The influence of a uniformly compressed floating elastic plate on the development of three-dimensional waves in a homogeneous fluid

A study is made in the linear formulation of the influence of a uniformly compressed floating elastic plate on the unsteady three-dimensional wave motion of a homogeneous fluid of finite depth. Waves are excited by a region of normal stresses which moves on the surface of the plate. Three-dimensional flexural-gravity waves were studied in [1, 2] without allowance for compressing forces. Plane waves under conditions of longitudinal compression were considered in [3, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–83, November–December, 1984.     

Evolution of three-dimensional gravitationally warped waves during the movement of a pressure zone of variable intensity

Three-dimensional, unestablished, gravitationally warped waves arising due to the motion of a harmonically time-varying pressure zone over a solid, thin plate floating on the surface of a homogeneous liquid of finite depth have been studied in the linear formulation. In the absence of a plate, three-dimensional waves are generated by the movement of a region of periodic perturbations, where established waves have been studied in [1, 2], and unestablished waves have been investigated in [3–5]. The evolution of three-dimensional, gravitationally warped waves formed during the motion of a constant load over a plate has been considered in [6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 54–60, September–October, 1986.     

Some features of flexural-gravity wave propagation in the presence of a crack in sheet ice

The effect of a crack in an ice sheet on the propagation of surface flexural-gravity waves in a basin of constant depth is analyzed. The ice sheet is simulated by two floating semi-infinite fragments of a thin elastic isotropic plate. As the boundary-contact conditions on the line of contact between the parts of the plate the conditions of continuity of displacements for arbitrary slopes simulating one ice-floe overlying on another and free-edge conditions (crack) are considered. The dependence of the amplitude characteristics of the perturbations on the thickness of the ice, its degree of compression, the incident wave frequency, the depth of the basin, and the form of the boundary-contact conditions is investigated. Problems of wave diffraction on inhomogeneities of an elastic plate were solved in [1, 2], and on a crack in the ice sheet in [3, 4].Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–150, March–April, 1996.     

Interaction of surface and flexural-gravity waves in ice cover with a vertical wall

The problem of the interaction of surface and flexural-gravity waves with a vertical barrier is solved in a two-dimensional formulation. It is assumed that the fluid is ideal and incompressible, has infinite depth, and is partially covered with ice. The ice cover is modeled by an elastic plate of constant thickness. The eigenfrequencies and eigenmodes of oscillation of the floating elastic ice plate, the deflection and deformation of ice, and the forces acting on the wall are determined.     

Radiation and Diffraction of Water Waves by a Submerged Body with Ice Cover in Finite Depth

This paper presents the three-dimensional Green-function method to predict the radiation and diffraction of water waves by a submerged body in water of uniform finite depth with an ice cover. The fluid is assumed to be perfect and irrotational, the ice is modelled as an elastic plate. The zero-speed Green function of finite depth satisfying the linearized covered-surface condition is derived in three dimensions, the numerical results for the Green function and its derivatives are given. The integral equations are established by distributing the source strength on the body surface, the radiation and diffraction problems are solved. A submerged sphere is taken as an example, the effects of the water depth and the flexural rigidity of ice cover on hydrodynamics are analysed, and the good agreement with the analytical solutions reveals that the present method is correct and reliable.

     

Solitary wave packets beneath a compressed ice cover

A family of plane solitary wave packets of a small (but finite) amplitude on the surface of an ideal incompressible fluid of finite depth beneath an ice cover is described. The solitary wave trains correspond to solutions of the two-dimensional system of Euler’s equations of an ideal incompressible fluid of the type of a traveling wave which decreases at infinity and has identical phase and group velocities. The ice cover is simulated by an elastic Kirchhoff-Love plate freely floating on the fluid surface in the compressed state.     

Three-dimensional internal waves in the near zone in the case of a load moving over the surface of a floating elastic plate

The effect of a thin elastic floating plate on the three-dimensional internal waves in the near zone of a moving region of constant pressure is studied with reference to a two-layer model of a liquid of finite depth. The dependence of the spatial distributions of the amplitudes of the wave disturbances due to the internal waves at the plateliquid interface and on the surface of density discontinuity on the rate of displacement of the pressure region and the characteristics of the plate is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 85–91, January–February, 1990.     

Motion of an External Load over a Semi-Infinite Ice Sheet in the Subcritical Regime

The three-dimensional problem of steady-state forced vibrations of fluid and semiinfinite ice sheet under the action of a local external load traveling along the rectilinear sheet edge at a constant velocity is considered. Two cases are analyzed. In the first case the fluid surface outside the ice sheet is free and in the second the fluid is confined by a rigid vertical wall and the ice sheet edge adjacent to the wall can be both clamped and free. The ice sheet is simulated by a thin elastic isotropic plate floating on the surface of fluid of finite depth. The load traveling velocity is assumed to be not higher than the minimum phase velocity of the flexural-gravity waves (subcritical regime). The solution to the linear problem is obtained by means of the integral Fourier transform and matching the expansions of the velocity potential in the vertical eigenfunctions. Examples of the numerical investigation of the ice sheet and fluid displacements are given.     

Influence of an elastic plate on the motion of an inhomogenfous fluid

The influence of a thin elastic isotropic plate on the wave motion of an inhomogeneous fluid originating under the effect of external periodic perturbations is investigated. The fluid density increases constantly with depth. Analogous problems have been examined for an inhomogeneous fluid without a plate in [1, 2] and with a plate on the surface of a homogeneous fluid in [3–5].Sevastopol'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–67, January–February, 1972.     

