Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

Results of modeling the dependence of the height of flexural-gravity waves in a floating plate with a sealed edge on the inclination of the bottom of the reservoir

На основании проведенных автором экспериментов по разрушению льда амфибийными судами на воздушной подушке (СВП) резонансным методом при их движении в сторону берега, их крупномасштабных моделей в полевых условиях, а также известных случаев разрушения ледяного покрова движущимися в сторону берега транспортными средствами показана возможность существенного увеличения толщины льда, разрушаемого у его береговой кромки. Приведены данные экспериментов, выполненных в опытовом бассейне на моделирующих ледяной покров упругих пленках, подтверждающие эти возможности. В работе представлены результаты моделирования зависимости высоты изгибно-гравитационных волн (ИГВ), возбуждаемых в плавающей пластине движущейся в направлении ее заделанной кромки нагрузкой, от угла наклона дна бассейна. Также приведены рекомендации по использованию наклонности дна для повышения эффективности разрушения ледяного покрова (увеличения толщины разрушаемого льда) путем возбуждения резонансных ИГВ. На основании выполненных экспериментов показано, что приближение к берегу ИГВ, возбуждаемых СВП с последующим их выходом на него, может значительно увеличить толщину разрушаемого льда за счет отрыва от берега его кромки. Отмечена возможность увеличения ледоразрушающей способности ИГВ благодаря последующим проходам судна вдоль нее из-за роста деформаций ослабленного таким образом льда, т.е повышения ледоразрушающей способности СВП. Based on the experiments conducted by the author on ice destruction by amphibious hovercraft using the resonant method when they move towards the shore, their large-scale models in the field, as well as known cases of ice cover destruction by vehicles moving towards the shore, the possibility of a significant increase in the thickness of ice destroyed at its coastal edge is shown. The data of experiments performed in the experimental pool on elastic films modeling the ice cover are presented, confirming these possibilities. The paper presents the results of modeling the dependence of the height of Flexural-gravitational waves excited in a floating plate by a load moving in the direction of its embedded edge on the angle of inclination of the pool bottom. Recommendations are also given for using the bottom slope to increase the efficiency of ice cover destruction (increasing the thickness of the destroyed ice) by exciting resonant Flexural-gravitational waves. Based on the performed experiments, it is shown that the approach to the shore of Flexural-gravitational waves excited by hovercraft with their subsequent exit to it can significantly increase the thickness of the destroyed ice due to the separation of its edge from the shore. The possibility of increasing laboratree the ability of the Flexural-gravitational waves during the next passages of the ship along it due to the growth of deformations weakened the ice.

Relaxation properties of the ice cover

The aim of this work was to generalize the known data and conduct experiments to determine the relaxation properties of ice in the ice cover under short-term (no more than 1 min.) loading. The problem lies in the fact that when one is solving applied problems of ice engineering, ice is often considered as an elastic isotropic material, and its stress-strain state (SSS) is studied in terms of the theory of bending of elastic plates. This does not allow performing theoretical calculations when resonant flexural gravity waves (IGW) are excited by moving loads, because under these conditions, the deflections of the ice increase to infinity and the known solutions become unusable. In fact, ice clearly manifests the properties of a quasi-isotropic medium, and the relationship between stresses and deformations is of a viscoelastic nature. It is noted in the work that, depending on the mode in which external loads act on the ice cover, its inelastic properties affect the nature of its behavior in different ways, while the viscoelastic properties of the ice cover are well described by the linear models of Maxwell or Kelvin-Voigt inelastic continuous media. The experimental material is duly processed and analysis is carried out of the results of experimental studies performed in the field by loading the ice cover with balanced loads using a specially made loading device, which was a frame with three supports. The design of the device made it possible to load the ice cover with balanced loads, which made it possible to exclude the influence of false elasticity of water on the results of experiments. For the rheological models of ice behavior indicated, the most probable ranges of changes in the relaxation times of stresses and deformations of the ice cover in the ice conditions considered are given. The results obtained can be used in theoretical studies of ice engineering problems.

