28 research outputs found
A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity
A nonlinear Schr\"odinger equation for the envelope of two dimensional
surface water waves on finite depth with non zero constant vorticity is
derived, and the influence of this constant vorticity on the well known
stability properties of weakly nonlinear wave packets is studied. It is
demonstrated that vorticity modifies significantly the modulational instability
properties of weakly nonlinear plane waves, namely the growth rate and
bandwidth. At third order we have shown the importance of the coupling between
the mean flow induced by the modulation and the vorticity. Furthermore, it is
shown that these plane wave solutions may be linearly stable to modulational
instability for an opposite shear current independently of the dimensionless
parameter kh, where k and h are the carrier wavenumber and depth respectively
The Basics of Water Waves Theory for Analogue Gravity
This chapter gives an introduction to the connection between the physics of
water waves and analogue gravity. Only a basic knowledge of fluid mechanics is
assumed as a prerequisite.Comment: 36 pages. Lecture Notes for the IX SIGRAV School on "Analogue
Gravity", Como (Italy), May 201
Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model
International audienceBeing considered as a prototype for description of oceanic rogue waves (RWs), the Peregrine breather solution of the nonlinear Schrodinger equation (NLS) has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave eld generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves
Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model
Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrödinger equation has been recently observed and intensely investigated experimentally in particular within the context of water
Numerical simulation of solitary gravity waves on deep water with constant vorticity
Essentially nonlinear dynamics of surface gravity waves on deep water with
constant vorticity is modeled using governing equations in conformal
coordinates. It is known that the dispersion relation of surface gravity waves
upon shear flow has two branches, with one of them being weakly dispersive for
long waves. The weakly nonlinear evolution of waves of this branch can be
described by the integrable Benjamin-Ono equation, which has soliton and
multi-soliton solutions, and solitons interact elastically. To what extent the
properties of such solitary waves obtained within the weakly nonlinear models
are preserved in the exact Euler equations is not known. Here, we investigate
the behaviour of this class of solitary waves without the restrictive
assumption of weak nonlinearity by using the exact Euler equations. Evolution
of localized initial perturbations leading to formation of one or multiple
solitary waves is modeled and properties of finite amplitude solitary waves are
discussed. It is shown that within exact equations two-soliton collisions are
inelastic and the waves receive phase shift after the interaction
Hydroelastic solitary waves with constant vorticity
In this work, two-dimensional hydroelastic solitary waves in the presence of constant vorticity are studied. Time-dependent conformal mapping techniques first developed for irrotational waves are applied subject to appropriate modification. An illustrative high-order Nonlinear Schrödinger Equation is presented to investigate whether a given envelope collapses into a singular point in finite time by using the virial theory. Travelling solitary waves on water of infinite depth are computed for different values of vorticity and new generalised solitary waves are discovered. The stabilities of these waves are examined numerically by using fully nonlinear time-dependent computations which confirm the virial theory analysis.</p
Capillary-gravity solitary waves on water of finite depth interacting with a linear shear current
The problem of two-dimensional capillary-gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter-propagating waves of different types: Korteweg de-Vries (KdV)-type long solitary waves and wave-packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left- and right-propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long-wave/short-wave interaction. Traveling-wave solutions and their dynamics in the full Euler equations are computed numerically using a time-dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wavetrains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations