An exact solution of a linearized problem of the radiation of monochromatic internal waves in a viscous fluid

The eigenvalue method is used to construct an exact solution of the linearized boundary-value problem of the generation of internal waves in an exponentially stratified fluid, when the source is part of a plan which vibrates along its surface. The spatial structure of the solution obtained describes two well-known types of wave beams-unimodal and bimodal. In the limiting cases the phase pattern of the waves is identical with well-known asymptotic forms and laboratory experiments. The exact solution is compared with the solution of the model problem of the generation of waves by force sources, constructed using homogeneous fluid theory. The phase patterns of the waves in both cases agree everywhere with the exception of critical angles, when the wave propagates along the radiating surface. The amplitudes of the radiated waves are the same only for certain ratios of the angles of inclination of the plane and the direction of propagation of the beams.     

有流存在下两层密度分层流体毛细重力波的三阶Stokes波解

以小振幅波理论为基础,利用奇异摄动方法研究了有背景流存在下两层密度成层状态下的毛细重力波,求得了两层密度成层状态下各层流体速度势的三阶解及毛细重力波波面位移的三阶Stokes波解,并讨论了毛细重力波的kelvin-Helmholtz的不稳定性.结果表明在有流存在的情况下,两层密度成层流体毛细重力波的一阶渐近解、频散关系,二阶渐近解及三阶渐近解不仅依赖于各层流体的厚度和密度,也依赖于表面张力和各层流体的背景流流场;毛细重力波的三阶解描述了背景流场与毛细重力波之问的三阶非线性相互作用.对于给定的波数k(实数)毛细重力波可能出现kelvin-Helmholtz不稳定性.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Anisotropic transient wave motions in a conducting fluid under uniform magnetic and current fields

An initial value investigation into the development of two-dimensional anisotropic surface waves generated by a harmonically oscillating pressure distribution acting on the undisturbed free surface of an inviscid, incompressible homogeneous and electrically conducting fluid is made in this paper in considerable detail. The problem is solved by the use of generalized function treatment in conjunction with asymptotic methods. An asymptotic solution of the problem related to some physically realistic pressure distributions is presented. It is shown that an ultimate steady state is set up in the limit. Two limiting cases such as (i) very deep fluid and (ii) very shallow fluid, which are of particular interest have been examined with some emphasis. Finally, the effects of the imposed magnetic and current fields as well as the surface tension on the wave motions has been examined in some detail. Additionally, it is shown that the present method of solution provides an interesting example of the applicability of the generalized function method in problems of magnetohydrodynamics     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

Process of establishing a plane-wave system on ice cover over a dipole moving uniformly in an ideal fluid column

We consider a planar evolution problem for perturbations of the ice cover by a dipole starting its uniform rectilinear horizontal motion in a column of an initially stationary fluid. Using asymptotic Fourier analysis, we show that at supercritical velocities, waves of two types form on the water–ice interface. We describe the process of establishing these waves during the dipole motion. We assume that the fluid is ideal and incompressible and its motion is potential. The ice cover is modeled by the Kirchhoff–Love plate.     

Evolution equation of propagation of surface gravity waves in the presence of bottom excitation

An evolution equation that describes the propagation of surface nonlinear dispersive waves in a fluid of finite depth under excitation of a bottom surface is derived. The method of solution is based on the method of power series and asymptotic analysis. On this basis, in a particular case, we investigate the influence of the bottom compliance in the form of a Winkler elastic base and a more general Pasternak base on the transport of wave energy.     

自由表面与粘性尾迹的相互作用

解析地研究了无限深不可压粘性流体中运动物体产生层流尾迹与自由表面波的相互作用．以定常的Oseen方程模拟受扰流动,对于小振幅自由表面波则采用线性化的运动学和动力学边界条件．在数学描述上,运动物体以Oseen极子模拟,受扰流场分解成表述粘性尾迹的无界奇异Oseen流和描述自由面效应的有界正则Oseen流之和．通过积分变换法,得到自由表面波的精确解．借助Lighthill的两步格式,导出了自由面波高带有附加校正项的渐近解．所得对称解显示了波动的振幅因粘性和潜深的存在而呈指数衰减．     

Weakly non-linear high frequency waves for a first order quasi-linear system involving source-like terms

Making use of the method of asymptotic expansion of multiple scales, a study of weakly non-linear, high frequency waves, through “homogeneous” media characterized by dissipative or dispersive hyperbolic systems of partial differential equations, is proposed. Within the present theoretical framework, asymptotic waves in a heat-conducting fluid are considered.     

Weakly non-linear high frequency waves for a first order quasi-linear system involving source-like terms

Making use of the method of asymptotic expansion of multiple scales, a study of weakly non-linear, high frequency waves, through “homogeneous” media characterized by dissipative or dispersive hyperbolic systems of partial differential equations, is proposed. Within the present theoretical framework, asymptotic waves in a heat-conducting fluid are considered.     

