Dispersion of interface waves in sediments with power-law shear speed profiles. II. Experimental observations and seismo-acoustic inversions.

The propagation of seismic interface waves is investigated in soft marine sediments in which the density is constant, the shear modulus is small, and the profile of shear speed c(s) versus depth z is of the power-law form c(s) (z) = c0z(v), in which c0 and v are constants (0< v < 1). Both the phase speed V and the group speed U of interface waves scale with frequency as f(v/(v -1)) and they obey the simple relation U= (1 - v) V. These relations are derived in a simple way using ray theory and the WKB method; a companion paper [O. A. Godin and D. M. F. Chapman, J. Acoust. Soc. Am. 110, 1890 (2001)] rigorously derives the same result from the solutions to the equations of motion. The frequency scaling is shown to exist in experimental data sets of interface wave phase speed and group speed. Approximate analytical formulas for the dispersion relations (phase and group speed versus frequency) enable direct inversion of the profile parameters c0 and v from the experimental data. In cases for which there is multi-mode dispersion data, the water-sediment density ratio can be determined as well. The theory applies to vertically polarized (P-SV) modes as well as to horizontally polarized (SH) modes (that is, Love waves).     

Exact and approximate results of non-extensive quantum statistics

We develop an analytical technique to derive explicit forms of thermodynamical quantities within the asymptotic approach to non-extensive quantum distribution functions. Using it, we find an expression for the number of particles in a boson system which we compare with other approximate scheme (i.e. factorization approach), and with the recently obtained exact result. To do this, we investigate the predictions on Bose-Einstein condensation and the blackbody radiation. We find that both approximation techniques give results similar to (up to ) the exact ones, making them a useful tool for computations. Because of the simplicity of the factorization approach formulae, it appears that this is the easiest way to handle with physical systems which might exhibit slight deviations from extensivity. Received 19 August 1999 and Received in final form 1 November 1999     

Classical charged fluids at equilibrium near an interface: Exact analytical density profiles and surface tension

The structure of equilibrium density profiles in an electrolyte in the vicinity of an interface with an insulating or conductive medium is of crucial importance in chemical physics and colloidal science. The Coulomb interaction is responsible for screening effects, and in dilute solutions the latter effects give rise to universal leading corrections to nonideality, which distinguish electrolyte from nonelectrolyte solutions. An example is provided by the excess surface tension for an air-water interface, which is determined by the excess particle density, and which was first calculated by Onsager and Samaras. Because of the discrepancy between the dielectric constants on both sides of the interface, every charge in the electrolyte interacts with an electrostatic image, and the Boltzmann factor associated with the corresponding self-energy has an essential singularity over the length scalel from the wall. Besides Coulomb interactions, short-range repulsions must be taken into account in order to prevent the collapse between charges with opposite signs or between each charge and its image when the solvent dielectric constant is lower than that of the continuous medium on the other side of the interface. For a dilute and weaklycoupled electrolyte,l is negligible with respect to the bulk Debye screening length ξ_D. In the framework of the grand-canonical ensemble, systematic partial resummations in Mayer diagrammatics allow one to exhibit that, in this regime, the exact density profiles at leading order are the same as if they were calculated in a partially-linearized mean-field theory, where the screened pair interaction obeys an inhomogeneous Debye equation. In the latter equation the effective screening length depends on the distancex from the interface: it varies very fast over the lengthl and tends to its bulk value over a few ξDs. The equation can be solved iteratively at any distancex, and the exact density profiles are calculated analytically up to first order in the coupling parameter l/ξ_D. They show the interplay between three effects: (1) the geometric repulsion from the interface associated with the deformation of screening clouds, (2) the polarization effects described by the images on the other side of the interface, (3) the interaction between each charge and the potential drop created by the electric layer which appears as soon as the fluid has not a charge-symmetric composition. Moreover, the expressions allow us to go beyond Onsager-Samaras theory: the surface tension is calculated for charge-asymmetric electrolytes and for any value of the ratio between the dielectric constants on both sides of the interface. Similar diagrammatic techniques also allow one to investigate the charge renormalization in the dipolar effective pair interaction along the interface with an insulating medium.     

Numerical and analytical calculations of the parameters of power-law spectra for deep water gravity waves

We determine the asymptotic behavior of the coupling coefficient for four-wave interactions of gravity waves in deep water in the limiting case when two wave vectors of interacting waves are small with respect to the other two (“long–short interactions”). It makes possible to find numerically dimensionless Kolmogorov constants for the power-law Kolmogorov–Zakharov spectra. The results obtained are crucially important for comparison of the weak turbulent theory with the experiments and natural observations.     

