Parabolic equation for nonlinear acoustic wave propagation in inhomogeneous moving media

A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. Published in Russian in Akusticheskiĭ Zhurnal, 2006, Vol. 52, No. 6, pp. 725–735. This article was translated by the authors.     

Extended characteristics method in nonlinear geometric acoustics

An asymptotic “extended characteristics method” is developed for solving nonlinear Riemann-type wave equations as applied to calculating the ray pattern of intense spatially modulated waves in weakly inhomogeneous media. The method makes it possible to avoid the singularity related to the foci of the initial wavefront, calculate the displacement in foci caused by the inhomogeneity of the medium, and thus calculate the ray pattern and intensity of the acoustic field. The beauty of the method is an exact nonlinear transfer equation for the field along the ray and the construction of its general solution for an arbitrary form of inhomogeneity. It is shown that the method is applicable to calculating the spatial structure of intense focused waves and wave beams outside the focal region in a nonlinear geometric acoustics approximation.     

Statistical characteristics of an intense wave behind a two-dimensional phase screen

An analysis of the parameters of nonlinear waves transmitted through a layer of a randomly inhomogeneous medium is carried out. The layer is modeled by a two-dimensional phase screen. Passing through the screen plane, the wave acquires a random phase shift. The wave front becomes distorted, and randomly located regions of ray convergence and divergence are formed, in which the nonlinear evolution of the wave alters profoundly. The problem is solved in the approximation of geometrical acoustics. The ray pattern of a plane wave transmitted through the regular screen is constructed. The solution that describes the spatial structure of the field and the evolution of an arbitrary temporal wave profile behind the screen is obtained. Statistical characteristics of the discontinuity amplitude are calculated for different distances from the screen. A random modulation is shown to result in a faster (in comparison with the case of a homogeneous medium) nonlinear attenuation of the wave and in the smoothing of the shock profile. The distribution function of the wave field parameters becomes broader because of random focusing effects.     

Structure of the focal region at a focusing with a time reversal of waves in an inhomogeneous medium

A focusing array with a time reversal of waves in an inhomogeneous medium is considered. It is shown that, at the focus of such an array, an oscillation trap can be formed. In a homogeneous medium, a wave first travels to the array focus, is focused, and then travels away from the focus, whereas, in an inhomogeneous medium, the wave does not travel at all. In the oscillation trap, an intense oscillation is formed, which arrives from nowhere and escapes to nowhere. The size of the oscillation trap is much smaller than that of the focal spot of the array in free space. The physical nature of this phenomenon and the possible areas of its practical application are discussed.     

Instability of oscillations in the flow moving through a random medium as the counterpart of the localization of waves in a passive random medium

This paper deals with waves propagating in a one-dimensional flow moving through a randomly layered medium. The flow velocity is assumed to be greater than the group velocity of the waves in the reference system of the flow. As a result, in the laboratory reference system, all the waves propagate in a single direction. Amplitudes of these waves moving through a randomly inhomogeneous medium are growing exponentially. This effect is analogous to the wave localization phenomenon in a randomly inhomogeneous passive medium. The only difference is that the wave propagation in a passive medium is described by the boundary value problem, while all the oscillations in a medium with flow propagate in a single direction and hence the corresponding problem is formulated in the form of the initial value Cauchy problem. In the former case exponentially decreasing solutions are realized in the direction of the wave incidence, while in the latter case (as in the case of parametric resonances) the exponentially increasing solutions are realized.     

Volume Fresnel reflection of electromagnetic waves in three-dimensionally inhomogeneous media

Eikonal approximation is used to derive equations describing propagation of monochromatic electromagnetic waves in a three-dimensionally inhomogeneous medium, including volume Fresnel reflection from inhomogeneities. The analysis is based on a locality principle. Separation into reflection and transmission effects is performed. The former effects are found to be isotropic, whereas the latter are anisotropic and depend on interference phenomena. Interference effects lead to violation of the Rytov law of polarization rotation. Brewster phenomena in layered and three-dimensionally inhomogeneous media are shown to occur under different conditions.     

Backscattering of electromagnetic waves by large-scale inhomogeneities in a periodically inhomogeneous medium

It is shown that backscattering of electromagnetic waves is possible in a periodically inhomogeneous medium by random inhomogeneities whose scale is greater than the wavelength. A small scattered field emerges in the case of appearance of the Bragg cavity when the periodic layer is a matching system for the incident wave. Scattering is effective even for inhomogeneities whose scale is much greater than the Fresnel radius of the inhomogeneous layer. The correlation radius of the scattered field can also be that large. Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 10, pp. 1230–1240, October, 1997.     

