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Periodic water waves of permanent form traveling at constant speed, the so-called Stokes waves, are studied in water of fixed finite depth using methods previously used in water of infinite depth. We apply our methods to waves of varying steepness over a range of fixed depths in order to determine how a number of physical quantities related to the waves change as the steepness of the waves increases. Finally, we examine the Taylor sign condition for these waves, as well as the complex singularities outside of their domain of definition when the waves are considered as a function of a conformal variable.
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a widely open problem). In this paper we prove the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. This is a very hard small divisors problem for Partial Differential Equations due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension with a very complicated geometry of the resonances. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equa
The forthcoming Wide Area Vista Extragalactic Survey (WAVES) on the 4-metre Multi-Object Spectroscopic Telescope (4MOST) has a key science goal of probing the halo mass function to lower limits than possible with previous surveys. For that purpose, in its Wide component, galaxies targetted by WAVES will be flux-limited to $Z<21.1$ mag and will cover the redshift range of $z<0.2$, at a spectroscopic success rate of $\sim95\%$. Meeting this completeness requirement, when the redshift is unknown a priori, is a challenge. We solve this problem with supervised machine learning to predict the probability of a galaxy falling within the WAVES-Wide redshift limit, rather than estimate each object's redshift. This is done by training an XGBoost tree-based classifier to decide if a galaxy should be a target or not. Our photometric data come from 9-band VST+VISTA observations, including KiDS+VIKING surveys. The redshift labels for calibration are derived from an extensive spectroscopic sample overlapping with KiDS and ancillary fields. Our current results indicate that with our approach, we should be able to achieve the completeness of $\sim95\%$, which is the WAVES success criterion.
Classical Trefftz methods approximate Helmholtz solutions using propagative plane waves and are subject to strong numerical instabilities. Evanescent plane wave bases can substantially mitigate this phenomenon. We propose a simple recipe to select such basis functions. We show that the numerical results obtained by the Ultraweak Variational Formulation (UWVF) greatly improve thanks to this choice. More details and examples will soon be available in [Galante, Moiola, Parolin 2026].
Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which allows us to obtain a description of a large class of uni-directional waves leading to the Ostrovsky equation and allowing for the presence of shear inertial oscillations and barotropic transport. Importantly, unlike the conventional derivations leading to the Ostrovsky equation, the constructed solutions do not impose the zero-mean constraint on the initial conditions for any variable in the problem formulation. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg-de Vries (KdV) equation but having a local amplitude and/or periodicity defect, and show that such initial conditions can lead to the emergence of bursts of large internal waves and shear currents. As a by-product of our study, we show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the
In this paper, we investigate the spectral stability of periodic traveling waves in the two dimensional gravity-capillary water wave problem. We derive a stability criterion based on an index function, whose sign determines the spectral stability of the waves. This result aligns with earlier formal analyses by Djordjević \& Redekopp [15] and Ablowitz \& Segur [1], which employed the nonlinear Schrödinger approximation in the modulational regime. In particular, we show that instability is excluded near spectral crossings away from the origin when the surface tension is positive and the inverse square of the Froude number $α\in(0,1),$ which results from the fact that the corresponding Krein signatures are identical. It is also shown that there exists $α_1 = (23 - 3\sqrt{41})/8$ and a curve $β: (α_1, 1]\rightarrow \mathbb{R}_{+},$ such that for any $α\in (α_1, 1]$, small amplitude periodic waves are spectrally stable when $β> β(α)$. These findings highlight the stabilizing effect of surface tension on periodic capillary-gravity waves.
