On acoustic radiation from a nonlinearly vibrating charged drop

An expression for the time-varying shape of an incompressible liquid drop immersed in a compressible dielectric medium is derived to the second-order approximation in drop vibration amplitude. It is shown that the acoustic radiation spectrum of the drop has a monopole component, which makes a considerable contribution to the integral radiation intensity. Its appearance is associated with the time variation of the zeroth-mode vibration amplitude, showing up in the second order of smallness.     

Electromagnetic radiation from linearly and nonlinearly oscillating charge drops

It has been shown that analytic calculations of the intensity of electromagnetic radiation from an oscillating charged drop in the approximation linear in the oscillation amplitude (small parameter is on the order of 0.1) give only the quadrupole component of the total radiation. The dipole component can only be obtained in calculations using higher-order approximations. Nevertheless, the intensity of the dipole radiation turns out to be substantially higher (by 14–15 orders of magnitude). This is because the decomposition of radiation from a system of charges into multipole components (differing even in the rates of decrease in the potential with the distance) is carried out using the expansion in a substantially smaller parameter, viz., the ratio of the size of the emitting system (in our case, a drop of radius ~10 μm) to the distance to the point of observation in the wave zone of the emission of radiation (emitted wavelength) of 100–1000 m. As a result, this second small parameter is on the order of 10^–7 to 10^–8. On the other hand, in accordance with the field theory, the ratio of intensities of quadrupole and dipole radiations is proportional to the squared ratio of the hydrodynamic velocity of the oscillating surface of a charged drop to the velocity of propagation of an electromagnetic signal in vacuum (velocity of light), which yields a ratio of 10^–14 to 10^–15.     

Quadrupole electromagnetic radiation of an oscillating charged droplet

Analytical calculations using the first order of smallness with respect to dimensionless amplitude of oscillations show that the intensity of electromagnetic radiation of a charged droplet is determined by time-dependent quadrupole moment.     

Electromagnetic radiation due to nonlinear oscillations of a charged drop

The nonlinear oscillations of a spherical charged drop are asymptotically analyzed under the conditions of a multimode initial deformation of its equilibrium shape. It is found that if the spectrum of initially excited modes contains two adjacent modes, the translation mode of oscillations is excited among others. In this case, the center of the drop’s charge oscillates about the equilibrium position, generating a dipole electromagnetic radiation. It is shown that the intensity of this radiation is many orders of magnitude higher than the intensity of the drop’s radiation, which arises in calculations of the first order of smallness and is related to the drop’s charged surface oscillations.     

Application of the nuclear liquid drop model to atomic and molecular physics problems

The liquid drop model is applied to describe some basic properties of atoms, homoatomic molecules, metallic clusters of atoms and fullerene molecules. Equilibrium atomic size, energy and polarizability of the atom are calculated. Collective modes of oscillations (dipole, quadrupole and monopole, or breathing, ones) are regarded. Electromagnetic radiation by an atom, passing through a barrier is studied. Equilibrium volume of a homoatomic molecule of two atoms, axes ratio, dissociation energy and the frequencies of the dipole oscillations are calculated. Models to describe some properties of clusters and fullerene molecules are proposed. The size of the metallic cluster, its energy and the frequency of dipole oscillations are calculated. The frequencies of the dipole and breathing mode oscillations of the fullerene molecules are obtained. The calculated frequency of the dipole oscillations was found to be in a rather good accord with the experimental one.     

Time reversal for a single spherical scatterer.

We show that the time reversal operator for a planar time reversal mirror (TRM) can have up to four distinct eigenvalues with a small spherical acoustic scatterer. Each eigenstate represents a resonance between the TRM and an induced scattering moment of the sphere. Their amplitude distributions on the TRM are orthogonal superpositions of the radiation patterns from a monopole and up to three orthogonal dipoles. The induced monopole moment is associated with the compressibility contrast between the sphere and the medium, while the dipole moments are associated with density contrast. The number of eigenstates is related to the number of orthogonal orientations of each induced multipole. For hard spheres (glass, metals) the contribution of the monopole moment to the eigenvalues is much greater than that of the dipole moments, leading to a single dominant eigenvalue. The other eigenvalues are much smaller, making it unlikely multiple eigenvalues could have been observed in previous experiments using hard materials. However, for soft materials such as wood, plastic, or air bubbles the eigenvalues are comparable in magnitude and should be observable. The presence of multiple eigenstates breaks the one-to-one correspondence between eigenstates and distinguishable scatterers discussed previously by Prada and Fink [Wave Motion 20, 151-163 (1994)]. However, eigenfunctions from separate scatterers would have different phases for their eigenfunctions, potentially restoring the ability to distinguish separate scatterers. Since relative magnitudes of the eigenvalues for a single scatterer are governed by the ratio of the compressibility contrast to the density contrast, measurement of the eigenvalue spectrum would provide information on the composition of the scatterer.     

