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.     

Average number and probability distribution of caustics in randomly inhomogeneous medium and behind a random phase screen

Formulas are derived for the average cross-sectional caustic density and the probability density of the distances to caustics behind a random phase screen. The constancy of the average number of caustics at great distances behind the screen as the probability density approaches zero is explained.Nizhny Novgorod Architecture and Construction Academy. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 4, pp. 471–478, April, 1994.     

Amplitude fluctuations in the theory of radio-wave refraction scattering

We consider some aspects of the theory of radio-wave scattering with respect to studies of radiation intensity fluctuations behind a random phase screen. The expressions for a piecewise approximation of the function of distribution of intensity fluctuations in the near zone and in the region of random focusing behind the screen are found. The expressions obtained are used to calculate the radio-wave scintillation index in these regions. The corresponding calculations are performed with allowance for the finite dimensions of external and internal scales of turbulent inhomogeneities of the phase screen. It is shown that the radio-wave refraction scattering, as a rule, is characterized by a pronounced random focusing of the radiation intensity behind the phase screen. However, under strongly developed turbulence of the radio-wave propagation medium (phase screen) the above phenomenon may be absent.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 9, pp. 1114–1124, September, 1996.This paper was supported by the Russian Foundation for Fundamental Research under project 95-02-03716.     

Statistics of caustics in an underwater sound channel

The numerical-analytical phase screen method is used to analyze the statistics of the density of caustics in an underwater sound channel with large-scale random inhomogeneities. Different cases of wave propagation direction with respect to the channel axis are considered, and the influence of the inhomogeneity correlation radius is investigated.     

Experimental investigation of the travel-time variance of an acoustic wave propagating through the grid-generated turbulence

An experimental technique for the investigation of the behaviour of acoustic wave propagation through a turbulent medium is discussed. The present study utilizes the ultrasonic travel-time technique to diagnose a grid-generated turbulence. Travel-time variance is studied versus mean flow velocity, travel distance and outer turbulence scale. The effect of thermal fluctuations, which result in fluctuations of sound speed, is studied using a heated-grid experiment. Experimental data obtained using ultrasonic technique confirm numerical and theoretical predictions of nonlinear increase of the travel-time variance with propagation distance, which could be connected to the occurrence of caustics. The effect of turbulent intensity on the travel-time variance and appearance of caustics is studied. It is demonstrated experimentally that the higher turbulence intensity leads to the shorter distance, at which the first caustic occurs. The probability density for caustics appearance is analysed against the measured wave amplitude fluctuations. The analysis reveals that the region of high-amplitude fluctuations corresponds to the region where the probability of formation of random caustics differs from zero. Experimental results are in very good agreement with theoretical and numerical predictions.     

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.     

Phase Fluctuations of Radio Waves in the Saturated-Scintillation Regime

We consider the problem of phase fluctuations of radio waves behind a strong phase screen and in an optically thick layer. It is shown that the phase-fluctuation distribution of the received radiation at an observation point located in the saturated-scintillation area behind a turbulent phase screen is almost identical to the normal distribution of phase fluctuations of the wave on the screen. Amplitude and phase fluctuations of the received radiation are uncorrelated both for single-point and space-diversity reception if, in the latter case, the distance between the observation points exceeds the spatial scale of the diffraction component of the complex field of the received signal. Expressions for the mean square and the structural function of phase fluctuations of radio waves behind a turbulent phase screen in the saturated-scintillation regime are obtained. It is shown that the structural function of phase fluctuations in the diffraction component of the scattered field, which is exactly the function that forms saturated scintillations of the received radiation, almost coincides with the structural function of phase fluctuations on the screen. It is also shown that the diffraction effects can be neglected and the geometric-optical approximation should be used when calculating statistical characteristics of phase fluctuations of a plane wave in the saturated-scintillation area in an optically thick layer with large-scale refractive-index irregularities. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 4, pp. 275–282, April 2005     

The horizontal structure of the sound field in the ocean

Results of calculating the horizontal structure of acoustic field in the ocean with a canonical sound velocity profile are presented. The calculation is performed in the framework of geometrical acoustics by combining the fields of water-path rays at every point with allowance for their phases and amplitudes. The field distribution at a fixed depth is found to be not very informative: within a cycle length, it contains 1–3 maxima, which are mainly caused by the caustics present at the given depth. The width of caustics is ～0.1–1 km. Between the caustics, as well as in their absence, the field amplitude is much smaller than that in the regions of caustics and varies depending on the phases and amplitudes of rays arriving at a given point. The comparison of the calculated horizontal field structure with that obtained from full-scale measurements for the regions between the caustics is difficult because of the possible fluctuations of sound propagation conditions.     

