Scattering of time-harmonic elastic waves by an elastic inclusion with quadratic nonlinearity

This paper considers the scattering of a plane, time-harmonic wave by an inclusion with heterogeneous nonlinear elastic properties embedded in an otherwise homogeneous linear elastic solid. When the inclusion and the surrounding matrix are both isotropic, the scattered second harmonic fields are obtained in terms of the Green's function of the surrounding medium. It is found that the second harmonic fields depend on two independent acoustic nonlinearity parameters related to the third order elastic constants. Solutions are also obtained when these two acoustic nonlinearity parameters are given as spatially random functions. An inverse procedure is developed to obtain the statistics of these two random functions from the measured forward and backscattered second harmonic fields.     

Computer simulation of acoustic nonlinear parameter tomography

I.IntroductionUltrasonicimaginghasbcenwide1yusedinthcfie1dofclinicaldiagnosis,becauseitcanvisualizethetissuetharacterizationandinternalstructureofbiologicalobjectsbyacousticwave.Usingconventionalultrasonicimagingtechnique,theimagesofacousticlinearparameterssuchassoundve1ocity,acousticimpedenccandattenuationcoefficientmaybeobtained.Thesehavebecometheeffectivemethodsofu1trasonicdiagnosis.How-ever,inordertoobscrvctheearlystageofcanccr,weintendtoobtainmoreaccurateandmorecompleteinformationaboutth…     

On reconstructing photon statistics by on/off detectors: Toward the multi-partite case

The propagation of pulses of the pump and its second harmonic in a quadratically nonlinear medium whose linear properties are characterized by a negative refractive index at the pump frequency and by a positive refractive index at the harmonic frequency is considered theoretically. In the case of a low intensity of the interacting waves, the pump and second-harmonic pulses propagate in the opposite directions, but sufficiently powerful pulses can form a simulton—a solitary two-frequency wave propagating in a certain direction as a single whole. Solutions to a set of equations are found which describe the steady-state propagation of a solitary wave and of a nonlinear periodic (cnoidal) wave.     

Nonlinear equations for quasi-monochromatic waves near a caustic in a dispersive dissipative medium with cubic or quadratic nonlinearity

The behavior of a quasi-monochromatic nonlinear wave near a caustic is considered. Nonlinear ordinary differential equations for a dispersive dissipative medium with a cubic or quadratic nonlinearity are derived. For the latter medium, nonstationary equations describing it near the caustic are presented with allowance for the dissipative dispersive terms. These equations yield ordinary ones for quasi-monochromatic waves. The amplitude of the second harmonic is expressed in terms of the squared amplitude of the first harmonic. The amplitude of the second harmonic, as well as the solution as a whole, increases near the caustic.     

Propagation of acoustic wave in viscoelastic medium permeated with air bubbles

Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and \"{U}berall H, {\em J. Acoust. Soc. Am}., 1978; 63: 1699--1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters, and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.     

Linear and nonlinear Biot waves in a noncohesive granular medium slab: transfer function, self-action, second harmonic generation

Experimental results are reported on second harmonic generation and self-action in a noncohesive granular medium supporting wave energy propagation both in the solid frame and in the saturating fluid. The acoustic transfer function of the probed granular slab can be separated into two main frequency regions: a low frequency region where the wave propagation is controlled by the solid skeleton elastic properties, and a higher frequency region where the behavior is dominantly due to the air saturating the beads. Experimental results agree well with a recently developed nonlinear Biot wave model applied to granular media. The linear transfer function, second harmonic generation, and self-action effect are studied as a function of bead diameter, compaction step, excitation amplitude, and frequency. This parametric study allows one to isolate different propagation regimes involving a range of described and interpreted linear and nonlinear processes that are encountered in granular media experiments. In particular, a theoretical interpretation is proposed for the observed strong self-action effect.     

一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

Experimental study of nonlinear acoustic effects in a granular medium

Results of a series of experimental studies of nonlinear acoustic effects in a granular medium are presented. Different effects observed in the experiments simultaneously testify that the nonlinearity of granular media is governed by the weakest intergrain contacts. The behavior of the observed dependences suggests that the distribution function of contact forces strongly increases in the range of forces much smaller than the mean force value, which is inaccessible for conventional experimental measuring techniques. For shear waves in a granular medium, the effects of demodulation and second harmonic generation with conversion to longitudinal waves are studied. These effects are caused by the nonlinear dilatancy of the medium, i.e., by the nonlinear law of its volume variation in the shear stress field. With the use of shear waves of different polarizations, the anisotropy of the nonlinearity of the medium is demonstrated. The observation of the cross-modulation effect shows that the nonlinearity-induced modulation components of the probe wave are much more sensitive to weak nonstationary perturbations of the medium, as compared to the linearly propagating fundamental harmonic. The nonlinear effects under study offer promise for diagnostic applications in laboratory measurements and in seismic monitoring systems.     

