Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution and the phase is not uniformly distributed between -π and π. The standard Rayleigh and uniform distributions are obtained for special values of the length and in the case of an infinite illumination length.     

Distribution of the probability of oscillations of the surface of liquid hydrogen in the turbulent regime

The probability distribution density of the deviation of the surface of liquid hydrogen from the equilibrium plane state in the system of capillary waves has been analyzed. It has been shown that this probability distribution density for the case of the excitation of the surface oscillations by low-frequency noise in the turbulent regime is well reproduced by a Gaussian. When the oscillations are excited by a low-frequency harmonic force, the stochastization of the waves occurs after several scattering events.     

Scattering of Rayleigh waves by a statistical inhomogeneity of the mass density

The problem of the scattering of a Rayleigh wave by a surface inhomogeneity of the mass density of an isotropic solid is solved in the Born approximation of perturbation theory. The inhomogeneity is statistical with a Gaussian correlation function in the plane parallel to the surface and is deterministic with an exponentially decaying dependence on the coordinate perpendicular to the surface. Expressions are derived for the displacement fields in the scattered longitudinal (P), transverse (SV and SH), and Rayleigh (R) waves at large distances from the inhomogeneity. The Rayleigh wave energy scattering coefficients are calculated as functions of the wavelength λ, the correlation length a of the inhomogeneity, the depth d of the defective layer, and the Poisson ratio of the medium, σ. The angular distribution of the scattered Rayleigh wave energy is determined. Asymptotic expressions are obtained for the scattering coefficient in various limiting cases with respect to the parameters a/λ and λ/d. The relation between the energies in the scattered P, SV, SH, and R waves is established. The resulting equations are used to calculate the scattering coefficients numerically over a wide range of variation of the parameters a/λ, λ/d, and σ; the results are presented in the form of graphs and a table. A physical pattern of the scattering process is constructed and used as a basis for interpreting the results of the study. Fiz. Tverd. Tela (St. Petersburg) 39, 267–274 (February 1997)     

Acousto-exciton interaction in a gas of 2D indirect dipolar excitons in the presence of disorder

A theory for the linear and quadratic responses of a 2D gas of indirect dipolar excitons to an external surface acoustic wave perturbation in the presence of a static random potential is considered. The theory is constructed both for high temperatures, definitely greater than the exciton gas condensation temperature, and at zero temperature by taking into account the Bose–Einstein condensation effects. The particle Green functions, the density–density correlation function, and the quadratic response function are calculated by the “cross” diagram technique. The results obtained are used to calculate the absorption of Rayleigh surface waves and the acoustic exciton gas drag by a Rayleigh wave. The damping of Bogoliubov excitations in an exciton condensate due to theirs scattering by a random potential has also been determined.     

Scattering of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces - study with the curvilinear coordinate method

We present a method giving the bi-static scattering coefficient of two-dimensional (2-D) perfectly conducting random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. This method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The Monte Carlo technique is applied and the bi-static scattering coefficient is estimated by averaging the scattering amplitudes over several realizations. The random surface is represented by a Gaussian stochastic process. Results are compared to published numerical and experimental data. Comparisons are conclusive.     

Interferometric determination of the s and d-wave scattering amplitudes in 87Rb

We demonstrate an interference method to determine the low-energy elastic scattering amplitudes of a quantum gas. We linearly accelerate two ultracold atomic clouds up to energies of 1.2 mK and observe the collision halo by direct imaging in free space. From the interference between s- and d- partial waves in the differential scattering pattern we extract the corresponding phase shifts. The method does not require knowledge of the atomic density. This allows us to infer accurate values for the s- and d-wave scattering amplitudes from the zero-energy limit up to the first Ramsauer minimum using only the van der Waals C6 coefficient as theoretical input. For the 87Rb triplet potential, the method reproduces the scattering length with an accuracy of 6%.     

One-phonon scattering of He from the LiF(001)〈100〉 Rayleigh mode

The one-phonon cross-section from the Rayleigh mode calculated with a point-ion lattice dynamical theory accounts for all salient inelastic scattering intensities seen in He-LiF(001)〈100〉 measurements published by Brusdeylins, Doak and Toennies [J. Chem. Phys. 75 (1981) 1784]. Scattering cross-sections are found by the Rayleigh method, wherein the repulsive part of the scattering potential is approximated as a hard-wall. The solution is in the form of a perturbation expansion in the elastic corrugation and phonon motion of the surface. For each phonon order the solution is infinite in the elastic corrugation. Resonant intensity enhancement occurs either through selective desorption or selective adsorption, both of which have amplitudes which explicitly appear in the perturbation expansion and can be separately calculated. Intensity enhancement also occurs by kinematical focusing. We find a new type of kinematical focusing in addition to that discussed by Benedek [Phys. Rev. Letters 35 (1975) 234].     