Scattering of flexural gravity waves by a pair of submerged vertical porous barriers located above a porous sea-bed

An analytical study is presented to investigate the scattering of oblique flexural gravity waves by a pair of totally submerged vertically placed porous barriers, located at some distance from each other, for a homogenous fluid flowing over a porous sea-bed. A thin ice-sheet, replacing the usual free surface, is considered as the upper surface where it is treated as a thin elastic plate. The complete analytical solution, under the assumption of small-amplitude theory and structural response, is acquired by employing eigenfunction expansion and least square method. Subsequently, computation for the reflection and transmission coefficients, energy loss and wave forces is carried out and discussed for different parameter values corresponding to the ice-sheet, porous sea-bed and porous barriers. This study establishes that the oscillatory behavior exhibited by the reflection of the waves. The vertical porous barriers are found to dissipate a significant portion of the wave energy when an increase in the inertial effect of the porous barriers takes place. The hydrodynamic force on the barriers also follows an oscillatory pattern and it increases when the length of the barrier is increased. It is demonstrated that, corresponding to various structural parameters, almost no reflection and full transmission take place for an impermeable sea-bed and also when only real porosity parameter of the porous sea-bed is considered. Further, variation in the elastic parameter of the floating ice-sheet is observed to command a considerable influence when the wave impinges upon the submerged vertical porous barriers.

Observing Wind-Forced Flexural-Gravity Waves in the Beaufort Sea and Their Relationship to Sea Ice Mechanics

We developed and deployed two inertial measurement units on mobile pack ice during a U.S. Navy drifting ice campaign in the Beaufort Sea. The ice camp was more than 1000 km from the nearest open water. The sensors were stationed on thick (>1 m) first- and multi–year ice to record 3-D accelerations at 10 Hz for one week during March 2020. During this time, gale-force winds exceeded 21 m per second for several hours during two separate wind events and reached a maximum of 25 m per second. Our observations show similar sets of wave bands were excited during both wind events. One band was centered on a period of ~14 s. Another band arrived several hours later and was centered on ~3.5-s. We find that the observed wave bands match a model dispersion curve for flexural gravity waves in ~1.2-m ice with a Young’s modulus of 3.5 GPa under compressive stresses of ~0.3 MPa. We further evaluate the bending stress and load cycles of the individual wave bands and their potential role in break-up of sea ice. This work demonstrates how observations of waves in sea ice using these and similar sensors can potentially be a valuable field-based tool for evaluating ice mechanics. In particular, this approach can be used to observe and describe the combined mechanical behavior of consolidated floes relevant for understanding sea ice mechanical processes and model development.

Laboratory Investigations of the Bending Rheology of Floating Saline Ice and Physical Mechanisms of Wave Damping in the HSVA Hamburg Ship Model Basin Ice Tank

An experimental investigation of flexural-gravity waves was performed in the Hamburg Ship Model Basin HSVA ice tank. Physical characteristics of the water-ice system were measured in several locations of the tank with a few sensors deployed in the water and on the ice during the tests. The three-dimensional motion of ice was measured with the optical system Qualisys; water pressure was measured by several pressure sensors mounted on the tank wall, in-plane deformations of the ice and the temperatures of the ice and water were measured by fiber optic sensors; and acoustic emissions were recorded with compressional crystal sensors. The experimental setup and selected results of the tests are discussed in this paper. Viscous-elastic model (Burgers material) is adopted to describe the dispersion and attenuation of waves propagating below the ice. The elastic modulus and the coefficient of viscosity are calculated using the experimental data. The results of the measurements demonstrated the dependence of wave characteristics from the variability of ice properties during the experiment caused by the brine drainage. We showed that the cyclic motion of the ice along the tank, imitating ice drift, and the generation of under ice turbulence cause an increase of wave damping. Recorded acoustic emissions demonstrated cyclic microcracking occurring with wave frequencies and accompanying bending deformations of the ice. This explains the viscous and anelastic rheology of the model ice.