Edge waves under initial stress

The frequency equation giving the phase velocity of a wave at the edge of a thick plate under initial stress is obtained. Some particular cases are discussed to derive (a) the velocity of edge waves in a thin plate and (b) the velocity of Rayleigh waves in a plate of infinite thickness under initial stress. The results are compared with those for zero initial stress.     

Diffraction of acoustic waves on the leading edge of a flat plate in a supersonic flow

A scheme is proposed for calculating the intensity of the acoustic wave field generated by diffraction of a beam of acoustic waves on a sharp leading edge of a flat plate in a supersonic flow. This wave field is shown to be a functional of the mass-flow amplitude distribution in the acoustic field at the level of the plate surface upstream of the latter. This distribution can be found on the basis of measurements. The discontinuity of the normal-to-plate component of the velocity perturbation on the plate edge plays an important role in determining mass-flow fluctuations along the plate. At large distances from the leading edge of the plate, where the diffraction wave on the boundary-layer edge degenerates into longitudinal acoustic waves, the amplitude of mass-flow fluctuations decreases with increasing distance from the leading edge and depends on wave orientation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 64–70, March–April, 2005.     

Numerical modeling of the receptivity of a supersonic boundary layer to acoustic disturbances

The receptivity of a supersonic (M_∞ = 6) boundary layer on a flat plate to acoustic disturbances is investigated on the basis of a numerical solution of the 2D Navier-Stokes equations. Numerical results obtained for fast and slow acoustic waves impinging on the plate at zero angle agree qualitatively with asymptotic theory. Calculations carried out for other angles of incidence of the acoustic waves reveal new features of the perturbation field in the neighborhood of the leading edge of the plate. It is shown that, due to visco-inviscid interaction, the shock formed near the leading edge may significantly affect the acoustic field and the receptivity.     

Parametric excitation of flexural-gravity edge waves in the fluid beneath an elastic ice sheet with a crack

The properties of elastic-gravity oscillations of deep water beneath a thin elastic plate with a crack are investigated in the paper. The dependence of the reflection and transition coefficients of the waves through the crack on wave frequency and incident angle are found. The shape of the fluid surface deformed by edge waves, propagating along the crack and decreasing exponentially away from the crack, is investigated in the vicinity of the crack. The asymptotic equations describing the parametric excitation of counterpropagating edge waves by flexural-gravity waves which hit the crack at normal incidence are derived.     

Action of periodic surface pressure on an ice cover in the vicinity of a vertical wall

This paper presents the solution of the linear hydroelastic problem for steady forced vibrations of a semi-infinite ice cover under the effect of localized external load. The ice cover is simulated by a viscoelastic thin plate, the thickness of the fluid layer is assumed to be small, and the shallow water theory is used. The fluid is limited by a solid vertical wall, and the rectilinear edge of the elastic plate adjacent to the wall can be both free and clamped. The solution is obtained with the help of the Fourier integral transform. The behavior of the ice cover is studied depending on the frequency of the external load and boundary conditions on the edge of the plate. It is shown that, in the case of a free edge of the plate, there are considerable deflections on the edge, which could be comparable with deflections at the center of the pressure impact region. It is established that, due to the existence of wave movements of the type of edge waves, the external load energy is transferred to larger distances along the free edge, and there are significant bending moments on the edge of the clamped plate, which can lead to fracture of the ice cover with sufficiently great intensity of the external load.     

正交各向异性平板开孔弹性波的衍射与动应力集中

采用各向异性平板弯曲波动方程及摄动方法，对正交各向异性无限平板开孔弹性波的衍射及动应力集中的问题进行了分析研究，得到了在稳态波作用下此种忆边界条件波动问题的渐近形式分析解。同时采用复变函数方法及保角映射技术，为求解正交各向异性无限平板开孔弹性波的衍射及动力集中问题提供了一种统一规范的分析方法。     

Drag of a Plate Planing on the Shallow Water with Formation of Waves

The plane problem of the plate planing at a constant velocity on the surface of a heavy, ideal, incompressible, finite-depth fluid is considered. The approximate, depth-independent expression for the force acting on the plate is derived from the linear distribution of the fluid velocity along the plate and the height of the flow stagnation point, without regard for jet formation near the leading edge. In this approximate formulation the plate drag depends on its velocity and the trailing edge immersion and does not depend on the planing angle. Experiments and numerical calculations in the exact formulation are performed in the near-critical flow regimes. It is shown that the wave patterns in the experiments and numerical calculations coincide, the formula for the drag being in agreement with the numerical experiments. An approximate criterion of the formation of waves going away from the plate in the forward direction is proposed.     