Spontaneous emission and elastic scattering of light from quantum-well excitons in a Fabry-Perot microcavity

A theory of spontaneous emission and elastic light scattering by quasi-two-dimensional excitons in a quantum well placed in a Fabry-Perot microcavity is developed. The problem is solved by means of electrodynamic Green’s functions with inclusion of fluctuations of the quantum-well width and cavity wall shape treated as a perturbation. General expressions are found in a zero approximation of perturbation theory (plane interfaces) for the radiative decay rates of quasi-two-dimensional excitons and for their energy shifts in the cavity. The boundary conditions for the electromagnetic field are taken into account through the coefficients of inward light reflection from the cavity walls. Resonance contributions to the scattering cross sections, which differ in the polarizations (p or s) of the incident and scattered waves, are derived in the lowest (Born) approximation in quantum-well width fluctuations. The spectral and angular dependences of elastic light scattering are studied numerically for Gaussian and exponential correlation functions. It is shown that the contribution from quantum-well width fluctuations to light scattering exceeds that due to single interfaces (surfaces) of a heterostructure by two orders of magnitude.     

On the applicability of the Lippmann-Schwinger approach to the scattering of particles by semiinfinite crystals

We present a formal approach, using integral equations of the Lippmann-Schwinger type, to the scattering of particles by semiinfinite periodic potentials. The two possible physical situations, scattering of free plane waves and scattering of Bloch waves, are treated with two different integral equations. The validity of this formalism to describe the particle wavefunction in the whole space is established, and its connection with the matching method is studied. The corresponding extinction theorems, one for each situation, are derived. Interesting relations among the probability amplitudes for both scattering problems are found.     

Analytical derivation of the stationarity and the ergodicity of a field scattered by a slightly rough random surface

In the framework of the small perturbation method, we present a new theoretical derivation of the statistical and spatial properties of a field scattered by a one-dimensional slightly rough random surface. The work concerns the intermediate field zone where the scattered field is reduced to the contribution of the progressive plane waves. The surface is assumed to be stationary, ergodic and Gaussian. First, from a statistical point of view, we demonstrate that under oblique incidence the scattered field is not stationary while it is strictly stationary under normal incidence. For an infinite surface, the scattered field modulus obeys to Hoyt law and the phase is not uniform. Second, from a spatial point of view, for a given altitude and under all incidences, we show that the scattered field is ergodic. Under oblique incidence, the phase is spatially uniform and the modulus is given by a Rayleigh law. Under normal incidence, the phase is not uniform and the modulus is given by a Hoyt law. Third, from a practical point of view, we show that the field measured by a directional antenna is ergodic and stationary if the angular transfer function of the antenna does not contain the specular direction.     

Anfachung und Dämpfung magneto-akustischer Wellen in MHD-Kanälen und ihr Einfluß auf die mittlere Stromdichte und die mittlere Hall-Feldstärke

Magneto-acoustic waves generated by fluctuations in the Hall parameter, the electric conductivity and the stream velocity are theoretically investigated in a weakly ionized plasma streaming across a strong external magnetic field and bearing a current flowing perpendicular to both magnetic field and stream velocity. The investigations hold for seeded rare gas plasmas at any degree of seed ionization but are resticted to waves propagating in parallel or antiparallel direction to the current density vector and in parallel or antiparallel direction to the stream velocity vector and to wave lengths which are small in comparsion to the interaction length which occurs as a characteristic wave length. The influence of these waves on the mean current density and the mean Hall field intensity is calculated in case of small amplitudes and low degree of seed ionization up to second order terms. Omitting Ohmic heating the dispersion equation can be solved exactly. A phase shift exists between the fluctuations in gas density and gas velocity. The phase velocity and the amplification rate depend on the wave length. Typical results are represented in a diagram. For both types of waves the phase velocity slightly rises with increasing wave length, while the amplification rate decreases. Waves propagating in opposite direction to the current density vector are amplified, if the electron velocity exceeds a critical value. They reduce the mean current density and the mean Hall field intensity. Waves propagating in opposite direction to the stream velocity vector are also amplified except for very high degrees of seed ionization. The threshold current density is greater than that for the waves of the first type approximately by the Hall parameter as factor. At extremely high degree of seed ionization the phase velocity is directed opposite to the direction occuring at weakly ionized seed. Waves of the second type decrease the mean current density, but increase the mean Hall field intensity.     