非线性声波方程二次谐波高阶近似解及实验验证

非线性声波方程的传统二次谐波二阶近似解不够精确,对测量材料的非线性系数造成较大误差。利用摄动法展开非线性声波方程得到一系列非齐次偏微分方程,根据低阶解的性质拟定高阶特解的形式,通过符号计算求出高次谐波特解,对所有二次谐波成分求和最终获得二次谐波的高阶近似解。在水中开展非线性声学实验验证高阶近似解,结果表明:二次谐波相对幅度随传播距离和输入信号均呈现先增加后减小的趋势,高阶近似解即使在传播距离不太短和输入信号不太弱的宽泛条件下也能更好地吻合实验结果。得到的高阶近似解不仅弥补了现有二次谐波幅度理论公式的不足,进行非线性声学实验时还可扩展测量范围,充分利用实验数据也能提高非线性系数的测量精度。      

PARAMETRIC SIMULTONS IN ONE-DIMENSIONAL NONLINEAR LATTICES

Parametric simultaneous solitary wave (simulton) excitations are shown to be possible in nonlinear lattices. Taking a one-dimensional diatomic lattice with a cubic potential as an example, we consider the nonlinear coupling between the upper cut-off mode of acoustic branch (as a fundamental wave) and the upper cut-off mode of optical branch (as a second harmonic wave). Based on a quasi-discreteness approach the Karamzin－Sukhorukov equations for two slowly varying amplitudes of the fundamental and the second harmonic waves in the lattice are derived when the condition of second harmonic generation is satisfied. The lattice simulton solutions are given explicitly and the results show that these lattice simultons can be nonpropagating when the wave vectors of the fundamental wave and the second harmonic waves are exactly at π/a (where a is the lattice constant) and zero, respectively.     

非线性有限束光声效应理论

当一束受正弦调制的激光入射于光声腔中的固体样品上时,由于非线性的光声效应,在光声腔中不仅能接收到基频成份的声信号,还能接收到其二次谐波成份。本文提出一个非线性热波束方程及其相应的非线性边界条件。借助于逐步近似法在光源为高斯径向分布的情形下,求解这一方程。利用Hankel变换获得这一方程的一级与二级近似解。解析结果表明,二次谐波的热波束仍然维持高斯径向分布,而其高斯半径比基频成份小。分析还表明,二次谐波的振幅不仅与线性热参数,而且也与非线性热参数有关。后者或许能提供样品的更多的有意义的信息。综合其各种特点, 关键词：     

A Note on Second Harmonic Resonance of a Magneto-Acoustic Wave in Planar Plasma-Filled Waveguide

The second harmonic generated by non-linear self interaction of a magneto-acoustic wave in a planar plasma-filled waveguide is a freely propagating mode. Coupled-mode equations for resonant nonlinear interaction of this second harmonic mode with the fundamental mode are derived. Using these equations the possibility of generation of the second harmonic is analysed.     

Interactive length of fundamental wave and second harmonic generated on the surface of anomalous dispersion medium

In this Letter, a new method is presented to calculate the interactive length between the fundamental wave and second harmonic generation(SHG) for the configuration of total internal reflection on the inner surface of a nonlinear crystal. Three independent experiments are designed to measure the bandwidths of this second harmonic wave. The theoretical expression of the intensity of SHG is obtained through a nonlinear coupled wave equation. The interactive length of this phase-matched SHG can be calculated mathematically by utilizing the measured bandwidths and the intensity equation. There is no existing method to obtain the interactive length either from theoretical calculations or by experimental measurement. This method can be applied to estimate the extremely short interactive volume in nonlinear processes.     

Nonlinear acoustic response through minute surface cracks: FEM simulation and experimentation

The second harmonic of a Rayleigh wave passing through a minute surface crack has been numerically analyzed by semi-explicit FEM including special elements which account for a nonlinear stress-strain relation at crack surfaces. Minute cracks perpendicular to a free, flat surface close under compressive stress when width of the crack opening is less than the longitudinal amplitude of the Rayleigh wave. Thereafter, compressive and shear stresses are partially transmitted through the closed cracks, whereas tensile and shear stresses are not transmitted through cracks that remain open. This leads to marked nonlinear ultrasonic response. Calculation was performed for an aluminum block having a surface crack. The transverse component of the Rayleigh wave propagating through the cracks shows distorted waveforms, making the second harmonic amplitude clearly noticeable. In an experiment, the second harmonic component of the leaky Rayleigh wave was detected for a simple crack model consisting of two aluminum blocks, by use of a PVDF line-focused transducer. The results of the experiment show that the second harmonic amplitude is a second-order function of the fundamental wave amplitude, and is more pronounced for low compressive stress applied to close the crack surfaces.     

