Two-dimensional water waves in the presence of a freely floating body: trapped modes and conditions for their absence

2015 ◽  
Vol 779 ◽  
pp. 684-700 ◽  
Author(s):  
Nikolay Kuznetsov
Keyword(s):  
Water Waves ◽  
Small Amplitude ◽  
Floating Body ◽  
Two Dimensional ◽  
Trapped Modes ◽  
Constant Depth ◽  
Time Harmonic ◽  
Equipartition Of Energy ◽  
Linear Setting ◽  
Inverse Procedure

The coupled motion is investigated for a mechanical system consisting of water and a body freely floating in it. Water occupies either a half-space or a layer of constant depth into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes. Under the assumption that the motion is of small amplitude near equilibrium, a linear setting is applicable, and for the time-harmonic oscillations it reduces to a spectral problem with the frequency of oscillations as the spectral parameter. Within this framework, it is shown that the total energy of the water motion is finite and the equipartition of energy holds for the whole system. On this basis two results are obtained. First, the so-called semi-inverse procedure is applied for the construction of a family of two-dimensional bodies trapping the heave mode. Second, it is proved that no wave modes can be trapped provided that their frequencies exceed a bound depending on the cylinder properties, whereas its geometry is subject to some restrictions and, in some cases, certain restrictions are imposed on the type of mode.

On the coupled time-harmonic motion of deep water and a freely floating body: trapped modes and uniqueness theorems

2012 ◽  
Vol 703 ◽  
pp. 142-162 ◽  
Author(s):  
Nikolay Kuznetsov ◽  
Oleg Motygin
Keyword(s):  
Small Amplitude ◽  
Velocity Potential ◽  
The Body ◽  
Floating Body ◽  
Trapped Modes ◽  
Dimensionless Quantity ◽  
Time Harmonic ◽  
Equipartition Of Energy ◽  
Zero Vector ◽  
Inverse Procedure

AbstractWe investigate the time-harmonic small-amplitude motion of the mechanical system that consists of water and a body freely floating in it; water occupies a half-space, whereas the body is either surface-piercing or totally submerged. As a mathematical model of this coupled motion, we consider a spectral problem (the spectral parameter is the frequency of oscillations), for which the following results are obtained. The total energy of the water motion is finite and the equipartition of energy holds for the whole system. For any value of frequency, infinitely many eigensolutions are constructed and each of them consists of a non-trivial velocity potential and the zero vector describing the motion of the body; the latter means that trapping bodies (infinitely many of them are found) are motionless although they float freely. They are surface-piercing, have axisymmetric submerged parts and are obtained by virtue of the so-called semi-inverse procedure. We also prove that certain restrictions on the body geometry (which are violated for the constructed trapping bodies) guarantee that the problem has only a trivial solution for frequencies that are sufficiently large being measured in terms of a certain dimensionless quantity.

On the coupled time-harmonic motion of a freely floating body and water covered by brash ice

2016 ◽  
Vol 795 ◽  
pp. 174-186 ◽  
Author(s):  
Nikolay Kuznetsov ◽  
Oleg Motygin
Keyword(s):  
Small Amplitude ◽  
Finite Interval ◽  
Finite Energy ◽  
Floating Body ◽  
Coupled Motion ◽  
Time Harmonic ◽  
All Solutions ◽  
Equipartition Of Energy ◽  
Linear Setting ◽  
Inverse Procedure

A mechanical system consisting of water covered by brash ice and a body freely floating near equilibrium is considered. The water occupies a half-space into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes of the coupled motion, which is assumed to be of small amplitude. The corresponding linear setting for time-harmonic oscillations reduces to a spectral problem whose parameter is the frequency. A constant that characterises the brash ice divides the set of frequencies into two subsets and the results obtained for each of these subsets are essentially different. For frequencies belonging to a finite interval adjacent to zero, the total energy of motion is finite and the equipartition of energy holds for the whole system. For every frequency from this interval, a family of motionless bodies trapping waves is constructed by virtue of the semi-inverse procedure. For sufficiently large frequencies outside of this interval, all solutions of finite energy are trivial.

