On collinear steady-state gravity waves with an infinite number of exact resonances

2019 ◽  
Vol 31 (12) ◽  
pp. 122109
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Infinite Number

On the steady-state resonant acoustic–gravity waves

2018 ◽  
Vol 849 ◽  
pp. 111-135 ◽  
Author(s):  
Xiaoyan Yang ◽  
Frederic Dias ◽  
Shijun Liao
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Infinite Number ◽  
Dynamic Pressure ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Analytic Approximation ◽  
Wave Component ◽  
Small Denominators ◽  
Acoustic Gravity Waves

The steady-state interaction of acoustic–gravity waves in an ocean of uniform depth is researched theoretically by means of the homotopy analysis method (HAM), an analytic approximation method for nonlinear problems. Considering compressibility, a hydroacoustic wave can be produced by the interaction of two progressive gravity waves with the same wavelength travelling in opposite directions, which contains an infinite number of small denominators in the framework of the classical analytic approximation methods, like perturbation methods. Using the HAM, the infinite number of small denominators are avoided once and for all by means of choosing a proper auxiliary linear operator. Besides, by choosing a proper ‘convergence-control parameter’, convergent series solutions of the steady-state acoustic–gravity waves are obtained in cases of both non-resonance and exact resonance. It is found, for the first time, that the steady-state resonant acoustic–gravity waves widely exist. In addition, the two primary wave components and the resonant hydroacoustic wave component might occupy most of wave energy. It is found that the dynamic pressure on the sea bottom caused by the resonant hydroacoustic wave component is much larger than that in the case of non-resonance, which might even trigger microseisms of the ocean floor. All of these might deepen our understanding and enrich our knowledge of acoustic–gravity waves.

scholarly journals Symmetry analysis of rotating fluid

2005 ◽  
Vol 47 (1) ◽  
pp. 65-74 ◽  
Author(s):  
K. Fakhar ◽  
Zu-Chi Chen ◽  
Xiaoda Ji
Keyword(s):  
Steady State ◽  
Infinite Number ◽  
Special Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Lie Theory ◽  
Rotating Fluid ◽  
Type Solution ◽  
Function Type ◽  
Basic Equations

AbstractThe machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

scholarly journals Generalized function treatment of the Alfvén-gravity wave problem

1983 ◽  
Vol 6 (2) ◽  
pp. 395-402
Author(s):  
L. Debnath ◽  
K. Vajravelu
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Function Method ◽  
Radiation Condition ◽  
Physical Significance ◽  
Generalized Function ◽  
Wave Problem ◽  
Physical Interest ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid

A study is made of the steady-state Alfvén-gravity waves in an inviscid incompressible electrically conducting fluid with an interface due to a harmonically oscillating pressure distribution acting on the interface. The generalized function method is employed to solve the problem in the fluid of infinite, finite and shallow depth. A unique solution of physical interest is derived by imposing the Sommerfeld radiation condition at infinity. Several limiting cases of physical interest are obtained from the present analysis. The physical significance of the solutions and their limiting cases are discussed.

Ideal planar fluid flow over a submerged obstacle: Review and extension

2021 ◽  
Vol 63 ◽  
pp. 377-419
Author(s):  
Larry K. Forbes ◽  
Stephen J. Walters ◽  
Graeme C. Hocking
Keyword(s):  
Fluid Flow ◽  
Free Surface ◽  
Steady State ◽  
Gravity Waves ◽  
Classical Problem ◽  
Time Dependent ◽  
Time Dependent Behaviour ◽  
Critical Overview ◽  
Dependent Behaviour ◽  
Obstacle Height

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour. doi:10.1017/S1446181121000341

Hopf Bifurcation in the Double-Glazing Problem With Conducting Boundaries

1987 ◽  
Vol 109 (4) ◽  
pp. 894-898 ◽  
Author(s):  
K. H. Winters
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Gravity Waves ◽  
Traveling Waves ◽  
Critical Rayleigh Number ◽  
Boussinesq Approximation ◽  
Time Dependent ◽  
Oscillatory Convection ◽  
Extended System ◽  
State Equations

Oscillatory convection has been observed in recent experiments in a square, air-filled cavity with differentially heated sidewalls and conducting horizontal surfaces. We show that the onset of the oscillatory convection occurs at a Hopf bifurcation in the steady-state equations for free convection in the Boussinesq approximation. The location of the bifurcation point is found by solving an extended system of steady-state equations. The predicted critical Rayleigh number and frequency at the onset of oscillations are in excellent agreement with the values measured recently and with those of a time-dependent simulation. Four other Hopf bifurcation points are found near the critical point and their presence supports a conjectured resonance between traveling waves in the boundary layers and interior gravity waves in the stratified core.