Reflection and transmission of regular water waves by a thin,floating plate

Measurements of the wave fields reflected and transmitted by a thin floating plastic plate are reported for regular incident waves over a range of incident periods (producing wavelengths comparable to the plate length) and steepnesses (ranging from mild to storm-like). Two different plastics are tested, with different densities and mechanical properties, and three different configurations are tested. The configurations include freely floating plates, loosely moored plates (to restrict drift), and plates with edge barriers (to restrict waves overwashing the plates). The wave fields reflected and transmitted by plates without barriers are shown to become irregular, as the incident waves become steeper, particularly for the denser plastic and the moored plate. Further, the proportion of energy transmitted by the plates without barriers is shown to decrease as the incident wave becomes steeper, and this is related to wave energy dissipation.     

Controlling the disturbances in a hypersonic flat-plate shock layer by unsteady action from the surface

The results of a wind-tunnel experiment on the joint action of periodic acoustic fast-mode disturbances of the outer flow and disturbances generated at the leading edge of a plate on the hypersonic (M_∞ = 21) viscous shock layer on the plate are presented. The possibility of positively controlling the intensity of density fluctuations in the plate shock layer by means of disturbances introduced from the leading edge is shown. Direct numerical simulation of the suppression (enhancement) of disturbances under the simultaneous action on the shock layer of the two-dimensional fast-mode acoustic waves in the outer flow and the source of two-dimensional suction/injection disturbances near the leading edge of the plate is performed under the experimental conditions. The experimental and calculated results are shown to be in good agreement.     

Edge waves in a shallow fluid beneath a fractured elastic plate

It is shown that a fracture in an elastic plate floating on the surface of a shallow liquid layer is a waveguide along which wave energy can be transported. The edge wave velocity is less than the velocity of flexural-gravity waves. The existence of an antisymmetric edge wave mode depends on the Poisson's ratio of the elastic plate. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 185–189, July–August, 1994.     

Fine structure of a stratified flow near a flat-plate surface

The pattern of disturbances arising during the motion of a strip along a horizontal surface in a continuously stratified fluid with identified upstream and attached internal waves, boundary layers, and edge singularities is calculated in the liner approximation. The flow pattern behind a flat plate moving with a constant velocity in a continuously stratified fluid is studied with the use of the optical schlieren technique; transformation of waves and finely structured elements of the flow with increasing plate velocity is analyzed. The calculated and experimentally observed patterns of internal waves at low velocities are demonstrated to be in good agreement. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 77–91, November–December, 2007.     

Acoustic receptivity of Mach 4.5 boundary layer with leading-edge bluntness

Boundary layer receptivity to two-dimensional slow and fast acoustic waves is investigated by solving Navier–Stokes equations for Mach 4.5 flow over a flat plate with a finite-thickness leading edge. Higher order spatial and temporal schemes are employed to obtain the solution whereby the flat-plate leading edge region is resolved by providing a sufficiently refined grid. The results show that the instability waves are generated in the leading edge region and that the boundary-layer is much more receptive to slow acoustic waves (by almost a factor of 20) as compared to the fast waves. Hence, this leading-edge receptivity mechanism is expected to be more relevant in the transition process for high Mach number flows where second mode instability is dominant. Computations are performed to investigate the effect of leading-edge thickness and it is found that bluntness tends to stabilize the boundary layer. Furthermore, the relative significance of fast acoustic waves is enhanced in the presence of bluntness. The effect of acoustic wave incidence angle is also studied and it is found that the receptivity of the boundary layer on the ‘windward’ side (with respect to the acoustic forcing) decreases by more than a factor of four when the incidence angle is increased from 0° to 45°. However, the receptivity coefficient for the ‘leeward’ side is found to vary relatively weakly with the incidence angle.      

