The transmission of a flexural-gravitational wave through several straight obstacles in a floating plate

Periodic wave processes in a thin elastic plate floating on the surface of an incompressible fluid of finite depth are studied. The plate completely covers the fluid surface and executes flexural oscillations under the action of gravitational waves in the fluid. The system of free oscillations in the plate is disrupted along a set of parallel lines. Rigid clamping of the plate, a sliding fastening and an infinitesimally narrow slit are considered as such disruptions. The apparatus used to construct the solution is quite general, and other disruptions in the elastic properties of a plate or its reinforcement, that are realized with linear boundary-contact conditions, can be treated in a similar way. The transmission and reflection of a harmonic flexural-gravitational wave, that is orthogonally incident on the inhomogeneities in the plate, are studied. Exact analytical representations of the wave fields in the plate and in fluid are obtained and the transmission and reflection coefficients for the incident flexural-gravitational wave are determined. The forces developed in the fastenings are found.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

The diffraction of plane gravitational waves by the edge of an ice cover

The diffraction of plane surface gravitational waves by the edge of an ice cover lying on the surface of an incompressible fluid of infinitely great depth is considered. The ice cover is simulated by a thin elastic plate. The wave reflection and transmission coefficients are determined when it interacts with the ice cover. A wave field is constructed in the fluid under the conditions that a periodic lumped force and a lumped moment act on the edge of the ice cover. It is shown that as the incident wavelength increases the reflection coefficient tends to zero and the transmission coefficient tends to unity.     

Vibrations of an elastic plate partially immersed in a liquid

The free vibrations of an elastic plate, placed orthogonally at the bottom of an unbounded open reservoir of constant depth, are considered. The reservoir is filled with a compressible liquid, the plate protrudes above the free surface and its rigidly fastened to the bottom. Gravitational effects are ignored. An analytical representation of the acoustic field in the liquid and the vibrational field in the plate is obtained. The natural frequencies and natural forms of vibration are calculated as a function of the height of the liquid level. An approximate approach to calculating the vibrations of the plate is considered.     

The action of an unsteady external load on a circular elastic plate floating on shallow water

A solution of the transient problem of the behaviour of a circular elastic plate floating on the free surface of a liquid under the action of an external load is constructed. It is assumed that the liquid is ideal and incompressible and that its depth is small compared with the radius of the plate. The compatible motion of the plate and the liquid is treated within the framework of linear theory. The flow of the liquid is assumed to be irrotational. The behaviour of the plate under different loads is investigated and it is shown that the bounded dimensions of the elastic plate have a substantial effect on its unsteady behaviour.     

Nonlinear Propagation of Wave Packets for Near-Critical Wave Numbers in a Liquid that Is Piecewise Nonuniform with Depth

This is a study of the evolution of finite amplitude wave packets at the contact surface between a semi-infinite liquid medium and a liquid located above it near a solid lid. The limiting case of near-critical wave numbers for propagation of wave packets in the liquid system is discussed. Perturbation expansions are obtained for the deviation of the contact surface from its unperturbed position and for wave numbers near the critical value. This is compared with the solution for the analogous limiting cases of one or two semi-infinite liquids. A version in which the wave numbers are far from critical is examined.     

Wave Propagation in a Layer of Magnetic Liquid

A linearized equation for the propagation of surface gravitational waves in a layer of magnetized liquid of finite depth is examined. The liquid is assumed to be inviscid, incompressible, and to possess magnetization properties in the absence of electrical conductivity, while the motion is assumed to be irrotational. Travelling wave solutions are obtained. The dependences of the phase and group velocities of the magnetic liquid on the magnetic parameters are studied. It is shown that for some values of the magnetic parameters there is an interval of short wavelengths for which the group velocity is negative, which indicates that the wave energy propagates in the negative direction.     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environments

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load in thermal environments are studied. The thermal effects and temperature-dependent material properties are taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varies in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived from Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Using the dispersion relation and integral transforms, exact integral solutions of the functionally graded plate under a point impact load in thermal environments are obtained. The influences of the volume fraction distributions and temperature field on the wave propagation and transient response of functionally graded plates are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and provide a theoretical basis for engineering applications.     

Scattering of guided SH-wave by a partly debonded circular cylinder in a traction free plate

A solution of the scattering problem of guided SH-wave by a partly debonded circular cylinder centered in a traction free plate has been set up. The plate is divided up into three regions with two imaginary planes perpendicular to the plate walls. In the central region where the partly debonded cylindrical obstacle is posted, the wave field is expanded into the cylindrical wave modes and Chebyshev polynomials. In the other two exterior regions the fields are expanded into the plate wave modes. A system of fundamental equations to solve the problem is obtained according to the traction free boundary condition on the plate walls and the continuity condition of the traction and the displacement across the imaginary planes. The approximate numerical method termed mode-matching technique is used to construct a matrix equation to obtain curves showing the coefficient of reflection and transmission versus the ratio of the cylinder’s radius to the plate’s half-thickness and the angular width of the debonded region. A comparison of the numerical results between the welded interface condition and the debonded interface condition is made, and the results are discussed.     