On the coupled time-harmonic motion of water and a body freely floating in it

2011 ◽  
Vol 679 ◽  
pp. 616-627 ◽  
Author(s):  
NIKOLAY KUZNETSOV ◽  
OLEG MOTYGIN
Keyword(s):  
Boundary Value Problem ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Water Layer ◽  
The Body ◽  
Body Motion ◽  
Constant Depth ◽  
Harmonic Motion ◽  
Time Harmonic ◽  
Equipartition Of Energy

We consider a spectral problem that describes the time-harmonic small-amplitude motion of the mechanical system that consists of a three-dimensional water layer of constant depth and a body (either surface-piercing or totally submerged), freely floating in it. This coupled boundary-value problem contains a spectral parameter – the frequency of oscillations – in the boundary conditions as well as in the equations governing the body motion. It is proved that the total energy of the water motion is finite and the equipartition of energy of the whole system is established. Under certain restrictions on body's geometry the problem is proved to have only a trivial solution for sufficiently large values of the frequency. The uniqueness frequencies are estimated from below.

scholarly journals Trapped modes around freely floating bodies in a two-layer fluid channel

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140396 ◽  
Author(s):  
Filipe S. Cal ◽  
Gonçalo A. S. Dias ◽  
Juha H. Videman
Keyword(s):  
Spectral Problem ◽  
Water Waves ◽  
Adjoint Operator ◽  
Trapped Modes ◽  
Floating Bodies ◽  
Floating Structures ◽  
Nonlinear Spectral Problem ◽  
Time Harmonic ◽  
Fluid Channel ◽  
Elimination Scheme

Unlike the trapping of time-harmonic water waves by fixed obstacles, the oscillation of freely floating structures gives rise to a complex nonlinear spectral problem. Still, through a convenient elimination scheme the system simplifies to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we present conditions guaranteeing the existence of trapped modes in a two-layer fluid channel. Numerous examples of floating bodies supporting trapped modes are given.

Note on the scattering of water waves by a vertical barrier

1966 ◽  
Vol 62 (3) ◽  
pp. 507-509 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
Alternative Approach

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

scholarly journals On the Benjamin–Lighthill conjecture for water waves with vorticity

2017 ◽  
Vol 825 ◽  
pp. 961-1001 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Water Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Constant Depth ◽  
Lipschitz Continuous ◽  
Vorticity Distribution ◽  
Inviscid Incompressible Fluid ◽  
Two Dimensional Flow ◽  
Wave Heights ◽  
Gravity Driven

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

Approximations to wave trapping by topography

1996 ◽  
Vol 325 ◽  
pp. 357-376 ◽  
Author(s):  
P. G. Chamberlain ◽  
D. Porter
Keyword(s):  
Water Waves ◽  
Two Dimensional ◽  
Trapped Modes ◽  
Wave Trapping ◽  
Mild Slope Equation ◽  
Precise Shape ◽  
Different Depths ◽  
Linear Water Waves ◽  
Local Irregularity

The trapping of linear water waves over two-dimensional topography is investigated by using the mild-slope approximation. Two types of bed profile are considered: a local irregularity in a horizontal bed and a shelf joining two horizontal bed sections at different depths. A number of results are derived concerning the existence of trapped modes and their multiplicity. It is found, for example, that the maximum number of modes which can exist depends only on the gross properties of the topography and not on its precise shape. A range of problems is solved numerically, to inform and illustrate the analysis, using both the mild-slope equation and the recently derived modified mild-slope equation.

scholarly journals On periodic geophysical water flows with discontinuous vorticity in the equatorial f -plane approximation

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170096 ◽  
Author(s):  
Calin Iulian Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Wave Speed ◽  
Laminar Flows ◽  
Local Bifurcation ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Water Flows

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f -plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ 1 adjacent to the surface situated above another layer of constant non-zero vorticity γ 2 ≠ γ 1 adjacent to the bed. For certain vorticities γ 1 , γ 2 , we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Fundamental singularities in the theory of water waves with surface tension

1970 ◽  
Vol 2 (3) ◽  
pp. 317-333 ◽  
Author(s):  
P. F. Rhodes-Robinson
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Water Waves ◽  
Potential Functions ◽  
Fundamental Wave ◽  
Harmonic Waves ◽  
Constant Depth ◽  
Wave Source ◽  
Time Harmonic ◽  
Effect Of Surface

In this paper the forms are obtained for the harmonic potential functions describing the fundamental wave-source and multipole singularities which pertain to the study of infinitesimal time-harmonic waves on the free surface of water when the effect of surface tension is included. Line and point singularities are considered for both the cases of infinite and finite constant depth of water. The method used is an extension of that which has been used to obtain these potentials in the absence of surface tension.

scholarly journals No steady water waves of small amplitude are supported by a shear flow with a still free surface

2013 ◽  
Vol 717 ◽  
pp. 523-534 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Finite Depth ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Positive Coefficients

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

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