Low frequency electrostatic nonlinear structures in an inhomogeneous magnetized non-Maxwellian electron–positron–ion plasma

The properties of low frequency (coupled acoustic and drift wave) nonlinear structures including solitary waves and double layers in an inhomogeneous magnetized electron–positron–ion (EPI) nonthermal plasma with density and temperature inhomogeneities are studied in a simplified way. The nonlinear differential equation derived here for the study of double layers in the inhomogeneous EPI plasma resembles with the modified KdV equation in the stationary frame. But the method used for the derivation of nonlinear differential equation is simple and consistent to give both the stationary solitary waves and double layers. Further, the illustrations show that superthermality κ, drift velocity and temperature inhomogeneity have significant effects on the amplitude, width, and existence range of the structures.     

Wave nonreciprocity in inhomogeneous gyrotropic media and multilayer systems: I. Inhomogeneous gyrotropic media

The propagation of electromagnetic waves in naturally (or structurally) gyrotropic inhomogeneous media is considered. A new mechanism of wave nonreciprocity is found, which is caused by the simultaneous presence of a gradient of one of the parameters of a medium and natural (or structural) gyrotropy. The problem of light transmission through a layer of a naturally gyrotropic inhomogeneous medium of finite thickness is solved by the method of addition of layers. Specific features of this nonreciprocity are considered. It is shown that such a system can operate as a one-sided reflector at certain angles of incidence of light. The mechanisms of enhancement of the nonreciprocity effects are investigated. It is shown that multiple reflections in a layer of finite thickness and diffraction of light by periodic inhomogeneities of a medium increase the nonreciprocity effects by several (from 2 to 5) orders of magnitude. This phenomenon, in turn, opens new fields of application of the nonreciprocity effects. Another interesting manifestation of wave nonreciprocity is revealed, which consists in asymmetry of the curve R(?):R(?)≠R(??) (R is the reflectance and ? is the angle of incidence).     

Wave Resonance in Media with Modular,Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.     

Scattering of an Alfvén wave by an inhomogeneity in the solar wind density

The interaction between a nonlinear Alfvén wave and intense inhomogeneities in the density of interplanetary plasma is considered in the magnetohydrodynamic (MHD) approximation. The cold-plasma approximation was used to carry out a more correct study of the interaction since the thermal pressure can introduce pronounced changes into the shape of specified inhomogeneities in the plasma density. Results of numerical solution of the well-known MHD equations are presented in the form of three films demonstrating different scenarios of development of the nonlinear dynamics. The films allow us to observe the dynamic evolution of the form of an Alfvén perturbation and the changes in the density inhomogeneities. For small-amplitude Alfvén waves this corresponds to the process of linear transformation by the density inhomogeneities, which does not lend itself to comprehensive analytical study. Numerical simulation reveals the phenomena of reflection from regions of sharp density variation, which are very sensitive to the spatial scales of the interacting objects. The same method is used to investigate the scattering of strong waves. After reversible changes in shape in a high-density region (where oscillations of the shock-wave front are attenuated), a moderate-amplitude Alfvén wave is recovered in a more rarefied medium. A strong scattered Alfvén wave brings about irreversible changes in the shape of the density inhomogeneity. The results obtained illustrate the process of interaction between Alfvén waves and strong density perturbations related to piston or explosive shock waves in the solar wind. State Pedagogikal University, Nizhny Novgorod, Russia; Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 2, pp. 152–163, February, 1998.     

Nonlinear Evolution of Driven Electron Plasma Oscillations in Inhomogeneous Plasmas

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Ponderomotive nongradient force acting on a relativistic particle crossing an inhomogeneous electromagnetic wave

The ponderomotive force acting on a relativistic charged particle crossing an inhomogeneous electromagnetic wave is investigated numerically and analytically. The initial velocity of the particle is perpendicular to the electric field vector of the wave and to the direction of its propagation. The wave has zero gradient in the direction of propagation and is inhomogeneous in both transverse directions. It is shown that the ponderomotive force acting on the particle is parallel to the wave vector. The magnitude of the force is determined not only by the extent of wave inhomogeneity in the direction of the translational motion of particle, but also by its inhomogeneity in the transverse direction. It is found that the trajectory of a particle is determined by the action of ponderomotive forces as well as by its drift in a nonuniform field.     