Interference between caustics of diffraction fields

The caustics of diffraction fields are structures that present a nonlinear phase distribution. The superposition of two caustics presents interesting irradiance distributions, which we analyze within the framework of catastrophe theory. This treatment permits the inclusion of the optical path difference involved in a parametric family, and by employing the geometric theory of diffraction we can analyze the interference patterns in a selective fashion. The theoretical predictions are verified experimentally with a Michelson interferometer.     

离轴位相奇点的动态传输

推导出高斯背景的离轴位相奇点光束的半屏衍射解析公式,详细研究了离轴位相奇点的动态传输.结果表明,半屏衍射,偏移量和传输距离都会影响位相奇点的传输行为和衍射场位相奇点的分布.改变偏移量和传输距离,衍射场伴随有位相奇点的产生,移动和湮没. 关键词： 奇点光学 离轴位相奇点光束 半屏衍射 光涡漩     

Behavior of heated particle gas and light-wave diffraction by phase screen in warm-ray model

On the basis of the analogy between the density evolution of a heated gas of noninteracting particles and the diffraction of monochromatic optical waves by a random phase screen, a formula is derived for the correlation function of intensity fluctuations behind the screen. A hypothesis on the statistical splitting of the geometric and diffraction means allows the multiplicity of the integral that determines this function to be reduced by half.     

Propagation of Gaussian background vortex beams diffracted at a half-plane screen

Taking the Gaussian background vortex beam with topological charge +2 as a typical example, a closed-form expression for vortex Gaussian beams passing through a half-plane screen is derived and used to study the propagation dynamics of on-axis and off-axis vortex diffracted beams, and to compare with the case of the free-space propagation. It is shown that there may exist many phase singularities or no phase singularity of vortex diffracted Gaussian beams in the diffraction field. Number and position of phase singularities are dependent on the vortex position at the source plane and propagation distance. The creation, motion and annihilation of phase singularities in the diffraction field may appear by varying the vortex position and propagation distance.     

Approximating moments of acoustic fields in random media

Abstract

Starting from a path integral representation of the fourth moment of an acoustic field, two types of approximations are considered in detail: a ‘zero-kink’ approximation and a ‘single-kink’ approximation. The expressions ‘zero-kink’ and ‘single-kink’ describe families of paths which are used to approximate path integrals over all continuous paths. The single-kink approximation reduces the propagation problem to a phase screen problem with fluctuations on the phase screen containing information about fluctuations along entire paths from source to receiver.     

Interfering modes and an axial wave in an underwater sound channel

Experiments on long-range propagation of low-frequency sound that were conducted starting from the mid-1980s indicate a complex character of propagation in an underwater sound channel, in which a source and a receiver are located close to the channel axis. A burst of energy propagating along the axis follows early arrivals, which are well described by the formulas of geometrical acoustics, in plots of acoustic intensity as a function of propagation time and hydrophone depth. This energy burst cannot be described using geometrical acoustics because of caustics with caustic beaks located near the channel axis. Very complex interference processes occur near these caustics. As the distance from the source grows, the dimensions of the interference vicinity increase and start to overlap producing a peculiar “axial wave.” For an arbitrary two-dimensional underwater sound channel, the axial wave can be represented as a sum of the first normal modes and a residue. This conclusion is based on the use of two representations for an acoustic field. The first of them includes the sum of ray components and an axial wave. The second representation consists of ray addends, the sum of the first normal modes, and a residue. Numerical results are obtained for a canonical profile of sound velocity at the frequency of 200 Hz for the distances of 1600–1650 km.     

Approximating moments of acoustic fields in random media

Starting from a path integral representation of the fourth moment of an acoustic field, two types of approximations are considered in detail: a 'zero-kink' approximation and a 'single-kink' approximation. The expressions 'zero-kink' and 'single-kink' describe families of paths which are used to approximate path integrals over all continuous paths. The single-kink approximation reduces the propagation problem to a phase screen problem with fluctuations on the phase screen containing information about fluctuations along entire paths from source to receiver.     

