Two-dimensional problem for internal waves generated by moving singular sources

When bodies move in a liquid with inhomogeneous density in a gravitational field waves are excited even at low velocities and in the absence of boundaries. They are the so-called internal waves (buoyancy waves), which play an important part in geophysical processes in the ocean and the atmosphere [1–4]. A method based on the replacement of the bodies by systems of point sources is now commonly used to calculate the fields of internal waves generated by moving bodies. However, even so the problems of the generation of waves by a point source and dipole are usually solved approximately or numerically [5–11]. In the present paper, we obtain exact results on the spectral distribution of the emitted waves and the total radiation energy per unit time for some of the simplest sources in the two-dimensional case for an incompressible fluid with exponential density stratification. The wave resistance is obtained simply by dividing the energy loss per unit time by the velocity of the source. In the final section, some results for the three-dimensional case are briefly formulated for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–83, March–April, 1981.     

Internal waves arising in the presence of the unsteady motion of a source in a continuously stratified fluid

A general approach to the investigation of linear internal and surface waves in a stably stratified fluid, arising from different types of perturbations, is presented in [1]. The methods of calculation of the internal waves generated by the motion of a mass source are developed for particular cases of steady horizontal motion in a continuously stratified fluid in [2] and arbitrary motion in an unbounded exponentially stratified fluid in [3]. Internal waves generated by other types of perturbations have also been investigated but only for particular cases of motion (see, for example, [4]). This paper presents a method for the calculation of unsteady, linear, internal gravity waves arising in an inviscid incompressible fluid with continuous stable stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1985.     

Parametric excitation of internal waves in a continuously stratified liquid

A study is made of the parametric excitation of internal waves in a continuously stratified liquid in a vessel executing oscillations in the vertical direction. It is shown that vertical oscillations of the vessel will excite oscillatory modes with eigenfrequencies equal to half of the oscillation frequency of the vessel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–169, September–October, 1982.I thank S. V. Nesterov for interest in the work.     

Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid

Beams of harmonic internal waves in a liquid with smoothly changing stratification are calculated in the Boussinesq approximation taking into account the effects of diffusion and viscosity. A procedure of local reduction of the beam in a medium with an arbitrary smooth stratification to the case of an exponentially stratified liquid is constructed. The coefficient of energy losses in the case of beam reflection on the critical level is calculated. Parameters of internal boundary flows with split scales of velocity and density that are formed by a wave beam on discontinuities of the buoyancy frequency and its higher derivatives are determined. Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 88–98, September–October, 1998.     

Propagation of solitary waves over underwater ridges

The propagation of solitary waves is investigated on the basis of a nonlinear system of equations of hyperbolic type describing the motion of the crest of a solitary wave over the surface of a liquid of variable depth [1]. The existence of solutions with discontinuities, the boundary conditions at which are introduced on the basis of [2, 3], is assumed. In the case of an infinite cylindrical ridge both solitary and periodic captured waves are found. Depending upon the height of the ridge and the parameters of the wave, the encounter between a uniform wave and a semi-infinite ridge yields qualitatively different solutions — continuous and discontinuous, where the primary wave is broken down by the ridge into several solitary waves. The amplitude of the wave may either increase or decrease over the ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–93, January–February, 1985.The author is grateful to A. G. Kulikovskii and A. A. Barmin for their interest in his work, useful discussions and valuable comments offered during the preparation of the article for the press.     

Influence of a magnetic field on the thermoacoustic stability of a high-temperature heat releasing gas

The conditions of thermoacoustic stability are found for a high-temperature electrically conducting gas with internal heat release in a constant magnetic field which transforms acoustic waves into fast and slow magnetoacoustic oscillations, and also introduces Joule dissipation. The investigation is by means of the energy balance method, and also by direct solution of the equations for small perturbations in the special case of wavelengths of the acoustic oscillations that are short compared with the inhomogeneity scales in the region of heat release. The limits of stability with respect to fast and slow magnetoacoustic oscillations are found.Translated from Izvestiya Akademii Nauk SSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1979.     

Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 55–62, November–December, 1973.     