A direct relationship between bending waves and transition conditions of floating plates

Ocean waves travel deep into ice fields in the polar regions, both affecting the formation of sea-ice and causing its break-up. Recently, it has been shown that a relatively simple linear water and bending wave theory can predict the decay rate of the wave energy travelling through fractured ice sheets and floes at the geophysically important wave periods of 6–15 s. That work used simple free-edge conditions. A possible improvement to the current model is to better represent the effective connection due to partially frozen cracks that occur in practice. The Wiener–Hopf technique gives explicit formulae for the velocity potential and surface deflection, expressed as series expansions over the modes of the elastic plate floating on water of finite depth, with the coefficients in the expansion given as functions of four constants. These constants are determined by a system of four linear equations, represented by a 4-by-4 matrix and a four-element vector. The elements of the matrix are given as explicit functions of relationship between edge conditions. General connections between ice sheets may be interpreted as a vertical and a rotational spring providing transition conditions for the shear force and the bending moment. The reflection and the transmission of waves can then be simply calculated as direct functions of the connection conditions. Conversely, reflected and transmitted waves allow complete characterization of the effective connection conditions at a material discontinuity.     

Nonlinear steady-state motion of an elastic plate floating on the surface of a liquid of infinite depth

The nonlinear problem of steady-state waves in an ideal fluid of infinite depth with a thin elastic plate floating on its surface is considered. The solution is found by a perturbation method. Three approximations are obtained. A case of branching of the solution is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 119–123, March–April, 1987.     

On the development of plane waves generated by a periodic pressure applied to the flow of a continuously stratified fluid

In the linear formulation, an investigation is made into the development of undamped (in time) plane waves generated by a. harmonically varying pressure applied to the free surface of an initially undisturbed flow of a continuously stratified fluid of finite depth. The cases of a homogeneous fluid and two-layer fluid are considered in [1–3]. Nonstationary waves in a continuously stratified flow generated by a time-independent pressure were investigated in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 99–104, July–August, 1980.     

Unsteady waves in a viscous fluid of finite depth

Here we study the plane and three-dimensional problems of unsteady waves which arise on the surface of a viscous fluid of finite depth under the influence of a velocity pulse applied on the bottom of the basin.The problem is considered as the simplest scheme for studying, with account for the effect of viscosity, the propagation of waves of the tsunami type which result from an underwater shock.Similar problems on the propagation of waves which arise from initial surface disturbances are considered in [1–9].     

Surface wave diffraction at a crack in sheet ice

The time-periodic motions of a liquid layer of finite depth beneath an ice sheet with a straight infinite crack having a periodic dependence on the horizontal coordinate in the direction of the crack are considered. The ice sheet is simulated by a thin elastic plate. It is assumed that the thickness of the plate changes abruptly across the crack. The problems of plane-wave diffraction at a crack, plane-wave diffraction atN cracks in a uniform ice sheet, and plane-wave reflection from a rigid wall are solved. The effect of the pre-existing state of stress of the ice sheet on the properties of the reflected waves is investigated. The condition of nontransmission of fix-frequency waves beneath the edge of the ice is obtained.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 93–102, March–April, 1993.     

Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.     

Oscillations of a cylindrical body submerged in a fluid with ice cover

The linear plane problem of oscillations of an elliptic cylinder in an ideal incompressible fluid of finite depth in the presence of an ice cover of finite length is solved. The ice cover is modeled by an elastic plate of constant thickness. The hydrodynamic loads acting on the body are determined as functions of the oscillation frequency and the positions of the cylinder and plate.     

Behavior of a semi-infinite ice cover under periodic dynamic impact

Oscillations of a semi-infinite ice cover in an ideal incompressible liquid of finite depth under local time-periodic axisymmetric load are considered. The ice cover is simulated by a thin elastic plate of constant thickness. An analytical solution of the problem is obtained using the Wiener–Hopf method. The asymptotic behavior of the amplitudes of oscillations of the plate and the liquid in the far field is studied. It is shown that the propagation of waves in the far field is uneven: in some directions, the waves propagate with a significantly greater amplitude.     

Behavior of a floating elastic plate during vibrations of a bottom segment

The Wiener-Hopf technique is used to obtain an analytical solution for the problem of vibrations of a floating semi-infinite elastic plate due to earthquake-induced vibrations of a bottom segment. An explicit solution is obtained ignoring the inertial term. The surface-wave amplitudes and ice-plate deflection are studied numerically as functions of the frequency and position of the vibrating bottom segment, ice thickness, and fluid depth.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 98–108, March–April, 2005.     

Behavior of a Semi-Infinite Ice Cover Under a Uniformly Moving Load

This paper consideres the behavior of a semi-infinite ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a load moving with constant velocity along the edge of the cover at some distance from it. The ice cover is modeled by a thin elastic plate of constant thickness. In a moving coordinate system, the deflection of the plate is assumed to be steady. An analytic solution of the problem is obtained using the Wiener–Hopf technique. The wave forces, the deflection of the plate, and the elevation of the free surface of the fluid at different velocities of the load are investigated.     

Plane Problem of Vibrations of an Elastic Floating Plate under Periodic External Loading

The Wiener–Hopf technique is used to construct an analytical solution of the problem of vibrations of a semiinfinite elastic floating plate under periodic external loading. The solution is obtained in explicit form ignoring draft. The dependences of the amplitudes of surface waves and iceplate deflection on the loading distribution and frequency, ice thickness, and liquid depth are studied numerically. It is established that for some types of acting load, no waves propagate in the plate and liquid and the plate vibrations are standing waves localized near the loading region. An example of such vibrations is given and a condition for the occurrence of localized vibrations is found.