Instability of nonlinear standing waves in front of a vertical wall

The standing waves formed in front of a vertical breakwater in Gdańsk North Port Harbour are examined. To simplify the wave-structure interaction problem, a laboratory experiment and mathematical model were designed. Standing waves in a limited space were generated under resonance conditions between a vertical wall and wave generator. Such slowly growing standing waves eventually become unstable and, consequently, create an impact impulse on the vertical wall. The dynamics of the vertical wall and hydrodynamics of the standing wave were measured and compared with a numerical model derived by variational calculus. The phenomenon of standing wave instability observed in nature was reproduced by laboratory experimentation and numerical simulation. The presented mechanism of breaking standing waves is more complicated in reality due to wave randomness and the multidirectional wave field.     

Standing waves for a two-way model system for water waves

In this paper, we prove the existence of a large family of nontrivial bifurcating standing waves for a model system which describes two-way propagation of water waves in a channel of finite depth or in the near shore zone. In particular, it is shown that, contrary to the classical standing gravity wave problem on a fluid layer of finite depth, the Lyapunov–Schmidt method applies to find the bifurcation equation. The bifurcation set is formed with the discrete union of Whitney's umbrellas in the three-dimensional space formed with 3 parameters representing the time-period and the wave length, and the average of wave amplitude.     

Viscous fluid flow induced by rotational-oscillatory motion of a porous sphere

Viscous fluid flow induced by rotational-oscillatorymotion of a porous sphere submerged in the fluid is determined. The Darcy formula for the viscous medium drag is supplementedwith a term that allows for the medium motion. The medium motion is also included in the boundary conditions. Exact analytical solutions are obtained for the time-dependent Brinkman equation in the region inside the sphere and for the Navier–Stokes equations outside the body. The existence of internal transverse waves in the fluid is shown; in these waves the velocity is perpendicular to the wave propagation direction. The waves are standing inside the sphere and traveling outside of it. The particular cases of low and high oscillation frequencies are considered.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Governing equations of envelopes created by nearly bichromatic waves on deep water

In this paper, the author derives the modified Schrödinger equation that governs the envelope created by nearly bichromatic waves, which are defined by the waves whose energy is almost concentrated in two closely approached wavenumbers. The stability of the solution of the modified Schrödinger equation for nearly bichromatic waves on deep water is discussed and the fact that the Benjamin–Feir instability occurs in a condition is shown. Moreover, the solutions of the modified Schrödinger equation for nearly bichromatic waves on deep water are obtained and, in a special case, the solution becomes the standing wave solution is shown.     

Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators

Phase waves rotating in a ring of unidirectionally coupled parametric oscillators are studied. The system has a pair of spatially uniform stable periodic solutions with a phase difference and an unstable quasiperiodic traveling phase wave solution. They are generated from the origin through a period doubling bifurcation and the Neimark?CSacker bifurcation, respectively. In transient states, phase waves rotating in a ring are generated, the duration of which increases exponentially with the number of oscillators (exponential transients). A power law distribution of the duration of randomly generated phase waves and the noise-sustained propagation of phase waves are also shown. These properties of transient phase waves are well described with a kinematical equation for the propagation of wave fronts. Further, the traveling phase wave is stabilized through a pitchfork bifurcation and changes into a standing wave through pinning. These bifurcations and exponential transient rotating waves are also shown in an autonomous system with averaging and a coupled map model, and they agree with each other.     

Wave attenuation in a vessel equipped with damper plates

The effect of vertical damper plates mounted at the center of a rectangular vessel normal or at an angle to a wave flow is experimentally investigated for the first mode of the standing surface waves excited at the parametric resonance. The variation of the resonance curves and the wave attenuation degree are discussed. The fluid depth effect on the wave motion damping is evaluated.     

Nonlinear faraday waves in a parametrically excited circular cylindrical container

IntroductionIn 1 83 1 ,Faraday[1]reportedhisexperimentalobservationofsurfacewavesindifferentfluidscoveringahorizontalplatesubjectedtoaverticalvibration ,andheobservedthesurfacestandingwavesoffluidsliketheteethofaveryshortcoarsecomb .Heremarksthatthesesurfacewaveshaveafrequencyequaltoonehalfthatoftheexcitation .ThisisthefamousFaradayexperiment.WedesignatethosefluidsurfacewavesformedbyverticallyexcitationandhaveafrequencyequaltoonehalfthatoftheexcitationasFaradaywaves.FollowingthisproblemMatth…     

Timoshenko beam theory: A perspective based on the wave-mechanics approach

This paper reviews Timoshenko beam theory from the point of view of wave mechanics. Vibration of beam structures can be studied in terms of either normal modes or propagating waves. The latter wave approach has two distinct features: first, it gives rise to clear physical understanding of beam vibration; second, it leads to exact methods for vibration analysis of beam structures, especially in the mid-frequency range. In this paper, the work on wave solutions of an infinite Timoshenko beam is first discussed. The work on the splitting effect of spinning on wave solutions is also reviewed. The wave is treated as constitutive components of standing waves (i.e. normal modes), and a discussion on how the wave components formulate various standing waves is presented. Finally, several numerical examples are presented to illustrate the pros and cons of using different wave approaches to tackle vibration analysis of finite-length Timoshenko beams.     

