On an invariant property of water waves

1971 ◽  
Vol 49 (2) ◽  
pp. 385-389 ◽  
Author(s):  
T. Brooke Benjamin ◽  
J. J. Mahony
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Horizontal Plane ◽  
Wave Motion ◽  
Surface Displacement ◽  
Heavy Liquid ◽  
Invariant Property ◽  
Finite Region ◽  
Free Wave ◽  
Free Surface Displacement

The discussion concerns free wave motions generated from rest in a finite region of an ocean of heavy liquid lying on a horizontal plane. It is shown that the horizontal fist moment of the free-surface displacement varies linearly with time. Hence, if the total volume displaced is not zero and therefore the centroid of the displacement is definable, the centroid travels with a constant horizontal velocity as the wave motion evolves. This conclusion holds exactly for waves of any amplitude and even remains applicable subsequent to the breaking of waves.

Nonlinear Helmholtz oscillations in harbours and coupled basins

1981 ◽  
Vol 104 ◽  
pp. 407-418 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Free Surface ◽  
Additional Assumption ◽  
Phase Plane ◽  
Surface Displacement ◽  
Elliptic Functions ◽  
Forced Oscillations ◽  
Amplitude Parameter ◽  
Spatial Uniformity ◽  
Open Sea ◽  
Free Surface Displacement

Free and forced oscillations in a basin that is connected through a narrow canal to either the open sea or a second basin are considered on the assumption that the spatial variation of the free-surface displacement is negligible. The free-surface displacement in the canal is allowed to be finite, subject only to the restriction (in addition to that implicit in the approximation of spatial uniformity) that the canal does not run dry. The resulting model yields a Hamiltonian pair of phase-plane equations for the free oscillations, which are integrated in terms of elliptic functions on the additional assumption that the kinetic energy of the motion in the basin(s) is negligible compared with that in the canal or otherwise through an expansion in an amplitude parameter. The corresponding model for forced oscillations that are limited by radiation damping yields a generalization of Duffing's equation for an oscillator with a soft spring, the solution of which is obtained as an expansion in the amplitude of the fundamental term in a Fourier expansion. Equivalent circuits are developed for the various models.

Oscillatory Free Surface Displacement of Finite Amplitude in a Small Orifice

2004 ◽  
Vol 126 (5) ◽  
pp. 818-826
Author(s):  
Brian J. Daniels ◽  
James A. Liburdy
Keyword(s):  
Free Surface ◽  
Potential Flow ◽  
Surface Displacement ◽  
Experimental Results ◽  
Significant Shift ◽  
Curvature Effect ◽  
Modal Frequency ◽  
Large Curvature ◽  
Curvature Effects ◽  
Free Surface Displacement

The oscillatory free-surface displacement in an orifice periodically driven at the inlet is studied. The predictions based on a potential flow analysis are investigated in light of viscous and large curvature effects. Viscous effects near the wall are estimated, as are surface viscous energy loss rates. The curvature effect on the modal frequency is shown to become large at the higher modal surface shapes. Experimental results are obtained using water for two orifice diameters, 794 and 1180 μm. Results of surface shapes and modal frequencies are compared to the predictions. Although modal shapes seem to be well predicted by the theory, the experimental results show a significant shift of the associated modal frequencies. A higher-order approximation of the surface curvature is presented, which shows that the modal frequency should, in fact, be reduced from potential flow predictions as is consistent with the large curvature effect. To account for the effect of finite surface displacements an empirical correlation for the modal frequencies is presented.

scholarly journals Three dimensional numerical simulations for non-breaking solitary wave interacting with a group of slender vertical cylinders

2009 ◽  
Vol 1 (1) ◽  
Author(s):  
Weihua Mo ◽  
Philip L.-F. Liu
Keyword(s):  
Free Surface ◽  
Numerical Model ◽  
Dynamic Pressure ◽  
Laboratory Data ◽  
Three Dimensional ◽  
Time History ◽  
Surface Displacement ◽  
Numerical Results ◽  
Finite Volume Scheme ◽  
Free Surface Displacement

AbstractIn this paper we validate a numerical model for-structure interaction by comparing numerical results with laboratory data. The numerical model is based on the Navier-Stokes(N-S) equations for an incompressible fluid. The N-S equations are solved by two-step projection finite volume scheme and the free surface displacements are tracked by the slender vertical piles. Numerical results are compared with the laboratory data and very good agreement is observed for the time history of free surface displacement, fluid particle velocity and force. The agreement for dynamic pressure on the cylinder is less satisfactory, which is primarily caused by instrument errors.