Efficient Generation of Customized Caustics: Nondiffracting Deltoid and Pentagram Beams

This study puts forward a powerful new strategy that allows highly efficient generation of the shaping transverse nondiffracting caustics that concentrate around the expected curve. First, under the condition of stationary phase approximation, as the typical example of caustic beams, the analytical formula of nondiffracting deltoid and pentagram caustics is theoretically derived. Based on the obtained analytical formula, the desired caustics of deltoid and pentagram are numerically simulated. Hence, their optical shape and the propagation characteristics of nondiffracting caustics can be investigated. Then, based on the given generating mechanism, the corresponding experimental system to experimentally explore the nondiffracting deltoid and pentagram caustics with good self-healing properties is constructed. Compared with the previous generating methods, the proposed method has a high light energy utilization.     

非热离子对非均匀碰撞热尘埃等离子体中三维孤波的影响

从理论上研究了非热离子、外部磁场、碰撞对非均匀热尘埃等离子体中三维非线性尘埃声孤波的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准的变系数Korteweg-de Vries(KdV)方程。然后把非标准KdV方程变为标准的变系数KdV方程,并且得到了标准的变系数KdV方程的近似解析解。由此解析解可以看出,非热离子的数目、碰撞、非均匀性、波的斜向传播、尘埃颗粒和非热离子的温度对三维非线性尘埃声孤波的振幅和宽度有很大的影响。外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与非热离子、波的斜向传播、尘埃颗粒的温度和非均匀性有关。     

Three-dimensional coupled-mode model and characteristics of low-frequency sound propagation in ocean waveguide with seamount topography

Large-scale topography, such as a seamount, substantially impacts low-frequency sound propagation in an ocean waveguide, limiting the application of low-frequency acoustic detecting techniques. A three-dimensional (3D) coupled-mode model is developed to calculate the acoustic field in an ocean waveguide with seamount topography and analyze the 3D effect. In this model, a correction is introduced in the bottom boundary, theoretically making the acoustic field satisfy the energy conservation. Furthermore, a large azimuth angle calculation range is obtained by using the operator theory and higher-order Padé approximation. Additionally, the model has advantages related to the coupling mode and parabolic equation theory. The couplings corresponding to the effects of range-dependent environment are fully considered, and the numerical implementation is kept feasible. After verifying the accuracy and reliability of the model, low-frequency sound propagation characteristics in the seamount environment are analyzed. The results indicate lateral variability in bathymetry can lead to out-of-plane effects such as the horizontal refraction phenomenon, while the coupling effect tends to restore the abnormal sound field and produces acoustic field diffraction behind the seamount. This model effectively considers the effects of the horizontal refraction and coupling, which are proportional to the scale of the seamount.     

湍流等离子体鞘套中高斯光束的传播特性分析

为了研究高斯光束在湍流等离子体鞘套中的传输特性,根据广义惠更斯-菲涅耳原理,采用基于快速傅里叶变换的功率谱反演法,用多随机相位屏来模拟湍流带来的影响.根据超声速飞行器绕流等离子体流场厚度在厘米级别的特点,光束在两个相位屏之间的传输过程中采用菲涅耳衍射积分的两次快速傅里叶变换算法(double fast Fourier transform algorithm),利用多随机相位屏模拟等离子体鞘套湍流对光束传输产生的影响,解决了多随机相位屏模拟湍流研究中的超短距离传输问题.当飞行高度为45 km,飞行速度为18马赫时,通过对超声速飞行器绕流等离子体流场的统计分析,发现在此飞行条件下折射率起伏方差的强度范围10~(–11)—10~(–14).对高斯光束在湍流等离子体流场中的传输特性进行了数值仿真.结果表明:在等离子体鞘套湍流中折射率起伏强度、波长、传输距离等都是影响高斯光束质量的重要因素.折射率方差越大,传输距离越长,光斑弥散越严重,光强起伏越大,光强减弱也越明显.光束的波长越长,高斯光束抑制湍流的能力越强,光斑弥散程度越小,光强起伏也越小.