驻定斜爆轰波并行数值模拟

采用多组分化学反应Euler方程组对驻定在高速飞行弹丸上的斜爆轰波流场进行了数值模拟。计算中分别采用TVD格式和基元反应模型,并基于并行编程模型MPI（message passing interface）实现了非结构网格上的并行计算,对流项和化学反应项用时间分裂法进行处理。计算结果表明并行计算能有效地提高计算速度,扩展计算规模,为进一步研究超驱爆轰推进技术奠定基础。     

Geometrical nonlinearity stabilizes a wave solid-state gyro

It was recognized long ago that quasi-harmonic standing waves in a thin-walled axisymmetric resonator, mounted on a rotating platform, are subject to a precession. This significant phenomenon is naturally associated with a concept of a solid-state wave gyro, or an inertial instrument used to measure angular rotation rate, as if any wave may be interpreted as a material particle moving in a rotating frame of reference. Because there are no typical mechanical parts, these wave sensors can be utilized with a lot of advantages. To run such a gyro in vita, one should excite and keep on by certain means a standing wave in the thin-walled axisymmetric resonator. Up to now, there are known two ways how to do it, and namely, using either external or parametric resonant mechanisms of excitation. Although both cases necessarily require an additional feedback control device in order to stabilize instable or other parasite oscillations of the resonator. This paper, following the study of nonlinear waves in a thin circular ring, demonstrates that the solid-state wave gyro may be naturally stabilized just at the expense of the geometrical nonlinearity by combining advantages of both the positional resonant excitation and the parametric resonance.     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

Structuring of suspended sediments in periodic vortex flow over a vortex ripple

The formation of spatially ordered structures in a suspended sediment under the action of two-dimensional standing surface gravity waves is studied experimentally for the first time in a rectangular vessel oscillating in the vertical direction. The parameters of the structured regions in vessels with individual vortex ripples and groups of ripples are found for the first and second wave modes. Isolated structured regions of the suspended sediment appear over the bottom topography and gradually reach the free surface. The corresponding spatial horizontal scales are determined by the sand ripple dimensions, while the vertical scale of the clouds increases with time. In all experiments, the structures formed remained unchanged during the whole interval of the fluid wave motion and disappeared when the parametric excitation of the waves stopped.     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.     

Flow of a free jet of rheologically complex fluid from a vibrating capillary

The propagation of finite-amplitude waves along a vibrating capillary jet is studied. The standing wave equation is derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1985.     

Breaking of Faraday waves and jet launch formation

Irregular and breaking Faraday waves are experimentally investigated. Among the irregular waves those with a small depression in the wave crest and periodic triplets are distinguished. In the case of breaking waves the mechanism of jet launch formation on the wave crest is considered. It is experimentally demonstrated that the breaking of standing waves in a rectangular reservoir starts with cavity collapse on the wave crest in process of formation. It is shown that jet launch from the wave crest is preceded by the initiation, development, and collapse of a cavity. A universal power-law dependence governing cavity collapse is obtained. A comparison of the experimental data with an analytical model suggests that cavity initiation is due to the nonlinearity of the waves themselves, namely, the presence of two small disturbances of the free surface traveling counter to one another and forming a cavity. The results obtained underline the importance of the initial stage of wave breaking.     

Breaking and cascade of internal gravity waves in a continuously stratified fluid

The evolution of a few large scale high frequency standing internal waves confined to a vertical plane is studied numerically. The growth of nonlinear interactions leads to a transfer of energy toward small vertical scales and lower frequencies: the result is a steep energy decrease due to wave breaking. Induced mixing is evaluated. A parametric forcing is also introduced in order to compare with laboratory experiments. Wave breaking also occurs but as opposed to the unforced case different phases are next observed: internal wave growth due to constructive forcing alternate with energy decrease.     

Effect of Inertia of Elastic Waves in Elastic Systems with Axial Symmetry

We show that a standing wave excited in an elastic circular ring behaves like a material body: if the moment of external forces directed along the symmetry axis of the ring is applied to the ring, then not only the ring itself but also the initially standing wave excited in it will come to the accelerated rotation. In this motion, this “standing wave” does not change its shape and performs accelerated precession relative to the ring. In this case, the acceleration of the wave with respect to the ring constitutes a certain fraction of the acceleration of the ring relative to the inertial space. The moment of momentum of precessing and traveling waves is calculated.     

Maximum wave steepness and instabilities of finite-amplitude standing gravity waves

We consider the profile of two-dimensional standing gravity waves and their linear stabilities. We have obtained the result that the maximum wave steepness is 0.61. The instability with maximum growth rate is three-dimensional for the wave steepness 0.1. Instabilities due to the four-wave resonance occur but those due to the five-wave resonance are not found for wave steepness smaller than 0.15.