Exact and approximate analysis of surface acoustic waves in an infinite elastic plate with a thin metal layer

For an accurate approximation of the effect of a thin layer over a finite substrate, we consider the displacements are continuous across the interface, while the stress components are obtained from derivatives of displacements. As a result, the stress boundary conditions are transformed to a relationship between stresses in the vicinity of the interface of two layers and density of the metal layer. Through this approximation, we eventually have four equations to solve for the surface acoustic wave dispersion relation of the two-layer structure. The approximate and accurate results are compared with good agreement for small thickness of the metal layer. These results are intended for periodic structures with separate solutions for electroded and unelectroded regions, which can be connected by the continuity boundary conditions for the analysis of the complete structure of typical surface acoustic wave resonators.     

Exact results in modeling planetary atmospheres—I. Gray atmospheres

An exact model is proposed for a gray, isotropically scattering planetary atmosphere in radiative equilibrium. The slab is illuminated on one side by a collimated beam and is bounded on the other side by an emitting and partially reflecting ground. We provide expressions for the incident and reflected fluxes on both boundary surfaces, as well as the temperature of the ground and the temperature distribution in the atmosphere, assuming the latter to be in local thermodynamic equilibrium. Tables and curves of the temperature distribution are included for various values of the optical thickness. Finally, semi-infinite atmospheres illuminated from the outside or by sources at infinity is dealt with.     

The surface shear viscosity of a planar liquid-vapor interface I. Theory

In this paper we develop a theory for the calculation of the surface shear viscosity of a planar liquid-vapor interface. The theory is an extension of the generalized hydrodynamics formalism, originally developed for the calculation of linear transport coefficients in isotropic bulk fluids. We develop an expression for the surface shear viscosity in terms of the actual shear viscosity profile in the interfacial region. We derive an expression for this profile in terms of the first four moments of the autocorrelation function of the transverse parallel velocity (the component of velocity parallel to k, which is the projection of k on to the interface). Finally, we calculate these moments for a planar liquid-vapor interface.     

Self-filtering unstable resonator: An approximate analytical model with comparison to computed and XeCl laser experimental results

Approximate analytical solutions for the lowest-order transverse mode of empty self-filtering unstable resonator have been deduced for magnification M 》 1, and compared with some experimental measurements on an X-ray preionized XeCl discharge laser.     

Dynamic stress around a cylindrical nano-inhomogeneity with an interface in a half-plane under anti-plane shear waves

Taking into account the size of the nanostructure, the effect of surface/interface stiffness on the dynamic stress around a cylindrical nano-inhomogeneity embedded in an elastic half-plane subjected to anti-plane shear waves is investigated. The boundary condition at the straight edge of the half-plane is traction free, which is satisfied by the image method. The analytical solutions of displacement fields are expressed by employing a wave function expansion method. The addition theorem for a cylindrical wave function is applied to accomplish the superposition of wave fields in the two half-planes. Analyses show that the effect of the interface properties on the dynamic stress is significantly related to the nano-scale distance between the straight edge and the center of the cylindrical nano-inhomogeneity. The frequency and incident angle of incident waves and the shear modulus ratio of the nano-inhomogeneity to matrix also show different effect on the dynamic stress distribution when the inhomogeneity shrinks to nano-scale. Comparison with the existing results is also given.     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. I. Exact results

A unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented. The well-known results of Mori and Zwanzig are recovered as special cases. Furthermore, it is shown that they differ only by the way in which the system is prepared at the initial time. Connections between this method and others recently developed are also discussed.     

Exact and approximate solutions of Riemann problems in non-linear elasticity

Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented.     

The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states

We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.     

Transient displacement induced in shear wave elastography: comparison between analytical results and ultrasound measurements

It is now accepted that an effective way to investigate the elastic properties of soft tissues is to generate a localized transient acoustic radiation force and to follow the associated displacements in the time/space domain. Shear waves induced by this stress field are particularly interesting in this kind of medium because they are governed by the shear elastic modulus mu, which is directly linked to the Young modulus, and spatial distribution and temporal evolution of the transient motion induced must therefore be obtained in detail. We report here a model based on the elastodynamic Green's function formalism to describe these displacements. 3D simulation of radiation force in homogenous elastic media was performed and the displacement curves computed at different radial distances for different temporal force profiles. Amplitude and duration of displacement were found to be reliable parameters to characterize the elastic properties of the medium. Experimental measurements were performed in a homogeneous agar-gelatin tissue-mimicking phantom, and two transducers were used to generate the radiation force and follow the induced displacements. Displacements obtained from different lateral locations around the applied force axis were then used to reconstruct the shear-wave propagation in a scan plane as a function of time. The experimental displacements/curves agreed with the theoretical profiles obtained by the elastodynamic Green's function formalism.     