The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper we additionally include linear ohmic resistors. Thus, the dissipation being taken into account leads to the existence of \mbox{shocks -- the} travelling waves with different asymptotically constant values of the voltage (the capacitor charge) and the current before and after the front of the wave. For the particular values of ohmic resistances (corresponding to strong dissipation) we obtain the analytic solution for the profile of a shock wave. Both the charge on a capacitor and current through the inductor are obtained as the functions of the time and space coordinate. In the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper we consider the nonlinear lossless transmission line. We formulate a simple wave approximation for such transmission line, which decouples left/right-going waves. The approximation can also be used for the lossy transmission line, which is considered in the first part of the paper, to describe the formation of the shock wave (but, of course, not the shock wave itsel
The solar wind upstream of Mars's bow shock can be described in terms of Alfvénic turbulence, with an incompressible energy cascade rate of $10^{-17}$ J m$^{-3}$ s$^{-1}$ at magnetohydrodynamics (MHD) scales. The solar wind has more Alfvén waves propagating outwards from the Sun (than inwards) and a median Alfvén ratio of $\sim0.33$. Newly ionized planetary protons associated with the extended hydrogen corona generate waves at the local proton cyclotron frequency. These 'proton cyclotron waves' (PCW) mostly correspond to fast magnetosonic waves, although the ion cyclotron (Alfvénic) wave mode is possible for large Interplanetary Magnetic Field cone angles. PCW do not show significant effects on the solar wind energy cascade rates at MHD scales but could affect smaller scales. The magnetosheath displays high amplitude wave activity, with high occurrence rate of Alfvén waves. Turbulence appears not fully developed in the magnetosheath, suggesting fluctuations do not have enough time to interact in this small-size region. Some studies suggest PCW affect turbulence in the magnetosheath. Overall, wave activity is reduced inside the magnetic pile-up region and the Martian ionosphere. How
The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations -- two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field -- are obtained exactly in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW's solitary wave edge amplitude and speed. The analytical results are favou
In dark-energy models where a scalar field is nonminimally coupled to the spacetime geometry, gravitational waves are expected to be supplemented with a scalar mode. Such scalar waves may interact with the standard tensor waves, thereby affecting their observed amplitude and polarization. Understanding the role of scalar waves is thus essential in order to design reliable gravitational-wave probes of dark energy and gravity beyond general relativity. In this article, we thoroughly investigate the propagation of scalar and tensor waves in the subset of Horndeski theories in which tensor waves propagate at the speed of light. We work at linear order in scalar and metric perturbations, in the eikonal regime, and for arbitrary scalar and spacetime backgrounds. We diagonalize the system of equations of motion and identify the physical tensor mode, which differs from the metric perturbation. We find that interactions between scalar and tensor waves generally depend on the scalar propagation speed. If the scalar waves are luminal or quasiluminal, then interactions are negligible. In the subluminal case, scalar-tensor interactions are effectively suppressed due to the incoherence of the wa
We investigate the gauge invariance of the second order gravitational waves induced by the first order scalar perturbations by following the Lie derivative method. It is shown explicitly that the second order gravitational waves are gauge invariant in the synchronous frame. In the gauge invariant framework, we derive the equation of motion of the second order gravitational waves and show that the second order gravitational waves are sourced from the first order scalar perturbations described well in the gauge invariant Newtonian frame. Since the observables of gravitational waves are measured in the synchronous frame, we define the energy density spectrum of the second order gravitational waves in terms of the gauge invariant synchronous variables. This way guarantees no fictitious tensor perturbations. It is shown that the gauge invariant energy density spectrum of the second order gravitational waves coincides with the one in the Newtonian gauge.
Solar flares are dramatic events in which magnetic reconnection in the corona leads to heating of plasma to tens of MK and acceleration of particles to high energies. They also centrally involve transport between the corona (where the magnetic reconnection occurs) and the lower solar atmosphere (where most energy is radiated from). There is substantial evidence for the presence of Alfvénic waves/turbulence in solar flares, for example, in the ubiquitous nonthermal broadening of flare spectral lines. The physical role that Alfvénic waves have in the flare has attracted considerable attention, especially since 2007-2010. This article reviews what spectroscopic observations reveal about the properties and importance of Alfvénic waves, turbulence and transport in solar flares; mechanisms for wave excitation by magnetic reconnection at high Lundquist numbers and braking of the sunward reconnection jet; and models of wave energy transport to the lower atmosphere and the resulting heating and dynamics. The article finishes with discussion of the outlook for new progress.
We consider a density-stratified fluid composed of two immiscible layers separated by a sharp interface. We study the regime of long internal waves interacting with modulated surface wave packets and describe their resonant interaction by a system of equations where the internal wave solves a high-order Benjamin-Ono (BO) equation coupled to a linear Schrödinger equation for the envelope of the free surface. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler's equations as described in Benjamin-Bridges [J. Fluid Mech. 333, 1997] and Craig-Guyenne-Kalisch [Comm. Pure Appl. Math. 58, 2005]. We also establish a local wellposedness result for a reduced BO-Schrödinger system using an approach developed by Linares-Ponce-Pilod [J. Diff. Eqs. 250, 2011].
