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

Abstract

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

Phase Structure of Fluid Fluctuations with a Floating Elastic Ice Plate under Nonlinear Interaction of Progressive Surface Waves

“…Wherein k1 > k2 > k3 > k4. Note that one of these resonance values k = = k1, obtained from the equation (4.1) when n = 3, coincides with the smaller of the two resonance values obtained in the case when the first approximation ζ1 is given in the form ζ1 = ζ11 [14].…”

Phase structure of fluid fluctuations with a floating elastic ice plate under nonlinear interaction of progressive surface waves

“…Отметим, что одно из этих резо-нансных значений k = k 1 , получаемое из уравнения (4.1) при n = 3, совпадает с меньшим из двух резонансных значений, получаемых в случае, когда пер-вое приближение ζ 1 задается в виде ζ 1 = ζ 11 [14].…”

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

“…In the present work, on the basis of the asymptotic expansion of the velocity potential obtained in [11] by the method of multiple scales [12] to within the quantities of third order smallness in the steepness of waves, we analyze the dependences of the distributions of components of the velocity of motion of liquid particles over the wavelength on the characteristics of the ice cover.…”