A resonant test-field model of gravity waves

1983 ◽  
Vol 132 ◽  
pp. 417-430 ◽  
Author(s):  
Bruce J. West
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Planck Equation ◽  
Steady State Solution ◽  
Test Field ◽  
Statistical Fluctuations ◽  
Resonant Interactions ◽  
Energy Spectral Density ◽  
Non Gaussian ◽  
Turbulent Wind Field

In this paper we propose an ‘irreversible’ resonant test-field (RTF) model to describe the statistical fluctuations of gravity waves on deep water driven by a turbulent wind field. The non-resonant interactions in the gravity-wave Hamiltonian are replaced by a Markov process in the equation of motion for the resonantly interacting gravity waves, i.e. Hamilton's equations are replaced by a Langevin equation for the RTF waves. The RTF models the irreversible energy-transfer process by a Fokker-Planck equation for the phase-space probability density, the exact steady-state solution of which is determined to be non-Gaussian. An H-theorem for the RTF predicts the monotonic approach to the asymptotic steady state near which the transport properties of the field are studied. The steady-state energy-spectral density is calculated (in some approximation) to be k−4.

A non-linear investigation of critical levels for internal atmospheric gravity waves

1971 ◽  
Vol 50 (3) ◽  
pp. 545-563 ◽  
Author(s):  
R. J. Breeding
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
Gravity Waves ◽  
Incident Wave ◽  
Critical Level ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Non Linear ◽  
Energy And Momentum ◽  
Almost All

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible. For Richardson numbers greater than 2·0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory–very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2·0 and greater than 0·25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0·25, over-reflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.

Steady-state resonance of multiple wave interactions in deep water

2014 ◽  
Vol 742 ◽  
pp. 664-700 ◽  
Author(s):  
Zeng Liu ◽  
Shi-Jun Liao
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Gravity Waves ◽  
Deep Water ◽  
Wave Energy ◽  
Two Dimensions ◽  
Physical Parameters ◽  
Wave System ◽  
State Resonance ◽  
Resonant Waves

AbstractThe steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

A general method for solving steady-state internal gravity wave problems

1972 ◽  
Vol 56 (4) ◽  
pp. 721-740 ◽  
Author(s):  
D. G. Hurley
Keyword(s):  
Steady State ◽  
Circular Cylinder ◽  
Gravity Waves ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Two Dimensional ◽  
Wave Problem ◽  
Wave Problems ◽  
General Method ◽  
Stratified Liquid

The paper describes a simple but general method for solving 'steady-state’ problems involving internal gravity waves in a stably stratified liquid. Under the assumption that the motion is two-dimensional and that the Brunt-Väisälä frequency is constant, the method is used to re-derive in a very simple way the solutions to problems where the boundary of the liquid is either a wedge or a circular cylinder. The method is then used to investigate the effect that a model of the continental shelf has on an incident train of internal gravity waves. The method involves analytic continuation in the frequency of the disturbance, and may well prove to be effective for other types of wave problem.

Growth and equilibrium of short gravity waves in a wind-wave tank

1977 ◽  
Vol 82 (4) ◽  
pp. 767-793 ◽  
Author(s):  
W. J. Plant ◽  
J. W. Wright
Keyword(s):  
Steady State ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Growth Rates ◽  
Energy Input ◽  
Wind Wave ◽  
Initial Growth ◽  
Wave Tank ◽  
Wind Speeds ◽  
Dominant Wave

Temporal and spatial development of short gravity waves in a linear wind-wave tank has been measured for wind speeds up to 15 m/s using microwave Doppler spectrometry. Surface waves of wavelength 4·1 cm, 9·8 cm, 16·5 cm and 36 cm were observed as a function of fetch, wind speed and wind duration. The waves grew exponentially from inception until they were about 10 dB smaller than their maximum height, and the temporal growth and spectral transport (spatial growth) rates were about equal when the wave amplitude was sufficiently small. The amplitude of a short gravity wave of fixed wavelength was found to decrease substantially at winds, fetches or durations greater than those at which the short gravity wave was approximately the dominant wave; such phenomena are sometimes referred to as overshoot. The dominant short gravity wave was observed to reach a maximum amplitude which depended only on wavelength, showing that wave breaking induced by an augmented wind drift cannot be the primary limitation to the wave height. Waves travelling against the wind were observed for wavelengths of 9·8 cm, 16·5 cm and 36 cm and were shown to be generated by the air flow at low wind speeds.Measured initial growth rates for 16·5 cm and 36 cm waves were greater than expected, suggesting the existence of a growth mechanism in addition to direct transfer from the wind via linear instability of the boundary-layer flow. Initial temporal growth rates and spectral transport rates were compared to yield an experimental determination of the magnitude of the sum of nonlinear interactions and dissipation in short gravity waves. If the steady-state energy input in the neighbourhood of the dominant wave occurs at the measured initial temporal growth rates, then most of the energy input is locally dissipated; relatively little is advected away. Calculated gravitycapillary nonlinear energy transfer rates match those determined from initial growth rates for 9·8 cm waves and the gravity–capillary wave interaction continues to be significant for waves as long as 16·5 cm. For longer waves the gravity–capillary interaction is too small to bring the short gravity wave to a steady state when it is the dominant wave of the wind-wave system.

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