Interaction of wave disturbances in an oscillating boundary layer

A theoretical analysis is made of the interaction of wave disturbances of small finite amplitude in a boundary layer in the case when the velocity distribution contains a periodic component that oscillates in time in accordance with a harmonic law. It is shown that it is in principle possible for there to be a four-wave synchronous (resonance) interaction in a cubic nonlinearity; equations are obtained for the amplitudes. Calculations made to test the effectiveness of the resonance phenomena have shown that the coupling coefficients are not sufficiently large for the superimposed oscillations to change significantly the nature of the interaction of the waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 154–158, September–October, 1980.We thank A. G. Volodin for assistance in the calculations.     

非线性海洋内波的理论、模型与计算

海水因盐度与温度的垂向差异造成密度层结现象, 进而由于海洋系统的内部扰动(如海潮流过局部隆起的海底地形)与外部扰动(如死水现象)造成等密面的波动, 这一现象称为“内波”. 内波在全球范围内大量存在, 尤其是在海峡入海口等密度层结现象较为明显和稳定的区域会有内波频繁活动. 海洋通常呈现“三明治”状的结构: 密度相对稳定的混合层与深水层, 以及位于中间密度连续过渡的密跃层. 密跃层的整体脉动对于海洋工程和海洋生态环境有重大的影响; 而密跃层内部的波动对于潜艇的非声探测(反过来说, 对于潜艇的隐身作战)具有潜在的应用价值. 而造成这些重大影响的根源在于内波在水平和垂直方向都具备传播能力, 这是有别于海洋表面波浪的关键之处.本文针对两类海洋密度模型-连续分层模型与间断分层模型, 从理论研究、数值模拟、实验室机理实验等方面论述了研究海洋内波的各类非线性模型(包括弱非线性的Korteweg-de Vries方程、Benjamin-Ono方程, Kadomtsev-Petviashvili方程等著名模型以及强非线性Miyata-Choi-Camassa方程、非线性势流理论、带密度变化的不可压缩Navier-Stokes方程等), 讨论各自的适用范围, 并借此探讨内波在海洋质量动量能量输运中所起的至关重要的作用.      

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Experimental study of surface waves with Faraday resonance excitation

The results of an investigation of standing two-dimensional gravity waves on the free surface of a homogeneous liquid, induced by the vertical oscillations of a rectangular vessel under Faraday resonance conditions, are presented. The frequency ranges of excitation are determined and resonance relationships for the second and third modes are obtained and analyzed. Nonlinearities of the waves generated, such as wave profile asymmetry and node oscillations, are evaluated. Wave breakdown and the onset of unstable oscillation modes are considered. Experimental results are compared with the theoretical data.The experimental studies [1–4], devoted to Faraday resonance, deal mainly with the conditions under which resonance arises and the frequency response.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–129, January–February, 1995.     

Generation of higher electron cyclotron frequency harmonics in plasma with a beam

We examine nonlinear excitation of the higher electron-cyclotron frequency harmonics for waves propagating perpendicular to an external uniform magnetic field in a Maxwell plasma for the case of low-density electron beam passage through the plasma. It is shown that the nonlinear excitation mechanism leads to the possibility of generating cyclotron harmonics for plasma parameters for which generation does not occur from the linear theory viewpoint. The nonlinear cyclotron harmonic generation increments are calculated for nonlinear scattering by the beam and plasma electrons of the high frequency longitudinal waves excited in the plasma by the beam.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 40–51, November–December, 1969.The author wishes to thank V. N. Tsytovich for posing the problem and for many discussions of the questions touched upon in the article.     