The vibrations and stability of a prestressed plate on an elastic foundation

The free vibrations of a transversely isotropic prestressed linear elastic half-space, localized close to a free surface, are considered. The free vibrations of a prestressed transversely isotropic infinite plate, lying on an elastic foundation, are also considered. The dispersion equation is analysed as a function of the wave numbers, the elastic properties of the foundation and of the plate and the values of the prestresses. The investigation is confined to cases when the initial stresses are less than the critical values, while the elastic waves do not penetrate into the depth of the foundation but are localized close to the free surface. The stability of the half-space and the plate on an elastic foundation is also considered. When analysing the vibrations and the stability of the plate, the results in the three-dimensional formulation of the problem are compared with the results of the two–dimensional Kirchhoff–Love and Timoshenko–Reissner models.     

Effect of sudden suction on the hydromagnetic natural convection from a vertical plate

The effects of step function change in suction velocity are investigated for a natural convection flow from a vertical porous flat plate of infinite length in the presence of transverse magnetic field for two cases namely (i) when the plate is suddenly raised to a uniform higher temperature, (ii) when the plate suddenly begins to generate a uniform heat flux at its surface. In (i) the coefficient of heat transfer becomes independent of the Hartmann numberM. In either case for a fixed time t except for correct steady state t→ ∞, the effect of the Hartmann number is to decrease both the velocity of the fluid and the skin-friction at the plate. In the correct steady state the velocity field and the skin-fiiction become independent of the Hartmann number.     

Evolution of flexure-gravitational stimulated vibrations of a floating nonuniformly compressed elastic plate

We consider the evolution of vibrations of a thin nonuniformly compressed elastic plate that floats on a surface of a homogeneous ideal incompressible fluid of finite depth, by using the method of integral transformations. Waves are caused by pressures of the type of traveling waves on the surface of the plate, which is harmonic in one of the directions. An analysis of conditions of perturbations of waves that form oscillations of the plate and the wave motion of the fluid is given. A study is given of the dependence of the structure and the number of waves from stressing forces and characteristics of perturbed pressures.Translated from Dinamicheskie Sistemy, No. 7, pp. 58–62, 1988.     

Effect of bottom topography on the unsteady behaviour of an elastic plate floating on shallow water

The unsteady behaviour of a thin elastic plate in the form of a strip of constant width and infinite length, floating on the surface of an ideal and incompressible liquid, is investigated within the limits of the linear shallow-water theory. The unsteady behaviour of the plate is caused by initial disturbances or an external load. The depth of the liquid under the plate is variable. It is assumed that all the flow characteristics are independent of the coordinate along the plate. The deflection of the plate is sought in the form of an expansion in eigenfunctions of the oscillations in a vacuum with time-varying amplitudes. The problem reduces to solving an infinite system of ordinary differential equations for the unknown amplitudes. The behaviour of the plate is investigated for different actions and shapes of bottom irregularities. It is shown that the bottom topography can have a considerable effect on the deformation of the plate.     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

Transverse Oscillations of a Cylindrical Vessel with a Two-Layer Liquid Separated by an Elastic Membrane

A solution is given for the transverse oscillations, under elastic forces, of a cylindrical vessel containing an ideal two-layer liquid with elastic membranes at the free and inner surfaces of the stratified liquid. The following limiting cases for the elastic membrane are examined: it lies only at the free surface of a uniform and two-layer liquid; it separates a two-layer liquid; it exists at both the free and the inner surfaces of a two-layer liquid. The numerically calculated dependences of the first eigenfrequency on the tension of the membrane, liquid densities, and filling depth are analyzed.     

Complementary Approximations to the Solution of a Problem in Water Waves

A surface wave is incident upon two vertical parallel barriersimmersed to a given depth beneath the surface of water of infinitedepth. The problem has an exact but complicated solution. Simplercomplementary approximations to the reflection coefficient arederived which may be computed easily. These approximations arefound to be very good for all wavelengths provided the ratioof separation to depth of the barrier is not too small. It isshown that there exist certain wavelengths for each barrierconfiguration for which total reflection of the incident waveoccurs.     

Remote stationary wave field generated by local perturbing sources in a flow of stratified fluid

A linear formulation is used to study the problem of stationary waves formed in a uniform flow of an inviscid incompressible vertically stratified fluid past a point source or a mass dipole. Formulas are derived representing the characteristics of the wave field in the form of the sum of single integrals. A method is developed for constructing complete asymptotic expansions of the integrals obtained for large distances from the wave generator, including uniform expansions near the leading fronts of the separate modes. Approximate solutions of the problem in question exist (/1–4/ et al.). The behaviour of the characteristics of the wave field near the leading fronts of internal waves was studied in /5, 6/. In the case of a deep liquid the asymptotic form uniform in the neighbourhood of the leading fronts is expressed in terms of Fresnel integrals /5/, and in the case of a liquid of finite depth by Airy functions /6/. Examples of the exact solution of the problem are given in /7/.