Mass transport in two-dimensional water waves

1991 ◽  
Vol 231 ◽  
pp. 395-415 ◽  
Author(s):  
Mohamed Iskandarani ◽  
Philip L.-F. Liu
Keyword(s):  
Mass Transport ◽  
Water Waves ◽  
Surface Displacement ◽  
Flow Region ◽  
Two Dimensional ◽  
Governing Equations ◽  
Free Surface Displacement ◽  
Inhomogeneous Flow ◽  
Flow Domain ◽  
Transport Flow

Mass transport in various kind of two-dimensional water waves is studied. The characteristics of the governing equations for the mass transport depend on the ratio of viscous lengthscale to the amplitude of the free-surface displacement. When this ratio is small, the nonlinearity is important and the mass transport flow acquires a boundary-layer character. Numerical schemes are developed to investigate mass transport in a partially reflected wave and above a hump in the seabed. When the mass transport is periodic in the horizontal direction, a spectral scheme based on a Fourier–Chebyshev expansion, is presented for the solution of the equations. For the ease of a hump on the seabed, the flow domain is divided into three regions. Using the spectral scheme, the mass transport in the uniform-depth regions is calculated first. and the results are used to compute the steady flow in the inhomogeneous flow region which encloses the hump on the seabed.

scholarly journals THE THEORY OF THE REFRACTION OF A SHORT CRESTED GAUSSIAN SEA SURFACE WITH APPLICATION TO THE NORTHERN NEW JERSEY COAST

2000 ◽  
Vol 1 (3) ◽  
pp. 8
Author(s):  
Willard J. Pierson, Jr. ◽  
John J. Tuttell ◽  
John A. Woolley
Keyword(s):  
Free Surface ◽  
New Jersey ◽  
Horizontal Plane ◽  
Wave Motion ◽  
Vertical Section ◽  
Sea Surface ◽  
Progressive Wave

The Thorndike Barnhart Dictionary (1951) defines a wave as a "moving ridge or swell of water." Almost everyone will agree to this definition. Milne-Thompson (1938) in Theoretical Hydrodynamics begins Chapter Fourteen on waves with the two paragraphs quoted in full below: "14.10 Wave motion. A wave motion of a liquid acted upon by gravity and having a free surface is a motion in which the elevation of the free surface above some chosen fixed horizontal plane varies with time. Taking the axis of x to be horizontal and the axis of z to be vertically upwards, a motion in which the vertical section of the free surface at time t is of the form z = a sin(mx - nt) (1), where a, m, n are constants, is called a simple harmonic progressive wave."

Equilibrium and oscillations of liquid fuel free surface in coaxial-cylindrical vessels under microgravity conditions

2021 ◽  
Author(s):  
Y. Zhaokai ◽  
A.N. Temnov
Keyword(s):  
Contact Angle ◽  
Free Surface ◽  
Liquid Fuel ◽  
Outer Space ◽  
Surface Displacement ◽  
Ideal Liquid ◽  
Intermolecular Forces ◽  
Hydrodynamic Characteristics ◽  
Free Surface Displacement ◽  
Microgravity Conditions

In the absence of significant mass forces, the behavior of liquid fuel under microgravity conditions is determined by surface tension forces, which are intermolecular forces at the interface of two phases. The paper posed and solved the problem of equilibrium and small oscillations of an ideal liquid under microgravity conditions, and also quantified the influence of various parameters: the contact angle α0, the Bond number, the ratio of the radii of the inner and outer walls of the vessel and the depth of the liquid. For the coaxial-cylindrical vessels, there were obtained expressions in the form of a Bessel series for the potential of the fluid velocities and the free surface displacement field. The study relies on the analytical and experimental data available in the literature and proves the reliability of the developed numerical algorithm. Findings of research show that for and r, with the physical state of the wetted surface being unchanged, the shape of the free surface tends to be flat and the contact angle has little effect on the intrinsic vibration frequency of the free surface of the liquid. The results obtained can be used to solve problems of determining the hydrodynamic characteristics of the movement of liquid fuel in outer space.

