An electromagnetic shear-stress gage for large-amplitude shear waves

In many materials, especially plastics, ceramics and rocks, large-amplitude shear-wave propagation studies could provide valuable information for the development of constitutive equations. A newly developed electromagnetic-gage configuration provides an output voltage which is directly related to the dynamic shear stress in the material. The electromagnetic shear-stress gage has been used to make direct measurements of shear-wave stresses in PMMA and Solenhofen limestone. Large-amplitude shear waves were obtained with a new plate-impact technique which generates shear waves by a controlled-reflection process. The configuration of the stress gage permits it to be used simultaneously with more conventional electromagnetic velocity gages, thus providing both types of data in one experiment.     

Damping of plasma waves

The mechanism of formation of a plateau of the electron velocity distribution function in monochromatic. plasma wave damping is discussed. It is shown that the distribution function is subject to strong modulation in a neighborhood of the phase velocity of the wave and that the steady state is established as a result of collisions. The collisionless damping obtained in the linear approximation in [1] is caused by resonance particles and depends on the electron velocity distribution function in the region

$$\upsilon _p - \sqrt {{{2e\Phi _0 } \mathord{\left/ {\vphantom {{2e\Phi _0 } m}} \right. \kern-\nulldelimiterspace} m}} \lesssim \upsilon _{p^| } \lesssim \upsilon + \sqrt {{{2e\Phi _0 } \mathord{\left/ {\vphantom {{2e\Phi _0 } m}} \right. \kern-\nulldelimiterspace} m}}$$     

Damping of drift waves

The damping of a monochromatic wave propagating at an angle to the magnetic field in an inhomogeneous plasma is studied. The nonlinear equations for resonance particles are solved in the drift approximation. The nonlinear damping decrement is calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp 53–55, January–February, 1970.The author is grateful to R. Z. Sagdeev for his interest in the work.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

Pseudolocalized three-dimensional solitary waves as quasi-particles

A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass (“wave mass”) of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that, at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics.     

Weak oblique collisions of interfacial solitary waves

A third order perturbation solution is developed to describe the interaction between two solitary waves approaching each other at an angle close to 180 ° on the interface between two immiscible inviscid homogeneous fluids. The solution is steady in the frame of reference moving with the point of intersection of the waves. To lowest order, the solution consists simply of the superposition of the undisturbed solitary waves. Second-order collision effects include interaction terms localized near the point of intersection and a phase shift in the solitary waves. In addition to corrections to the phase shift and localized interaction terms, third order effects are found to include a wave train that is stationary in the frame of reference moving with the point of intersection of the solitary waves. The amplitudes of the wave train and localized interaction terms diminish with distance from the point of intersection, and the solitary waves recover their initial shape asymptotically long after the collision. Thus, the only long-term effect of the collision is a phase shift.     

The stability of finite-amplitude interfacial solitary waves

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.     

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

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.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Correlation between theoretical and experimental solitary waves

Experimental data on surface solitary waves generated by five methods are given. These data and literature information show that at amplitudes 0.2<a/h<0.6 (h is the initial depth of the liquid), experimental solitary waves are in good agreement with their theoretical analogs obtained using the complete model of liquid potential flow. Some discrepancy is observed in the range of small amplitudes. The reasons why free solitary waves of theoretically limiting amplitude have not been realized in experiments are discussed, and an example of a forced wave of nearly limiting amplitude is given. The previously established fact that during evolution from the state of rest, undular waves break when the propagation speed of their leading front reaches the limiting speed of propagation of a solitary wave is confirmed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 44–52, May–June, 1999.     

Cubic non-linearity and longitudinal surface solitary waves

It is shown that for some seismic media both quadratic and cubic non-linearities should be taken into account in the governing equation for longitudinal waves. The new equation is obtained to account for non-linear surface waves in a medium surrounding a non-linearly elastic rod. Exact solutions of the equation allow us to describe simultaneous propagation of tensile and compressive localized strain waves. Various interactions between these waves give rise to both the multi-bump and “Mexican hat” localized wave structures closer to the surface waves recently observed in experiments.     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

Finite-amplitude solitary waves in a two-layer fluid

Second-mode nonlinear internal waves at a thin interface between homogeneous layers of immiscible fluids of different densities have been studied theoretically and experimentally. A mathematical model is proposed to describe the generation, interaction, and decay of solitary internal waves which arise during intrusion of a fluid with intermediate density into the interlayer. An exact solution which specifies the shape of solitary waves symmetric about the unperturbed interface is constructed, and the limiting transition for finite-amplitude waves at the interlayer thickness vanishing is substantiated. The fine structure of the flow in the vicinity of a solitary wave and its effect on horizontal mass transfer during propagation of short intrusions have been studied experimentally. It is shown that, with friction at the interfaces taken into account, the mathematical model adequately describes the variation in the phase and amplitude characteristics of solitary waves during their propagation.     

Interactions of atmospheric solitary waves of different modes

The interactions of atmospheric solitary waves with different modes are investigated by a perturbation method. The model considered in this paper consists of a lower layer with exponential density profile and an infinitely deep upper layer with constant density. The analysis show that the waves obey the Benjamin-Ono equation before and after interaction, and the main effect of the interaction is the phase shifts for each wave. The project supported by the National and Shanghai Education Commission of Science Foundation     

Instability of solitary waves for generalized Boussinesq equations

An equation of Boussinesq-type of the formu _tt -u _xx +(f(u)+u_xx)_xx=0 is considered. It is shown that a traveling wave may be stable or unstable, depending on the range of the wave's speed of propagation and on the nonlinearity. Sharp conditions to that effect are given.This research is supported in part by NSF Grant DMS 90-23864.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.