Rayleigh wave attenuation due to scattering by stationary defects

We obtain expressions for the energy spectrum widths of Rayleigh waves arising because of their scattering by point and distributed defects of the surface, as well as by the edge dislocations on the surface and by the grooves of a random lattice in the surface plane. The calculations are valid when the defect density is small. Under certain conditions, our results coincide with the results of other authors who studied the scattering of Rayleigh waves by point defects and by the grooves of a random lattice. The calculations are based on the Keldysh diagram technique modified for the case of semibounded media.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 241–257, May, 2005.     

Thermodynamic Energy of Semi-Infinite Solids with a Two-Dimensional Hot Electron Gas near the Surface

Expressions for the thermodynamic energy of semi-infinite solids with a hot electron layer near the stress-free surface are derived for films and for large samples. Conditions are determined under which this energy is associated with the hot electron excitation energy, with the nonequilibrium Rayleigh waves energy, and with the equilibrium bulk phonon energy. The calculations use the Keldysh diagram technique.     

Propagating and Standing Rayleigh Waves near Rivet Chains Connecting Kirchhoff Plates

We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy.

     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

Symmetric and anti-symmetric vibrations in micropolar thermoelastic materials plate with voids

The problem of symmetric and anti-symmetric vibrations in micropolar thermoelastic plate with voids has been investigated. The dispersive frequency equations are obtained for different surface waves propagating in the plate. The velocity curves are depicted for the symmetric and anti-symmetric vibrations, plate, Rayleigh and flexural waves. It is found that there exist two modes in the solution of frequency equation for the surface waves in micropolar thermoelastic plate with voids. We have observed that the first modes of velocity ratios of corresponding surface waves are lesser than those of second mode of vibration.     

Rayleigh wave scattering by a plane strip in a deep ocean

A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.     

Asymmetric fermi surfaces for magnetic schrödinger operators

We consider Schrödinger operators with periodic magnetic field having zero flux through a fundamental cell of the period lattice. We show that, for a generic small magnetic field and a generic small Fermi energy, the corresponding Fermi surface is convex and not invariant under inversion in any point.     

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)     

Loops of Energy Bands for Bloch Waves in Optical Lattices

We consider stationary Bloch waves in a Bose–Einstein condensate placed in a periodic potential for varying strengths of inter‐atomic interactions. Bifurcations of the stationary states are known to occur in this context. These bifurcations generate loops in the energy bands of the Bloch waves near the ends and the center of the Brillouin zone. Using the method of Lyapunov–Schmidt reductions, we show that these bifurcations are of the supercritical pitchfork type. We also characterize the change in stability of the stationary states across the bifurcation point. Analytical results are illustrated by numerical computations.     

Nonequilibrium Statistical Thermodynamics of Two Dissipative Processes in Solids: Energy Losses of Channeled Particles at Low Velocities and Ultrasound Absorption by Conduction Electrons

We use methods of nonequilibrium statistical thermodynamics to investigate two dissipative processes in solids. We find the electron energy losses of a particle moving in a planar channeling regime and the sound absorption coefficient in metals under electron impurity scattering. The oscillator model is used to analyze the contribution from the effect of electron entrainment by a moving scattering center to the total dissipated energy. We investigate the frequency and temperature dependence of energy losses and also the dependence of the absorption coefficient on the sound wave vector.     

表面效应对偏场下介电高弹体表面波传播的影响

采用表面薄层模型考察偏场下介电高弹体的表面效应,针对不同边界情形,建立一阶等效边界条件.基于有限变形电弹性体的线性增量理论,利用Stroh公式和Ting方法,给出等效边界条件的严格推导过程.进一步利用Stroh公式,获得了偏场下具有表面效应的介电高弹体中表面波的频散方程.以可压缩Neo-Hookean介电高弹体为例,分析了表面效应对预变形和电学偏场作用下的介电高弹体表面波传播特性的影响.结果表明,通过施加适当的偏场,可以调控和优化纳米声表器件的性能.     

Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

Complex behavior in biosystems: an information-theoretic approach

The particular question of transport of vibrational energy in biosystems is considered within the scope of Fröhlich–Davydov's model. It is shown that Davydov's solitary waves, strongly damped in near equilibrium conditions, can display long-range propagation when travelling in Fröhlich's condensate. The latter consists in the emergence of a self-organized dissipative structure (in Prigogine's sense), resembling a nonequilibrium Bose–Einstein-like condensation in the low-lying in frequency modes of vibration, once a critical level of pumping of metabolic energy is achieved.     

An explicit finite‐difference method for the approximate solutions of a generic class of anharmonic dissipative nonlinear media

We develop an explicit finite‐difference method to approximate solutions of modified, Fermi–Pasta–Ulam media, which consider the presence of parameters, such as external damping, relativistic mass, a coefficient for the nonlinear term, and a coefficient of coupling in the case of discrete systems. We propose discrete expressions to approximate consistently the total energy of the system and the average energy flux, and prove that the discrete rate of change of energy is a consistent estimate of its continuous counterpart. The method is thoroughly tested in the energy domain, and our results show that the method gives an approximately constant energy in the case of conservative systems, which fluctuates within a narrow margin of error that may be attributed to truncation errors. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Waves from a Deep Source of Longitudinal Waves in an Elastic Half Space with a Free Boundary

The nonstationary propagation of waves on the surface of an elastic half space from a deep expansion source (model of an explosion in a half space) is examined. Exact solutions are obtained in the form of integrals with finite limits and the general solution is calculated. Algebraic expressions are obtained for the Rayleigh wave. The transition of Rayleigh waves at the surface of the half space is studied. Calculations of Rayleigh waves from discontinuous pulsed sources are presented.     

Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures

We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations. Communicated by Vincent Rivasseau Submitted: January 4, 2006 Accepted: January 31, 2006     

On the nonlinear stability and the existence of selective decay states of 3D quasi-geostrophic potential vorticity equation

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.     

Relativistic Scott correction for atoms and molecules

We prove the first correction to the leading Thomas‐Fermi energy for the ground state energy of atoms and molecules in a model where the kinetic energy of the electrons is treated relativistically. The leading Thomas‐Fermi energy, established in [25], as well as the correction given here, are of semiclassical nature. Our result on atoms and molecules is proved from a general semiclassical estimate for relativistic operators with potentials with Coulomb‐like singularities. This semiclassical estimate is obtained using the coherent state calculus introduced in [36]. The paper contains a unified treatment of the relativistic as well as the nonrelativistic case. © 2009 Wiley Periodicals, Inc.     

De Branges Spaces of Entire Functions Symmetric About the Origin

We define and investigate the class of symmetric and the class of semibounded de Branges spaces of entire functions. A construction is made which assigns to each symmetric de Branges space a semibounded one. By employing operator theoretic tools it is shown that every semibounded de Branges space can be obtained in this way, and which symmetric spaces give rise to the same semibounded space. Those subclasses of Hermite-Biehler functions are determined which correspond to symmetric or semibounded, respectively, nondegenerated de Branges spaces. The above assignment is determined in terms of the respective generating Hermite-Biehler functions.     

Propagation of surface electromagnetic waves similar to Rayleigh waves in the case of Leontovich boundary conditions

If on a surface Σ bounding an electromagnetic field, the Leontovich boundary conditions with a pure imaginary exponent are fulfilled, then surface electromagnetic waves propagating along Σ may exist. These waves have much in common with the Rayleigh waves in elasticity theory. In the paper, the ray theory of this electromagnetic analog of the Rayleigh waves is constructed. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 5–19.