Analyticity of the resolvent for elastic waves in a perturbed isotropic half space

In this article the analytic and asymptotic properties of the resolvent for elastic waves in a three dimensional domain perturbed from the isotropic half space R ³₊ are studied. In this case, the asymptotic expansion of the resolvent at the origin has logarithmic terms. This property is totally different from the case of the whole 3‐dimensional space. The existence of the surface waves like the Rayleigh waves makes this difference. As an application of the asymptotic properties of the resolvent, the rate of the local energy decay estimates for the dynamical equations is obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Creeping waves on a highly elongated body of revolution

Creeping waves play an important role in diffraction by a smooth convex body and give an asymptotic of the diffracted field in the shadow. Known results obtained by the boundary-layer method do not allow us to explain some of the properties of creeping waves on highly elongated bodies. In this paper, creeping waves on highly elongated bodies are studied in the case where the binormal curvature of the surface is asymptotically large. The asymptotics derived contains solutions of a differential equation of the Heun type. The analysis of the dispersion equation for the surface waves is carried out numerically. It is discovered that the magnetic creeping wave travels along the surface of a highly elongated body with much less attenuation than predicated by the usual theory. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 22–34. Translated by I. V. Andronov.     

Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions

The asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem on a junction of rectangles: a thin rectangle with a width of ? > 0 and a rectangle with unit dimensions, is studied. In addition to asymptotic formulas for the main series of eigenvalues (in the low-frequency region), other series with stable characteristics are found in the medium-frequency region and explicit formulas for the correction terms are derived. In the framework of the linear theory of surface waves, the results of this work describe the effect of wave localization in shallow water.     

Propagation of Surface Waves at High Frequencies

An asymptotic theory is presented for the analysis of surfacewave propagation at high frequencies. The theory is developedfor scalar surface waves satisfying an impedance boundary conditionon a surface, which may be curved and, whose impedance may bevariable. A surface eikonal equation is derived for the phaseof the surface wave field, and it is shown that the wave fieldpropagates over the surface along the surface rays, which arethe characteristics of the surface eikonal equation. The wavefield in space is found by solving certain eikonal and transportequations with the aid of complex rays. The theory is then appliedto several examples: axial waves on a circular cylinder, sphericallysymmetric waves on a sphere, waves on a circular cone with avariable impedance, and waves on the plane boundary of an inhomogeneousmedium. In each case it is found that the asymptotic expansionof the exact solution agrees with the asymptotic solution.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Waves in an inhomogeneous fluid in the presence of a dock : PMM vol. 39, n 6, 1975, pp. 1140–1142

We investigate the propagation of waves generated by oscillations of a section of the bottom of a tank through a two-layer fluid, in the presence of a dock. Wave motions in an inhomogeneous fluid generated by displacement of a section of the bottom of a tank were studied in [1] where the upper surface of the fluid was assumed either to be completely free, or completely covered with ice. In the present paper we use the method given in [2] to investigate a similar problem under the assumption that the fluid surface is partly covered with an immovable rigid plate. The expressions obtained for the velocity potential are used to determine the form of the free surface and of the interface. We show that when the fluid is inhomogeneous, the wave amplitude on the free surface increases, while the presence of a plate reduces the amplitude of the surface waves, as well as of the internal waves in the region between the plate and the oscillating section of the bottom.     

Local Decay of Acoustic Waves in the Low Mach Number Limits on General Unbounded Domains Under Slip Boundary Conditions

We discuss several aspects of the problem of propagation and dispersion of acoustic waves arising in the low Mach number asymptotic limits of compressible fluid systems. A general approach is proposed based on analysis of the spectral measures associated to the corresponding wave propagator. In particular, the local decay estimates based on a result of Tosio Kato and on RAGE theorem are obtained as limit cases. The approach is applied to problems on domains their shape may vary with the Mach number.     

Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous,Time Independent Medium

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves for a single space variable in a homogeneous, time independent medium. This theory extends the results previously presented by A. Majda and R. Rosales, under similar hypotheses, to the case where waves break and shocks form. Similarly the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable, is included as a special case. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 ? 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. These are the same equations derived by Majda and Rosales previously. However, the waves are displaced relative to the positions prescribed by them.     

Exact solutions of the linear problem of the steady-state waves created by a dipole in a flow of a stratified fluid

The problem of the steady-state waves which are formed when there is uniform flow of a non-viscous, incompressible, vertically stratified fluid round a dipole is considered in a linear formulation. Using the analytical properties of the solutions, two formulae are obtained for the vertical displacement field in the form of series of single integrals taken over the spectral curves. These formulae are simpler than those which have been previously proposed /1/ since the integrands do not contain special functions with logarithmic singularities and enable one to simplify the numerical analysis of the close domain of the wave field in which the asymptotic forms /2–4/ are applicable /5/.