Attenuation of a Stoneley wave and higher Lamb modes due to the scattering by two-dimensional irregularities of the walls of a fluid-filled borehole

Attenuation of Stoneley waves and higher Lamb modes propagating along an irregular surface of a fluid-filled borehole is investigated. This problem generalizes the problem on the attenuation of Rayleigh waves by an irregular surface of an empty borehole [10]. The technique used to evaluate the attenuation coefficient is based on the perturbation method (surface irregularity heights are considered to be small in comparison with the wavelength) and the mean field method. As a result, an expression is obtained for the partial coefficients of the eigenmode attenuation due to the scattering of eigenmodes by the irregularities of the borehole walls into the same or other eigenmodes, as well as into the bulk longitudinal and transverse waves. The frequency-dependent behavior of the partial attenuation coefficients of both Stoneley waves and higher modes is analyzed against the ratio between the irregularity correlation length and the borehole radius for different correlation functions of irregularities.     

TM plane wave scattering and diffraction from a randomly rough half-plane: (part II) An evaluation of the diffraction kernel

In a previous paper (part I), it has been shown that a random wavefield from a randomly rough half-plane for a TM plane wave incidence is written in terms of a Wiener-Hermite expansion with three types of Fourier integrals. This paper studies a concrete representation of the random wavefield by an approximate evaluation of such Fourier integrals, and statistical properties of scattering and diffraction. For a Gaussian roughness spectrum, intensities of the coherent wavefield and the first-order incoherent wavefield are calculated and shown in figures. It is then found that the coherent scattering intensity decreases in the illumination side, but is almost invariant in the shadow side. The incoherent scattering intensity spreads widely in the illumination side, and have ripples at near the grazing angle. Moreover, a major peak at near the antispecular direction, and associated ripples appear in the shadow side. The incoherent scattering intensity increases rapidly at near the random half-plane. These new phenomena for the incoherent scattering are caused by couplings between TM guided waves supported by a slightly random surface and edge diffracted waves excited by a plane wave incidence or by free guided waves on a flat plane without any roughness.     

Behaviour of scattering from a rough surface at small grazing angles

The particular problem of wave scattering at low grazing angles is of great interest because of its importance for the long-distance propagation of radio waves along the Earth's surface, radar observation of near surface objects, as well as solving many other fundamental and applied problems of remote sensing. One of the main questions is: how do the scattering amplitude and specific cross section behave for extremely small grazing angles? We consider the process of wave scattering by a statistically rough surface with the Neumann boundary condition. This model corresponds to sound scattering from a perfectly 'hard' surface (for example, the interface between air and the sea surface) or 'vertically' polarized electromagnetic waves scattered by a perfectly conducting one-dimensional (i.e. cylindrical) surface when the magnetic field vector is directed along the generating line of this cylindrical surface. We assume that the surface roughness is sufficiently small (in the sense of the Rayleigh parameter) and the surface is rigorously statistically homogeneous and therefore, infinite. We confine ourselves only to the first-order approximation of small perturbation theory and therefore consider every act of wave scattering in the Born approximation when the Bragg scattering process takes place. Only one resonant Fourier component of surface roughness is responsible for the scattering in a given direction. However, we take into account the attenuation of incident and scattered waves due to the multiple scattering processes on the path 'before' and 'after' a scattering event in a given direction. Also we consider every one of these multiple scattering events only in the Born approximation. The main result we have obtained is that for small grazing angles the scattering cross section of the diffuse component decreases as the second power of the grazing angles with respect to the incident and scattered directions, and as the fourth power of the grazing angle for the backscattering (radar) situation. Generalizing our results from plane-wave scattering to finite beams allows us to obtain the criterion on the beamwidth. For sufficiently narrow beams the multiple scattering processes do not play any role because of a short 'interaction path', and only single Bragg scattering determines the scattering amplitude (which does not tend to zero for small grazing angles). However, for sufficiently wide beams the result obtained for infinite plane waves becomes valid: due to the above-mentioned multiple scattering processes, the scattering amplitude tends to zero for small grazing angles. Consequently, the behaviour of the scattering cross section for small grazing angles depends on the radiation pattern width of the transmitting and receiving antennae: for sufficiently wide beams the scattering cross section decreases to zero at small grazing angles, but for narrow beams it tends to the finite non-zero value.     