Influence of Magnetic-Field and Plasma-Density Inhomogeneities on the Propagation and Amplification of Radio Waves in the Solar Corona

We analyze the maser generation of millisecond spikes of the solar radio emission at the cyclotron resonance of a fast extraordinary wave in an inhomogeneous medium. It is shown that the magnetic-field inhomogeneity with parameters typical of the solar corona drastically reduces the time of electromagnetic-wave amplification, which is explained by the fact that these waves leave the resonance region in the wave-vector space. As a result, an unstable electron distribution can be formed. The efficient generation of radiation becomes possible only in such local regions where the influence of the magnetic-field inhomogeneity is compensated by small-scale inhomogeneities of the plasma density with typical scales ranging from tens to hundreds of kilometers. Taking the effect of inhomogeneous medium into account allows us to explain spatial and temporal characteristics of the spikes.     

Self-action of short pulses in nonhomogeneous graded-index light guides

The weak nonlinear process of propagation of short pulses in graded-index light guides that are weakly inhomogeneous in the longitudinal direction and slightly bent is investigated by means of a consistent asymptotic method. The process as a whole is proved to be three-scale in respect to a small parameter related to the magnitude of nonlinearity. The phase of the most rapid process and transverse distribution of the wave field are expressed explicitly in terms of a certain Sturm-Liouville problem. For a pulse envelope the nonlinear Schrödinger equation is derived, its coefficients depending on the longitudinal coordinate. The existence of a guaranteed interval of conservation of concentration of the pulse envelope is ascertained. For a class of very smooth inhomogeneities formulae are obtained describing the variation of the amplitude and width of the pulse during propagation.     

Generation of pulses upon nonresonant acousto-electromagnetic interaction

New mechanisms of generation of acoustic and electromagnetic soliton-like pulses in an optoelastic medium upon nonlinear nonresonant interaction of the polarization components of an electromagnetic field with acoustic oscillations in the medium are considered. It is shown that the acousto-electromagnetic interaction in such a system may lead to the formation of coherent soliton excitations in a thin crystal plate. It is found that a modulation instability occurs in an extended medium, which is caused by the spatial effects and leads to the generation of transverse sound waves. The evolution of a light field in a one-dimensional extended periodic optoelastic medium is also considered. It is shown that acoustic and electromagnetic solitons can be generated due to the mixing of direct and backward optical waves and their nonresonant interaction with a sound wave.     

Propagation of lower hybrid resonance waves in depleted-plasma regions in the upper auroral ionosphere

We study theoretically the propagation of lower-hybrid resonance (LHR) waves in the auroral ionospheric plasma. The ray-tracing technique is used to study the properties of LHR wave propagation with account of a large-scale inhomogeneity both along and across the geomagnetic field. It is shown that wave refraction in such an inhomogeneous medium can result in direct transformation of LHR waves whose wave normals make large angles with the geomagnetic field into whistler-mode waves, whose wave vectors are close to the geomagnetic-field direction and which can therefore pass through the ionosphere to the ground. The parameters of LHR waves which can thus be transformed into whistler-mode waves are found. The transformation process considered can be important for interpreting ground-based observations of ELF waves. Deceased. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 52, No. 4, pp. 279–289, April 2009.     

Reflection–refraction of attenuated waves at the interface between a thermo-poroelastic medium and a thermoelastic medium

Phenomenon of reflection and refraction is considered at the plane interface between a thermoelastic medium and thermo-poroelastic medium. Both the media are isotropic and behave dissipative to wave propagation. Incident wave in thermo-poroelastic medium is considered inhomogeneous with deviation allowed between the directions of propagation and maximum attenuation. For this incidence, four attenuated waves reflect back in thermo-poroelastic medium and three waves refract to the continuing thermoelastic medium. Each of these reflected/refracted waves is inhomogeneous and propagates with a phase shift. The propagation characteristics (velocity, attenuation, inhomogeneity, phase shift, amplitude, energy) of reflected and refracted waves are calculated as functions of propagation direction and inhomogeneity of the incident wave. Variations in these propagation characteristics with the incident direction are illustrated through a numerical example.