Parametric excitation of waves on a liquid surface in the presence of an insoluble film

The presence of a film of surface-active agents leads to a change of the force acting on the surface of a liquid. This change does not lead to a simple decrease of the surface tension , and it is connected with the appearance of tangential forces acting on the free surface of the fluid [1]. The stability of the free surface of a liquid with a film of a surface-active agent in a variable gravitational field is examined. The linear formulation of the problem is solved. A solution is sought in the form of a series in powers of the small viscosity by the method of Laplace transforms in time and Fourier transforms in the x and y variables (the xy-plane coincides with the undisturbed liquid surface). An integrodifferential equation of the second-order with periodic coefficients is derived for the displacement of the surface from the equilibrium position. The solution is found by the method of averaging [2]. It is shown that the excitation energy should not be less than the energy dissipated in the system. It is shown that the presence of the film substantially increases the threshold of the instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–162, September–October, 1976.The authors thank G. Z. Gershuna for calling their attention to this problem.     

Parametric excitation of waves at an interface

Experiments on the parametric excitation of waves at a fluid interface show a strong disagreement with theoretical results [1–3], since the latter do not take into account the influence of the second medium. This proves to be especially important at low frequencies. Thus, for a water-air interface with an excitation frequency = 60 sec^–1 the contribution amounts to 10%,and with = 30 sec^–1, even 20%. In this paper the stability of the interface of two viscous, incompressible fluids of finite depth in a variable gravity field is considered. The problem is put in the linear form by making an expansion with respect to the small viscosity and is solved by taking the Laplace transform with respect to time. A second-order integrodifferential equation with periodic coefficients is obtained for the deviation of the interface from the equilibrium position; its solution is sought by the method of averaging [4]. It is shown that the presence of the second fluid significantly raises the threshold of instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1977.     

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.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

A note on the generation of surface waves by inviscid shear flows

The propagation of finite-amplitude wave disturbances in a parallel shear flow over a flexible plane boundary is studied. It is assumed that the nonlinear effects dominate over the viscous effects in the shear-flow critical layer, so that no jump of the Reynolds stress exists across the critical layer and the wave-generation mechanism of Miles cannot be operative. It is demonstrated that waves of certain wavelengths can amplify, if direct-resonance conditions are met. Explicit results are presented for a boundary layer modelling the flow of air over water and an interpretation of the wave-growth mechanism is given.     

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distance- to-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the freesurface gravity waves can readily be recovered by the present results as the inertial surface disappears.     

Nonlinear waves on the free surface of an electrically conducting liquid

Gravity waves of small but finite amplitude on the free surface of an electrically conducting liquid are examined. For the waves whose propagation is described by the Korteweg-de Vries equation (in the absence of a magnetic field), equations are derived. In addition to nonlinearity and dispersion, these equations include the influence of applied magnetic fields. As an example, the effect of magnetic damping on a solitary wave is presented.     

Parametric instability of gravity waves on the surface of deep water

We propose to investigate the characteristics of the parametric generation of gravity waves on the surface of a body of deep water. The threshold conditions for the onset of generation are determined, and the results are compared with the experimental data. The singularities of the excitation of parametric oscillations in a resonator are noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 182–185, May–June, 1972.The authors are grateful to V. N. Pshenichnikov for assisting with the experiments.     

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.     

Parametric excitation of waves on a free boundary of a horizontal fluid layer

We consider the dynamical stability of horizontal fluid layer, performing harmonic oscillations in vertical direction. The continued fractions approach allowed us to avoid the conventional restriction to the case of small viscosity and almost-resonant frequencies. Our numerical results cover a wide range of the parameters (viscosity, amplitude and frequency of the oscillation, and depth of the layer). To cite this article: V.I. Yudovich et al., C. R. Mecanique 332 (2004).     

Threshold of parametric excitation of waves on a fluid surface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 61–67, July–August, 1988.     

Chaos of liquid surface waves in a vessel under vertical excitation with slowly modulated amplitude

Free surface waves in a cylinder of liquid under vertical excitation with slowly modulated amplitude are investigated in the current paper. It is shown by both theoretical analysis and numerical simulation that chaos may occur even for a single mode with modulation which can be used to explain Gollub and Meyer's experiment. The implied resonant mechanism accounting for this phenomenon is further elucidated.     

Non-linear oscillations of an inviscid liquid column under zero-gravity

Summary For a frictionless cylindrical liquid bridge of finite length in zero-gravity the non-linear natural frequencies are determined for large free liquid surface amplitudes. The non-linear boundary conditions have been expanded up to terms of third order. It was found that the system exhibits a softening oscillation behavior, indicating that with increasing wave amplitudes the natural frequencies exhibit decreased magnitude as compared to linear theory. In addition higher modes and shorter liquid bridges show increased softening character.

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Nichtlineare Schwingungen einer reibungsfreien Flüssigkeitssäule unter Schwerelosigkeit

Übersicht Das nichtlineare Verhalten der Oberflächenschwingungen einer reibungsfreien Flüssigkeitssäule endlicher Länge wurde im schwerelosen Raum untersucht. Die nichtlinearen Randbedingungen wurden in Reihen entwickelt, deren Glieder bis zur dritten Ordnung in der Auswertung berücksichtigt wurden. Dabei wurde die Oberflächenauslenkung als Funktion der Frequenz bestimmt und unterlineares Verhalten gefunden. Mit zunehmender Flüssigkeitsamplitude verringerten sich die Eigenfrequenzen im Vergleich zur linearen Theorie. Höhere Eigenfrequenzen und Säulen kleiner Länge zeigen wesentlich stärkere Unterlinearitäten.

     

Statistical fluctuations of the charged surface of an inviscid fluid

A fluctuation mechanism of origin of the deformation of a charged liquid surface, which explains the experimental facts, is proposed. The Hamilton equations that describe the dynamics of an inviscid fluid with electrically charged free surface are formulated. Solutions of these equations, which describe the evolution of the free surface, are found. The spectra of the surface disturbances are investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 117–124, May–June, 1992.     

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.     

Progressive surface waves on a liquid of non-uniform depth

The propagation of harmonic progressive waves on the surface of a homogeneous liquid of non-uniform depth is considered. Particular attention is devoted to the action of the Reynolds stresses in generating a mean or time-independent component in the flow.