Hydroelastic behavior of a floating plate in waves

The behavior of a floating semiinfinite plate in surface waves incident normally to the edge of the plate is studied. We used an analytical solution of this problem obtained earlier by the Wiener–Hopf technique. In this paper, we study the distributions of displacements, deformations, and pressure over the plate as functions of the dimensionless parameters of the problem (reduced rigidity and depth) and their asymptotic distributions for large and small wavelengths.     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

环形薄板二维驻波的研究

在理论上和实验上对环形薄板二维驻波波节图形(克拉尼图形) 进行了研究. 通过在极坐标下对垂直板面方向小振动方程进行分离变量, 求解出环形薄板小振动方程在外边界悬空时分别在两种内边界条件, 即内边界悬空和内边界简支下的解析解的简正模式, 并计算了在第一种边条件下几种共振模式的径向波速近似值, 以及两种边条件下的圆形驻波波节线的半径和薄板的弹性模量. 发现通过调节环形薄板上点振动源的频率, 可精确控制薄板上出现的克拉尼图形. 实验上观察到了仅有圆形波节线, 仅有辐射状波节线, 以及两种波节线同时存在3 种简正模式的情形, 且波节线的数量可严格控制. 理论结果跟实验符合得很好.      

Wave structure in the radial film flow with a circular hydraulic jump

 A circular hydraulic jump is commonly seen when a circular liquid jet impinges on a horizontal plate. Measurements of the film thickness, jump radius and the wave structure for various jet Reynolds numbers are reported. Film thickness measurements are made using an electrical contact method for regions both upstream and downstream of the jump over circular plates without a barrier at the edge. The jump radius and the separation bubble length are measured for various flow rates, plate edge conditions, and radii. Flow visualization using high-speed photography is used to study wave structure and transition. Waves on the jet amplify in the film region upstream of the jump. At high flow rates, the waves amplify enough to cause three-dimensional breakdown and what seems like transition to turbulence. This surface wave induced transition is different from the traditional route and can be exploited to enhance heat and mass transfer rates. Received: 25 April 2000/Accepted: 1 June 2001     

On bending waves localized along the edge of a plate

We consider bending waves localized along the edge of a plate using nonclassical equations of motion and boundary conditions.Institute of Mechanics, Armenian Academy of Sciences, Erevan. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 61–68, February, 1994.     

Use of equivalent body forces for acoustic emission from a crack in a plate

A method to determine acoustic emission of surface waves from a crack near the free edge of a plate, is presented, in terms of the function f(t), which defines the time dependence of the crack opening process, the crack opening volume per unit thickness of the plate, and the elastic constants of the plate. The determination of the time-varying displacement is based on the use of equivalent body forces, which are shown to be two double forces. The acoustic emission of the crack, or the equivalent radiation from the double forces, has been obtained by a novel use of the elastodynamic reciprocity theorem. It is of interest that the normal surface-wave displacement at a position x₀ of the free edge comes out as depending on df/dt evaluated at x₀ for t > x₀/c_R, where c_R is the velocity of surface waves on the free edge.     

Generation of the tollmien–schlichting wave in a supersonic boundary layer by two sinusoidal acoustic waves

The problem is solved using parabolized equations of stability for threedimensional perturbations of a compressible boundary layer on a flat plate. Nonlinearity is taken into account by quadratic terms that are most significant in estimates of the viscous critical layer of the stability theory. These terms are determined by the total field of two acoustic perturbations, and the equations become linear and inhomogeneous. The calculations are performed for one acoustic wave being stationary in the reference system fitted to the plate for Mach numbers M=2 and 5. Solutions are presented, which are identified very accurately with Tollmien–Schlichting waves at a rather large distance from the plate edge.     

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.