In the first part of this paper (mainly a review) we present general and formal (simple) introductions to the ordinary gaussian waves and to the Bessel waves, by explicitly separating the cases of the beams from the cases of the pulses; and, finally, an analogous introduction is presented for the Localized Waves (LW), pulses or beams. Always we stress the very different characteristics of the gaussian with respect to the Bessel waves and to the LWs, showing the numerous and important properties of the latter w.r.t. the former ones: Properties that may find application in all fields in which an essential role is played by a wave-equation (like electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). In the second part of this paper (namely, in its Appendix), we recall at first how, in the seventies and eighties, the geometrical methods of Special Relativity (SR) had predicted --in the sense below specified-- the existence of the most interesting LWs, i.e., of the X-shaped pulses. At last, in connection with the circumstance that the X-shaped waves are endowed with Superluminal group-velocities (as carefully discussed in the first
We present a new method to study harmonic waves in the low ionosphere (60 - 90 km) by detecting their effects on reflection of very low frequency (VLF) radio waves. Our procedure is based on amplitude analysis of reflected VLF radio waves recorded in real time, which yields an insight into the dynamics of the ionosphere at heights where VLF radio waves are being reflected. The method was applied to perturbations induced by the solar terminator motions at sunrises and sunsets. The obtained results show that typical perturbation frequencies found to exist in higher regions of the atmosphere are also present in the lower ionosphere, which indicates a global nature of the considered oscillations. In our model atmosphere, they turn out to be the acoustic and gravity waves with comparatively short and long periods, respectively.
The existence of gravitational radiation is a natural prediction of any relativistic description of the gravitational interaction. In this chapter, we focus on gravitational waves, as predicted by Einstein's general theory of relativity. First, we introduce those mathematical concepts that are necessary to properly formulate the physical theory, such as the notions of manifold, vector, tensor, metric, connection and curvature. Second, we motivate, formulate and then discuss Einstein's equation, which relates the geometry of spacetime to its matter content. Gravitational waves are later introduced as solutions of the linearized Einstein equation around flat spacetime. These waves are shown to propagate at the speed of light and to possess two polarization states. Gravitational waves can interact with matter, allowing for their direct detection by means of laser interferometers. Finally, Einstein's quadrupole formulas are derived and used to show that nonspherical compact objects moving at relativistic speeds are powerful gravitational wave sources.
Using numerical modeling investigated interaction of solitary waves (solitons) of the regularized long wave equation. For reception the stable model of the nonlinear medium are used methods of the linear prediction and progressive approximation. By modeling was determined that depending on ratio of velocities of the solitons and the form of highest derivatives balance is possible self-organization of the medium nonequilibrium state as formation of shock waves and stable on the form solitary waves, created as a result of full or partial mutual penetration of the solitons. Is possible also aggregation of the solitons in third wave. The shock waves can pass into other possible resonance state as a wave front with stable amplitude, which precedes developing in singularity negative front.
A certain class of exact solutions of Einstein Maxwell spacetime in general relativity is discussed which demonstrates at the level of theory that, when certain parametric resonance condition is met, the interaction of electromagnetic field with a gravitational wave will display certain Liapounov instability and lead to exponential amplification of a gravitational wave train described by certain Newman-Penrose component of the Weyl curvature. In some way akin to a free electron laser in electromagnetic theory, by the conversion of electromagnetic energy into gravitational energy in a coherent way, the feasibility of generating a pulsed laser like intense beam of gravitational wave is displayed.
Heating of magnetized turbulent plasma is calculated in the framework of Burgers turbulence [A.M. Polyakov, Phys. Rev. E. 52, 6183 (1995)]. Explicit formula for the energy flux of Alfven waves along the magnetic field is presented. The Alfven waves are considered as intermediary between the turbulent energy and the heat. The derived results are related to a wave channel of heating of the solar corona. If we incorporate amplification of Alfven waves by shear flow the suggested model of heating can be applied to analysis of the missing viscosity of accretion discs and to reveal why the quasars are the most powerful sources of light in the universe. We suppose that the Langevin-Burgers approach to turbulence we have applied in the current work can be also helpful for other systems where we have intensive interaction between a stochastic turbulent system and waves and can be used in many multidisciplinary researches in hydrodynamics and MHD.
Plasma outflow from a gravitational potential well with cosmic rays and self-excited Alfvén waves with cooling and wave damping is studied in the hydrodynamics regime. We study outflows in the presence of cosmic ray and Alfvén waves including the effect of cooling and wave damping. We seek physically allowable steady-state subsonic-supersonic transonic solutions. We adopted a multi-fluid hydrodynamical model for the cosmic ray plasma system. Thermal plasma, cosmic rays, and self-excited Alfvén waves are treated as fluids. Interactions such as cosmic-ray streaming instability, cooling, and wave damping were fully taken into account. We considered one-dimensional geometry and explored steady-state solutions. The model is reduced to a set of ordinary differential equations, which we solved for subsonic-supersonic transonic solutions with given boundary conditions at the base of the gravitational potential well. We find that physically allowable subsonic-supersonic transonic solutions exist for a wide range of parameters. We studied the three-fluid system (considering only forward-propagating Alfvén waves) in detail. We examined the cases with and without cosmic ray diffusion separatel