A theoretical approach to local attenuation measurements of shear waves by MRE. Preliminary experimental results

An essential highlight of the presented method is the employment of Magnetic Resonance Elastography (MRE) for local measurements of the attenuation of elastic shear waves introduced into a biological sample. Such a measurement can be accomplished by combining the MRE method with those methods, in which collective displacement of spins is induced by external physical factors, such as variable electric field, strong magnetic field gradient or longitudinal elastic wave. A theoretical basis of the method involving external factors and results of preliminary experiments have been presented in this paper.     

Dispersion and attenuation of surface shear acoustic waves with horizontal polarization on the free statistically rough surface of the hexagonal crystal

Expressions for dispersion of the phase velocity and inverse damping depth of surface acoustic waves with shear horizontal polarization are derived in an analytical form within perturbation theory using the modified mean-field method for the Z-cut hexagonal crystal with a free statically rough surface. Both two-and one-dimensionally rough surfaces are considered. The one-dimensionally rough surface is considered as a special case of the two-dimensionally rough surface. It is shown that shear surface waves with horizontal polarization cannot exist on the flat surface of the Z-cut hexagonal crystal. The derived expressions are studied analytically and numerically in the entire frequency range accessible in perturbation theory. The long-wavelength limit (most interesting from the experimental point of view) is considered, where the wavelength is much longer than the roughness correlation radius. The conditions for the existence of SH-polarized waves are determined for both roughness types. It is shown that dispersion and attenuation of SH polarized waves are qualitatively similar in character to those we considered previously for an isotropic medium.     

Anisotropy measured with shear and Rayleigh waves in rolled plates

The Rayleigh wave velocity is derived for propagation on polycrystalline metal plates with weakly orthorhombic anisotropy modelling the rolling texture. Use is made of the texture weakness to solve the velocity in a perturbation scheme. For the fully orthorhombic case, there was found no simple relation between the velocity difference of Rayleigh waves propagating along the principal axes of texture and the birefringence of shear waves propagated in the thickness direction. However, for the cubic metals the result is reduced to their linear relation to the substitution of crystallite orientation distribution function expanded with spherical harmonics. This relation has been obtained by Sayers. The application of texture-independent measurement of principal stress difference is discussed.     

Computed epitaxial monolayer structures: I. One-dimensional model: Comparison of computed and analytical results

An exact computational procedure is developed which, when applied to Frank and van der Merwe's one-dimensional dislocation model, yields the equilibrium structures of a one-dimensional atom chain with elastic interatomic forces and sinusoidal substrate potential. This forms the first part of the development of a general procedure for calculating the equilibrium structures of two-dimensional monolayers with elastic interatomic forces and substrate potentials of more complex character. The minimum energy principle is applied to distinguish the stable structures from the computed equilibrium ones. The results aie in close agreement with the analytical approximate solutions of Frank and van der Merwe. The atomic displacements and limiting misfits agree almost perfectly. The curves of lowest energy against misfit are likewise in excellent agreement for long chains. For relatively short chains, containing only few dislocations, the curves are composed of segments, one segment for each additional dislocation. The discreteness of dislocations in finite chains is also borne out in other properties. The calculations further show (i) that, for a finite chain with odd numbers of dislocations and atoms, the stable configuration is one with the central atom on a potential crest and (ii) that the cusped minima present in the interfacial energy curve of thick bicrystals disappear when complete or partial accommodation of lattice parameters is energetically possible as in thin films.     

Enhanced Compression of Fundamental Solitons in Dispersion Decreasing Fibers with Negative Third-order Dispersion I.Basic Principles

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Enhanced Compression of Fundamental Solitons in Dispersion Decreasing Fibers with Negative Third-order Dispersion I.Basic Principles

Compression of ultrashort fundamental solitons in dispersion decreasing fibers (DDFs) with negative third-order dispersion (TOD) is investigated by solving the generalized nonlinear Schrdinger equation numerically. We have shown that, in contrast to the degradation of soliton compression due to the combined effects of positive TOD and Raman self-scattering (RSS), the combined effects of negative TOD and RSS can significantly enhance soliton compression in DDF′s. The enhancement is due to RSS induced redshifting of the soliton wavelength and the negative TOD induced decrease of second-order dispersion toward the longer wavelength.