magnitude, but different signs, and for the nonlinear setting, we prove that the excess kinetic energy is negative.
- Flow of a periodic interfacial travelling water wavePublication . Cal, Filipe; Dias, GonçaloWe consider a symmetric periodic travelling wave propagating at the interface between two homogeneous, incompressible, irrotational and inviscid fluids bounded by horizontal planes. For interfacial waves of small amplitude, we present a formula for the interface wave depending on the pressure at the rigid lid and at the flat bottom, and, for the general non-linear case, we derive a lower bound for the interfacial wave height. Under certain conditions imposed on the horizontal component of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we study the monotonicity of the horizontal component of the velocity field along the streamlines and also analyze the monotonicity of the pressure along horizontal lines throughout the fluid in both layers, and along the boundary of the domain, between the crest and the trough. Finally, based on the behavior of the velocity field components, we build a pictorial description of the particle paths in both layers.