Scattering of an Acoustic Field by Refraction–Density Inhomogeneities with a Small Wave Size and Solution of the Problem of Direct Scattering in an Inhomogeneous Medium

The paper considers acoustic wave scattering by inhomogeneities with a small wave size using the Green’s function apparatus, which makes it possible universally to take into account both the refraction and density components of an inhomogeneity. Estimates for the multipole components of a field scattered by a nonresonance inhomogeneity are presented. For an inhomogeneity with small dimensions, it suffices to consider only monopole and dipole scattering. These conclusions are confirmed by an analysis of the field scattered by a circular cylinder with a small wave radius. The results are used to numerically simulate a Lippmann–Schwinger equation. The form of the discretized matrix Green’s function for identical values of the spatial arguments is presented. This makes it possible to take into account multiple scattering processes within a discretization element with a small wave size. Its use automatically fulfills the relations between the phase and amplitude of secondary acoustic field sources.     

多分量方位远探测声波测井的理论与应用

基于线性拟合的利用岩石动态模量预测静态模量方法存在经验性强且精度不准的问题,利用实验测量的岩石纵波速度、孔隙度和体密度作为输入参数,基于多基因遗传规划方法,不断进化模型得到静态模量关于三个参数的解析表达式,对储层砂岩的静态模量进行了预测。结果表明,多基因遗传规划方法得到的模量表达式更为精确,此外,由于多基因遗传规划方法中横波速度不必要作为计算过程中的输入参数,因此避免了横波速度测量不准确或缺失而带来的静态模量计算误差。     

Moving finite-size dipole emission

Emission of a finite-size arbitrarily moving dipole is considered. It is shown that the dipole radiation intensity can be expressed in terms of the dipole moment of the system for wavelengths longer than the dipole size. In this case, the system radiation intensity is much lower than that of the unit charge. For wavelengths much shorter than the dipole size, the dipole radiation intensity is equal to the sum of radiation intensities of two charges. For wavelengths comparable to the dipole size, interference minima and maxima arise in the angular distribution of the radiation intensity.     

Nonlinear capillary vibrations of a charged drop placed in a dielectric medium: Single-mode initial deformation of the drop shape

The nonlinear vibrations of the equilibrium spherical shape of a charged drop placed in a perfect incompressible dielectric medium are asymptotically calculated in the second-order approximation in single-mode initial deformation of the drop surface. The drop is assumed to be a perfect incompressible liquid. It is shown that the nonlinear vibration amplitudes, as well as the energy distribution between nonlinearly excited modes, depend significantly on the parameter ρ, where ρ is the ratio of the environmental density to that of the drop. It is also demonstrated that an increase in ρ raises the amplitude of the highest of the vibration modes excited due to second-order nonlinear interaction. In the second order of smallness, the amplitude of the zeroth mode is independent of the density ratio. As ρ grows, the effect of the self-charge of the drop, the interfacial tension, and the permittivity of the environment on the nonlinear oscillations increases.     

The interaction of transverse domain walls

The interaction between transverse domain walls is calculated analytically using a multipole expansion up to third order. Starting from an analytical expression for the magnetization in the wall, the monopole, dipole, and quadrupole moments are derived and their impact on the interaction is investigated using the surface and volume charges. The surface charges are important for the dipole moment while the volume charges constitute the monopole and quadrupole moments. For domain walls that are situated in different wires it is found that there is a strong deviation from the interaction of two monopoles. This deviation is caused by the interaction of the monopole of the wall in the first wire with the dipole of the wall in the second wire and vice versa. The dipole-dipole and the quadrupole-monopole interactions are found to be also of considerable size and non-negligible. A comparison with micromagnetic simulations shows a good agreement.     

Vortex sound due to a flexible boundary backed by a cavity in a low mach number mean flow

Low frequency sound radiated due to the unsteady motion of an inviscid vortex in the proximity of a flexible membrane backed by an airtight cavity on an otherwise rigid plane is investigated theoretically. Results show that both monopole and dipole are created but the latter is important only when the vortex is traversing over the membrane. The monopole results from the membrane vibration and the dipole from the transverse motion of the vortex. It is also found that these sound fields tend to counteract each other. The increase in the mean flow speed in general results in a stronger acoustic power radiation, but sound attenuation may be possible if the membrane-cavity system is weak compared with the mean flow momentum.     

声诱导电磁场的赫兹矢量表示与多极声电测井模拟

在假设声场不受电磁场影响的前提下，将Pride声电耦合方程组化为具有电流源的麦克斯韦方程组.与空间位置固定的电流源产生的电磁场不同，孔隙地层中声波诱导的电磁场是由空间波动的电流源产生的.通过引入赫兹矢量，将求解麦克斯韦方程组问题转化为求解关于赫兹矢量的非齐次矢量赫姆霍兹方程组.通过求解该方程组，得出电磁场表达式.利用此方法，针对声电效应测井，分别计算了由单极声源、偶极声源、四极声源激发的井内声场及其诱导电磁场的全波波形. 关键词： 孔隙介质 诱导电磁场 测井 多极声源     

On nonlinear resonant four-mode interaction between capillary vibrations of a charged drop

Energy transfer from higher modes of capillary vibrations of an incompressible liquid charged drop to the lowest fundamental mode under four-mode resonance is studied. The resonance appears when the problem of nonlinear axisymmetric capillary vibration of a drop is solved in the third-order approximation in amplitude of the multimode initial deformation of the equilibrium shape of the drop. Although the resonant interaction mentioned above builds up the fundamental mode even in the first order of smallness, its amplitude turns out to be comparable to a quadratic (in small parameter) correction arising from nonresonant nonlinear interaction, since the associated numerical coefficients are small.     