Internal Travelling Waves in the Limit of a Discontinuously Stratified Fluid

We consider internal travelling waves in a perfect stratified fluid, in the singular limit case when smooth stratifications approach a discontinuous two-layer profile. Our analysis concerns two-dimensional waves of small amplitude, propagating in an infinite horizontal strip of finite depth. The problems with smooth or discontinuous stratification are formulated as a unifying spatial evolution problem, where the stratification ρ plays the role of a functional parameter. The vector field is not smooth with respect to ρ, but has some weak continuity. When the Froude number is close to a critical value, we reduce the problem to one on a center manifold in a neighborhood of the trivial state independent of ρ (for the usual topology). Considering a weaker topology, we prove the continuity in ρ of the center manifold. Then the small solutions are described by an ordinary differential equation in ?², which depends continuously on ρ in the C ^k norm.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Stable nonlinear waves in a poiseuille flow

Nonlinear Tolmin-Schlichting waves are studied [1–8]. The investigation is carried out by means of a modified Stuart-Watson method [1–3]. In the case of a rigid regime of excitation terms to the fifth order are taken into account in expansions with respect to the amplitude of self-excited oscillations. The stability of self-excited oscillations with respect to two- and three-dimensional disturbances is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 40–45, September–October, 1978.The author thanks S. Ya. Gertsenshtein for attention to the work and discussion of the results.     

Effect of acoustic oscillations on the stability of a plane jet

The problem of the effect of acoustic oscillations on the stability of a compressible ideal-fluid jet flow is examined in the case of a plane jet with standing acoustic waves superimposed across it. The method of dividing the motion into fast and slow with allowance for nonlinear acoustic effects is employed. The acoustic oscillations are found to affect the growth rate of unstable hydrodynamic disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 54–60, July–August, 1991.     

Approximate method of solving the problem of characteristic oscillations of a liquid in cavities of revolution

The problem of the characteristic oscillations of a liquid in axisymmetric cavities of rotation has been fairly fully studied [1–5], its solution in the general case being found by the variational method. Analysis of numerical results using the variational method shows that to achieve acceptable accuracy it is necessary to retain an appreciable number of coordinate functions, which entails the solution of a matrix eigenvalue problem of high order, this applying especially to the case when it is necessary to determine several eigenfrequencies and the shapes of the oscillations. In the present paper, a method proposed earlier by Shmakov [6] is developed, the velocity potential being sought in the form of a sun of two potentials. The first (base) potential is a solution to the problem of the characteristic oscillations of a liquid in a cavity whose free surface coincides with the free surface of the original cavity, and the second (correcting) potential is chosen in the form of a system of harmonic functions, this system being complete and orthogonal on the wetted surface of the cavity. Cavities of revolution are analyzed as examples, and a detailed investigation of numerical results is made for a spherical cavity. The numerical analysis shows that a sufficiently accurate result in the determination of a frequency is obtained if one term of the base problem is retained and only the correcting potential is used to make this more accurate. As a result, it is only necessary to solve an algebraic equation of first degree in the square of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–8, September–October, 1983.     

Effect of periodic bottom roughness on gravitational waves in a liquid

The effect of a rigid bottom of periodic form on small periodic oscillations of the free surface of a liquid is considered with the assumption of low amplitude roughness. The methodologically most significant study in this direction, [1], will be utilized. In [1] the steady-state problem for flow over an arbitrarily rough bottom was studied. Other studies have recently appeared on small free oscillations above a rough bottom. Essentially these have considered the effect of underwater obstacles and cavities on surface waves in the shallow-water approximation (for example, [2], [3]). Liquid oscillations in a layer of arbitrary depth slowly varying with length were considered in [4]. However, these results cannot be applied to the study of resonant interaction of gravitational waves with a periodically curved bottom.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 43–48, July–August, 1984.     

Development of three-dimensional disturbances in a boundary layer with pressure gradient

The results of an experimental investigation of the three-dimensional stability of a boundary layer with a pressure gradient are presented. A low-turbulence subsonic wind tunnel was employed. The development of a three-dimensional wave packet of oscillations harmonic in time in the boundary layer on a model wing is studied. The amplitudephase distributions of the pulsations in the wave packet are subjected to a Fourier analysis. Spectral (with respect to the wave numbers) decomposition of the oscillations enables the flow stability with respect to plane waves with different directions of propagation to be examined. The results are compared with the corresponding data obtained in flat plate experiments. The effect of the pressure gradient on the development of the three-dimensional spectral components of the disturbances and the dispersion properties of the flow is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 85–91, May–June, 1988.