Strongly coherent dynamics of stochastic waves causes abnormal sea states

2020 ◽  
Author(s):  
Alexey Slunyaev
Keyword(s):  
Numerical Simulations ◽  
Water Waves ◽  
Spectral Decomposition ◽  
Stochastic Dynamics ◽  
Surface Displacement ◽  
Sea Waves ◽  
Geophysical Research ◽  
Free Wave ◽  
Coherent Dynamics ◽  
Stochastic Waves

<p>The dynamic kurtosis (i.e., produced by the free wave component) is shown to contribute essentially to the abnormally large values of the full kurtosis of the surface displacement, according to the direct numerical simulations of realistic directional sea waves within the HOSM framework. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic approach is inapplicable. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves. They strongly violate the classic dispersion relation and hence lead to a greater spread of the actual wave frequencies for given wavenumbers.</p><p>The research by is supported by the RSF grant No. 16-17-00041.</p><p>Slunyaev, A. Kokorina, The method of spectral decomposition into free and bound wave components. Numerical simulations of the 3D sea wave states. Geophysical Research Abstracts, V. 21, EGU2019-546 (2019).</p><p>A.V. Slunyaev, A.V. Kokorina, Spectral decomposition of simulated sea waves into free and bound wave components. Proc. VII Int. Conf. “Frontiers of Nonlinear Physics”, 189-190 (2019).</p><p>Slunyaev, A. Kokorina, I. Didenkulova, Statistics of free and bound components of deep-water waves. Proc. 14th Int. MEDCOAST Congress on Coastal and Marine Sciences, Engineering, Management and Conservation (Ed. E. Ozhan), Vol. 2, 775-786 (2019).</p><p>Slunyaev, Strongly coherent dynamics of stochastic waves causes abnormal sea states. arXiv: 1911.11532 (2019).</p>

On free-surface oscillations in a rotating paraboloid

1963 ◽  
Vol 17 (2) ◽  
pp. 257-266 ◽  
Author(s):  
John W. Miles ◽  
F. K. Ball
Keyword(s):  
Free Surface ◽  
Second Harmonic ◽  
Harmonic Component ◽  
Surface Displacement ◽  
Wave Number ◽  
Rigid Displacement ◽  
Axisymmetric Solution ◽  
Axisymmetric Mode ◽  
Azimuthal Wave ◽  
Free Surface Displacement

Lamb's analysis of small-amplitude, shallow-water oscillations in a rotating paraboloid, interpreted by him in the inconsistent context of an approximately plane free surface, is re-interpreted to obtain results that are valid for $0 \le \omega^2l|2g \; \textless \;1$ (ω = rotational speed, l = latus rectum of paraboloid); no equilibrium is possible for ω2l/2g > 1. It is shown that the frequencies of the dominant modes for the azimuthal wave numbers 0 (axisymmetric motion) and 1 are independent of ω for an observer in a non-rotating reference frame and that the frequencies of all other axisymmetric modes are decreased by rotation (Lamb concluded that they would be increased). An axisymmetric mode of zero frequency, which was over-looked by Lamb, also is found.Exact solutions to the non-linear equations of motion, which reduce to the aforementioned dominant modes for small amplitudes, are determined. The axisymmetric solution is inferred from similarity considerations and is found to contain all harmonics of the fundamental frequency. The finite motion of azimuthal wave-number 1 is a quasi-rigid displacement of the liquid and is found to be simple harmonic except for a second-harmonic component of the free-surface displacement (but the horizontal velocity at a given point remains simple harmonic).

BEM Modeling of Viscous Motion of Surface Water Waves

2004 ◽  
Author(s):  
Mirmosadegh Jamali
Keyword(s):  
Numerical Modeling ◽  
Free Surface ◽  
Surface Wave ◽  
Water Waves ◽  
Wave Motion ◽  
Equations Of Motion ◽  
Solid Surfaces ◽  
Finite Difference Technique ◽  
Surface Water Waves ◽  
Motion Characteristics

This study is concerned with numerical modeling of viscous surface wave motion using boundary element method (BEM). The equations of motion for thin boundary layers at the solid surfaces are coupled with the potential flow in the bulk of the fluid, and a mixed BEM-finite difference technique is used to obtain the surface wave motion characteristics including the decay rate. The technique is presented for a standing surface wave motion. The extension to other free surface problems is discussed.

scholarly journals Weak formulation of free-surface wave equations

2001 ◽  
Vol 5 (2) ◽  
pp. 75-85
Author(s):  
A. D. Sneyd
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Surface Displacement ◽  
Wave Dispersion ◽  
The Body ◽  
Weak Formulation ◽  
Free Surface Displacement ◽  
Free Surface Wave

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, η=∑n=1∞αn(t)En where the αn(t) are time-dependent coefficients. For a tank with vertical sides the En are eigenfunctions of the eigenvalue problem, ∇2+λ2E=0,  ∇E⋅n^=0 on the tank side walls. Evolution equations for the αn(t) can be obtained by taking inner products of the linearised equation of motion, ρ∂v∂t=−1ρ∇P+F with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.

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