Scattering of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces – study with the curvilinear coordinate method

We present a method giving the bi-static scattering coefficient of two-dimensional (2-D) perfectly conducting random rough surface illuminated by a plane wave. The theory is based on Maxwell's equations written in a nonorthogonal coordinate system. This method leads to an eigenvalue system. The scattered field is expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. The boundary conditions allow the scattering amplitudes to be determined. The Monte Carlo technique is applied and the bi-static scattering coefficient is estimated by averaging the scattering amplitudes over several realizations. The random surface is represented by a Gaussian stochastic process. Results are compared to published numerical and experimental data. Comparisons are conclusive.     

Viscosity and Diffusion Effects at the Boundary Surface of Viscous Fluid and Thermoelastic Diffusive Solid Medium

This paper concentrates on the wave motion at the interface of viscous compressible fluid half-space and homogeneous isotropic, generalized thermoelastic diffusive half-space. The wave solutions in both the fluid and thermoelastic diffusive half-spaces have been investigated; and the complex dispersion equation of leaky Rayleigh wave motion have been derived. The phase velocity and attenuation coefficient of leaky Rayleigh waves have been computed from the complex dispersion equation by using the Muller's method. The amplitudes of displacements, temperature change and concentration have been obtained. The effects of viscosity and diffusion on phase velocity and attenuation coefficient of leaky Rayleigh waves motion for different theories of thermoelastic diffusion have been depicted graphically. The magnitude of heat and mass diffusion flux vectors for different theories of thermoelastic diffusion have also been computed and represented graphically.     

Fast beating null strip during the reflection of pulsed Gaussian beams incident at the Rayleigh angle

It is well known that harmonic bounded Gaussian beams undergo a transformation into two bounded beams upon reflection on a solid immersed in a liquid. The effect is known as the Schoch effect and can be found at the Rayleigh angle for thick plates and at the different Lamb angles for thin plates. Here, a study is made on the effect of pulsed Gaussian beams reflected on solids. It is found experimentally that the Rayleigh wave phenomenon still generates two reflected bounded beams, whereas Lamb wave phenomena do not generate this effect. This fact may be explained intuitively by realizing that the Rayleigh phenomenon is a coincidental phenomenon that is generated in situ, whereas the Lamb wave phenomenon is a non-coincidental phenomenon that is generated only after incident sound is influenced by both sides of a thin plate. Another explanation is the fact that Rayleigh waves are not dispersive, whereas stimulation and propagation of Lamb waves is frequency dependent. A pulse contains many frequencies and therefore only a fraction of the incident pulse is transformed into a Lamb wave. In this paper, numerical simulations are performed that show that actually the Schoch effect does occur neither for Rayleigh waves, nor for Lamb waves. As a matter of fact, a pulse, incident at the Rayleigh angle, generates two reflected lobes with a null zone of a different kind. The null zone is beating several times during the passage of each pulse. This results in a 'null zone' having a lower mean intensity than any of the two lobes, still less outspoken than for the case of harmonic incident bounded beams. This effect does only occur for Rayleigh wave generation and is much less outspoken for Lamb wave generation.     

Rayleigh wave attenuation due to the scattering by two-dimensional irregularities of the walls of an empty borehole

Attenuation of the Rayleigh waves propagating along an irregular surface of an empty borehole is investigated. This problem generalizes the problem on the attenuation of Rayleigh waves by an irregular boundary of a half-space. The technique used to evaluate the attenuation coefficient is based on the perturbation method and the mean field method. As a result, an expression is obtained that relates the partial attenuation coefficients of the surface Rayleigh wave to the scattering by the irregular surface of an empty borehole into the bulk longitudinal and transverse waves (the R → P and R → S processes) and into the surface Rayleigh waves (the R → R processes). The frequency-dependent behavior of the partial attenuation coefficients is analyzed for different correlation functions of irregularities.     

Extreme waves statistics for the Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h ? 1, the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical nonlinear Schrödinger equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h ～ 1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical nonlinear Schrödinger equation.     

Relaxation of probability characteristics of the brownian motion of a nonlinear oscillator

We consider an oscillator with nonlinear elasticity and nonlinear damping under the action of a Gaussian delta-correlated random force. The oscillator is treated as a Brownian particle in the corresponding potential profile. We analyze the problem using the analytical-numerical method based on solving the chain of differential equations for the statistical moments, which is broken in a certain manner. For the case of nonlinear elasticity, we find the dependence of the relaxation times of the mean values and variances of both the coordinates and velocities on the system parameters and noise intensity. By analogy, the relaxation of the probability characteristics of the oscillation amplitude is studied for a system with nonlinear damping. In both cases, the evolution of the Gaussian or Rayleigh probability distributions is described on the basis of the moment relaxation. Nizhny Novgorod Architectural and Construction University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 4, pp. 468–478, September, 2000.