Influence of viscosity on aerodynamic sound emission in free space

The theory of aerodynamic sound is reformulated with account taken of the influence of viscosity in the source flow on the sound emission in free space. This is based on the Ffowcs Williams form of the Lighthill equation. The source flow is assumed to be determined by a localized vorticity field and characterized by low Mach numbers and high Reynolds numbers. It is found that the acoustic pressure emitted by the viscous vortex motion is composed of a quadrupole and two kinds of monopole. The Reynolds stress in the source flow is decomposed into isotropic and non-isotropic parts. The non-isotropic part leads to the quadrupole wave derived by Möhring, while the isotropic part is related to one of the monopole-like waves radiated when the total kinetic energy changes. The other monopole wave is associated with entropy production by viscous dissipation of the kinetic energy. All three components are influenced by viscosity.     

Nonlinear oscillations of a drop moving at a constant velocity in an insulating medium subjected to an electrostatic field

Nonlinear asymptotic calculations of the second order of smallness in the amplitude of the initial deformation of an ideally conducting liquid drop show that the laminar flow of an ideal conducting incompressible dielectric liquid flowing about the drop in an external electrostatic field parallel to the flow causes oscillation mode’s interaction in the first and second orders of smallness. Both linear and nonlinear interactions between the oscillation modes of the drop excite modes that are absent in the spectrum of modes governing the initial deformation of the drop’s equilibrium shape. In the second order of smallness, the mode interaction decreases the electrostatic field strength, as well as the velocity and density of the environment, that are critical for development of instability of the drop against the polarization charge.     

相控圆弧阵声源在TI地层井孔中产生的声场

研究了相控圆弧阵声源在各向异性地层井孔中产生的声场的特征,重点分析了该声源对于地层横波各向异性的方位敏感性,旨在探索一种新的地层声学各向异性测量方法。结果显示:相控圆弧阵声源在地层中可激发多种方位阶数的声场,主要成分为单极子波场和偶极子波场;在HTI地层竖直井中,方位角不同的圆弧阵声源激发的波动的速度不同,相控圆弧阵声源对于地层横波速度的方位各向异性的敏感性主要是它所激发的弯曲波的贡献。相比于传统的正交偶极子声波测井而言,采用相控圆弧阵声源可以评价井附近地层的周向非均质性,可以提高井周测量覆盖次数,提高反演结果的可靠性。另外,相控圆弧阵声源兼具备多极子声源的功能,优化的相位控制方式可以实现方位各向异性的最佳测量。      

Time-reversal analysis for scatterer characterization

A new application of time-reversal processing of wave scattering data permits characterization of scatterers by analyzing the number and nature of the singular functions (or eigenfunctions) associated with individual scatterers when they have multiple contributions from monopole, dipole, and/or quadrupole scattering terms. We discuss acoustic, elastic, and electromagnetic scattering problems for low frequencies. Specific examples for electromagnetic scattering from one of a number of small conducting spheres show that each sphere can have up to six distinct time-reversal eigenfunctions associated with it.     

反射波测井的菲涅尔带效应

基于反射波的单井远探测是声波测井发展的一个重要里程碑,可以有效探测距离井数米到数十米的异常反射体如裂缝,极大提升了声波测井的径向探测距离,在碳酸盐岩缝洞储层勘探开发有重要的作用。本文系统考察了有限尺寸裂缝的反射强度与声源频率和反射体距离的变化特征。模拟结果表明,无论是单极子还是偶极子,反射波的幅度都呈现出菲涅尔带效应。在一定距离内,反射波幅度随裂缝某一方向如宽度增大而增大,当宽带增加一定尺寸后,反射波幅度有一定的程度下降,然后小幅震荡。总之,反射波幅度与反射体尺寸不是简单的正相关,因此,基于反射波能量来评价反射体体积和储量是有条件的。     

The field of an equivalent multipole composite radiator in a waveguide

The possibility of replacing a spatially extended volume low-frequency sound source, which forms a three-dimensional directional field in the far-field zone, by an equivalent multipole source is considered. Developing the ideas of L.M. Brekhovskikh, who obtained the solution for a monopole, exact and simplified models of the source field are constructed in the form of superpositions of multipoles of various orders. The models are considered in the integral, mode, and ray approximations applied to the unbounded space and homogeneous and plane-layered waveguides. The results of computations are presented, and the conclusion is made that the contribution of the vertical dipole and quadrupole components to the interference structure of the field amplitude in a waveguide decreases with an increase in the distance from the source. The phase structure is found to be stable regardless of the type of the complex multipole source.