Interaction of Surface Waves in a Basin with Floating Broken Ice

2009

J Appl Mech Tech Phy

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Equations for three nonlinear approximations of a wave perturbation in a homogeneous ideal incompressible fluid covered by a thin elastic plate are obtained using the method of multiple scales and taking into account that the acceleration of vertical flexural displacements of the plate is nonlinear. Based on the obtained equations, asymptotic expansions up third-order terms are constructed for the fluid velocity potential and the perturbations of the plate-fluid interface (plate bending) caused by a traveling periodic wave of finite amplitude. The wave characteristics are analyzed as functions of the elastic modulus and thickness of the plate and the length and tilt of the initial fundamental harmonic wave.Introduction. Wave perturbations in a fluid with a floating elastic plate in a linear formulation were studied in [1][2][3][4][5][6][7]. Finite-amplitude waves in a homogeneous fluid with an elastic ice plate neglecting the nonlinearity of vertical displacements of the plate were investigated in [8,9]. In [10], the characteristics of progressive surface finite-amplitude waves were analyzed as functions of the thickness and nonlinearity of vertical vibrations of an absolutely flexible plate (broken ice). In the present work, the effect of a floating elastic plate on the propagation of periodic finite-amplitude waves is studied using the method of multiscale asymptotic expansion and taking into account that the acceleration of vertical flexural displacements of the plate is nonlinear.Formulation of the Problem. Let a thin elastic plate float on the surface of a homogeneous ideal incompressible fluid filling an unbounded pool of constant depth H. Assuming that the fluid motion is potential and the plate performs vibrations without separation, we examine the effect of the elasticity and thickness of the plate on the propagation of finite-amplitude flexural-gravity waves. In the dimensionless variables x = kx 1 , z = kz 1 , and t = √ kg t 1 (k is the wavenumber), the problem reduces to solving the Laplace equation

Section: Analysis Ofmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Finite-amplitude waves in a homogeneous fluid with a floating elastic plate

2009

J Appl Mech Tech Phy

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“…The velocity of the translational motion of liquid in the direction of propagation of waves of finite amplitude predicted by the Stokes theory [2] was analyzed in [3][4][5] and [6][7][8] for basins with free surface of infinite and finite depths, respectively. In [9], the same problem was solved for water with floating broken ice without quantitative analysis of the distributions of components of the velocity of liquid particles over the wavelength. The dependences of the components of the orbital velocity of motion of particles of homogeneous liquid with open surface on the wave number and steepness of running periodic waves with finite amplitude were investigated in [10].…”

Section: Introductionmentioning

confidence: 98%

Velocities of motion of liquid particles under the floating ice cover in the case of propagation of periodic waves of finite amplitude

2011

Phys Oceanogr

By using the velocity potential obtained by the method of multiscale asymptotic expansions to within the quantities of the third order of smallness, we study the dependences of the components of the velocity of motion of a homogeneous liquid under the floating ice cover on the thickness of the cover and its modulus of elasticity in the process of propagation of periodic waves of finite amplitude. It is shown that the presence of broken ice leads to a decrease in the moduli of components of the velocity of liquid particles and the phase delay of generated oscillations. The effect of the elasticity of ice becomes more pronounced as the wavelength of the initial harmonic decreases and manifests itself in the increase in the maximum values of the components of velocity and in the phase shift of oscillations in the direction of propagation of waves.

“…1b. Figures 2a and 2bshows the isolines of k 1 and k 2 on the plane H, Q 1 for h = 0.plate is absolutely flexible ((Å = 0, Q = 0), then solution (3.7) for a fluid of finite depth is valid for any values of the wave number[9]. a bFig.…”

mentioning

confidence: 96%

“…The oscillations of a floating elastic plate were studied in a linear formulation in [2,7,8,11,14] disregarding the compressive force and in [1,6,10,[12][13][14] taking the compressive force into account. The nonlinear oscillations of a floating absolutely flexible plate were studied in [9]. The finite-amplitude bending oscillations of a floating elastic plate were analyzed in [3,4] disregarding the fact that the acceleration of vertical displacement is nonlinear.…”

mentioning

confidence: 99%

Nonlinear oscillations of a floating elastic plate

2011

Int Appl Mech

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The multi-scales method is used to derive third-order equations of gravitational bending oscillations of a thin elastic plate floating on the surface of a homogeneous perfect incompressible fluid of finite depth. The equations incorporate the compressive force and nonlinear acceleration of vertical displacements of the plate. Based on these equations, the deflection of the plate and the velocity potential of the fluid induced by a traveling periodic wave of finite amplitude are expanded into asymptotic series to terms of the third order of smallness. The dependence of the oscillation characteristics on the elastic modulus and thickness of the plate, compressive force, the initial length and steepness of the wave is analyzed Keywords: thin elastic plate, fluid of finite depth, gravitational bending oscillations, initial wave length, wave steepnessIntroduction. The oscillations of a floating elastic plate were studied in a linear formulation in [2,7,8,11,14] disregarding the compressive force and in [1,6,10,[12][13][14] taking the compressive force into account. The nonlinear oscillations of a floating absolutely flexible plate were studied in [9]. The finite-amplitude bending oscillations of a floating elastic plate were analyzed in [3,4] disregarding the fact that the acceleration of vertical displacement is nonlinear.We will study the oscillations of a longitudinally compressed elastic plate floating on the surface of a homogeneous perfect incompressible fluid of constant depth in which a periodic wave of finite amplitude propagates. We will use the method of multiple-scale asymptotic expansions [5] and take into account the fact that the acceleration of the vertical displacements of the plate is nonlinear.1. Problem Formulation. Governing Equations. Consider a thin elastic plate floating on the surface of a homogeneous perfect incompressible fluid of constant depth. The plate and fluid are unbounded in the horizontal directions. Let the oscillations of the plate be nonlinear and the motion of the fluid be potential. With dimensionless variables x kx = 1 , z kz = 1 ,