Wave processes in microinhomogeneous elastic media with hysteretic nonlinearity and relaxation

The nonlinear propagation of an initially harmonic acoustic wave in a microinhomogeneous medium containing defects with quadratic hysteretic nonlinearity and relaxation is studied by the perturbation method. The frequency dependences of the effective nonlinearity parameters are determined for the self-action of the quasi-harmonic acoustic wave and the higher harmonic generation processes.     

An Intense Wave in Defective Media Containing Both Quadratic and Modular Nonlinearities: Shock Waves,Harmonics, and Nondestructive Testing

The observed nonclassical power-law dependence of the amplitude of the second harmonic wave on the amplitude of a harmonic pump wave is explained as a phenomenon associated with two types of nonlinearity in a structurally inhomogeneous medium. An approach to solving the inverse problem of determining the nonlinearity parameters and the exponent in the above-mentioned dependence is demonstrated. To describe the effects of strongly pronounced nonlinearity, equations containing a double nonlinearity and generalizing the Hopf and Burgers equations are proposed. The possibility of their exact linearization is demonstrated. The profiles, spectral composition, and average wave intensity in such doubly nonlinear media are calculated. The shape of the shock front is found, and its width is estimated. The wave energy losses that depend on both nonlinearity parameters—quadratic and modular—are calculated.     

New nonlinear intensity Kerr effect in the polar geometry

The problem of the electromagnetic wave reflection on the second optical harmonic from a semi-infinite optically isotropic magnetic medium is considered for the uniform magnetization direction corresponding to the polar Kerr effect. In the first approximation in magnetization, the method of Green’s tensor functions is used to derive expressions for the complex amplitudes of the wave fields for the incidence of s-and p-polarized waves (as well as of their superposition) on the medium. It is shown that, in the latter case, the nonlinear polar Kerr effect is of the intensity type. The dependences of the intensity effect on the angle of incidence of the inducing wave and on the angle of its polarization, which were obtained in numerical experiments, are presented. A comparative analysis of the linear and nonlinear intensity Kerr effects is carried out.     

Second Harmonic Generation of Lamb Wave in Numerical Perspective

The influences of phase and group velocity matching on cumulative second harmonic generation of Lamb waves are investigated in numerical perspective.Finite element simulations of nonlinear Lamb wave propagation are performed for Lamb wave mode pairs with exact and approximate phase velocity matching,with and without group velocity matching,respectively.The evolution of time-domain second harmonic Lamb waves is analyzed with the propagation distance.The amplitudes of primary and second harmonic waves are calculated to characterize the acoustic nonlinearity.The results verify that phase velocity matching is necessary for generation of the cumulative second harmonic Lamb wave in numerical perspective,while group velocity matching is demonstrated to not be a necessary condition.     

On the possibility of soliton-like double-frequency propagation of femtosecond pulses in an optical fiber under conditions of second harmonic generation

A soliton-like mode of double-frequency propagation of femtosecond pulses during second harmonic generation (SHG) in optical fibers is predicted on the basis of computer simulation. The essence of this mode is that, under certain conditions, the central part of the pulses of both the fundamental and the second harmonic waves propagate without a change in the intensity of either wave. This mode can be realized by either of two schemes of the SHG process. In the first scheme, the second harmonic enters a nonlinear medium with some definite input amplitude and definite phase shift with respect to the fundamental wave. In the other scheme, the second harmonic has a zero input amplitude and a phase shift is introduced into the interacting waves in particular cross sections of the medium.     

Nonlinear wave processes in porous waterlike media containing a system of capillaries partially filled with a viscous liquid

A nonlinear dynamic state equation of waterlike porous material that contains a system of cylindrical capillaries partially filled with viscous liquid was received. It is shown that an acoustic nonlinearity of such media contains the relaxation elastic and inelastic components due to the nonlinear dependence of the capillary and viscous pressure in fluid on the capillary diameter. For the medium, theoretical study of such nonlinear phenomena as generation of the second harmonic and a difference frequency wave, self-demodulation of high-frequency pulses as well as the change in the propagation velocity and absorption coefficient of a test wave being under an action of static loading have been carried out. The frequency dependences of medium nonlinearity parameters for these processes were determined.