The inverse problem based on a full dispersive wave equation

The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation.First,a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory,and the dispersion characteristics are discussed.Second,based on the full dispersive wave equation,an inverse problem for determining the four unknown coefficients of wave equation is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves,and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequencies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.     

Concrete damage diagnosed by using non-classical nonlinear acoustic method

<正>It is known that the strength of concrete is seriously affected by damage and cracking.In this paper,six concrete samples under different damage levels are studied.The experimental results show a linear dependence of the resonance frequency shift on strain amplitude at the fundamental frequency,and approximate quadratic dependence of the amplitudes of the second and third harmonics on strain amplitude at the fundamental frequency as well.In addition,the amplitude of the third harmonics is shown to increase with the increase of damage level,which is even higher than that of the second harmonics in samples with higher damage levels.These are three properties of non-classical nonlinear acoustics.The nonlinear parameters increase from 10~6 to 10~8 with damage level,and are more sensitive to the damage level of the concrete than the linear parameters obtained by using traditional acoustics methods.So,this method based on non-classical nonlinear acoustics may provide a better means of non-destructive testing(NDT) of concrete and other porous materials.     

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.     

THE REFLECTION AND REFRACTION OF A LARGE AMPLITUDE PLANE SOUND WAVE IN TWO DIMENSIONS

By using the successive approximation method,a wave equation,in which the second order ofapproximation is taken into account,has been derived from the basic equations of hydrodynamicsin Lagrangian coordinates.The reflection and refraction of a large amplitude plane sound waveon the plane interface of two non-mixing liquids are studied.It is interesting to point out that thereflection and refraction coefficients of second approximation not only depend on parameters ofmedia,but also relate to the position of sound source.It is found that the amplitudes of the secondorder reflected wave and refracted wave do not keep constant at the wavefront anymore as inlinear acoustics.Hence there are reflected wave and refracted wave which propagate in the directionother than that of waves,travelling along the directions defined by laws of reflection and refractionin linear acoustics.The nonlinear effects have to be taken into account in order to understand thephysical meaning of the results reported by Muir et al..     

Three-dimensional nonlinear theory of travelling wave tubes and simulation

A three-dimensional (3D) nonlinear theory of travelling wave tubes (TWTs) is developed, which includes a fundamental radio frequency (RF) and harmonics. When the instantaneous bandwidth exceeds an octave, the harmonic is generated and the mutual coupling between the harmonic and the fundamental RF can be observed in TWTs due to nonlinear interaction between the electron beam and the RF. At low frequencies the harmonic has an obvious effect.Based upon Tien‘‘s disc model, a plastic 3D super-particle model is proposed to improve the nonlinear analysis of TWTs.Numerical results employing a periodic magnetic focusing field are presented.     

Particle Simulation of Electrostatic Waves Driven by an Electron Beam

A one-dimensional electrostatic particle-in-cell simulation is performed to study electrostatic wave excitation due to an electron beam in a plasma system. The excited fundamental and harmonic waves are analyzed with the fast Fourier transformation and the wavelet transformation. The second harmonic is suggested to be generated by wave-wave coupling during the nonlinear evolution, which involves forward propagating and backward propagating Langmuir waves. Furthermore, the background electrons may be heated and accelerated by the electrostatic waves.     

Theoretical Development of Nonlinear Spring Models for the Second Harmonics on an Interface between Two Solids

Based on the exact solutions for the second-harmonic generations of the fundamental longitudinal and transverse waves propagating normally through a thin elastic layer between two solids, the approximate representations termed as ＇nonlinear spring models＇ relating the stresses and displacements on both sides of the interface are rigorously developed by asymptotic expansions of the wave fields for an elastic layer in the limit of small thickness to wavelength ratio. The applicability for the so-called nonlinear spring models is numerically analyzed by comparison with exact solutions for the second harmonic wave reflections. The present nonlinear spring models lay a theoretical foundation to evaluate the interracial properties by nonlinear acoustic waves.     

Double-Parameter Solutions of Projective Riccati Equations and Their Applications

The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.     

Fifth—Order Harmonic Generation using a Coherent Controlled Two—Pulsed Optical Field

We have experimentally studied the characteristics of fifth-order harmonic radiation produced by two coherent femtosecond laser pulses with a changeable relative phase.The intensities of harmonic generation are found to increase with the coherent degree.In one optical Period,the temporal variation of harmonics exhibits an asymmetric characteristic,which is interpreted in terms of ionization theory and the deformation of the wave packets of fundamental field contribution to harmonic generation.     

Bifurcation and solitary waves of the nonlinear wave equation with quartic polynomial potential

For the nonlinear wave equation with quartic polynomial potential, bifurcation and solitary waves are investigated. Based on the bifurcation and the energy integral of the two-dimensional dynamical system satisfied by the travelling waves, it is very interesting to find different sufficient and necessary conditions in terms of the bifurcation parameter for the existence and coexistence of bright, dark solitary waves and shock waves. The method of direct integration is developed to give all types of solitary wave solutions. Our method is simpler than other newly developed ones. Some results are similar to those obtained recently for the combined KdV－mKdV equation.     

一种全频散波方程的反问题

基于一种全频散波方程研究了对于谐波和波包的反问题。首先根据Mindlin理论建立了描述无耗散微结构线性固体中波传播模型一一一种全频散波方程,并讨论了其频散特性。然后基于该全频散波方程,提出了利用四种不同谐波的频率和相应波数确定波方程四个未知系数的反问题,并用严格的数学理论论证了此反问题。研究证明,通过测量同一种无耗散微结构线性固体中传播的四种不同谐波的频率和相应波数,在正常频散和反常频散情况下可唯一地确定波方程的未知系数,即材料的未知参数。      

Generation of Elastic Waves by a Magnetic Field Associated with Reversible Rotations in Triaxial Magnets

In the first-order anharmonicity approximation of Hooke's law, the amplitudes of elastic waves, absorption coefficients, wave numbers of the fundamental wave and of the first harmonic generated by an alternating magnetic field with a preset orientation relative to the basis axes of a crystal having arbitrary dimensions are calculated for multidomain magnets with rigidly fixed domain boundaries in terms of concentrations of magnet phases and magnetic structure parameters with allowance for the wave equation and angular momenta.     

材料非线性衰减系数的二次谐波测量方法研究

采用有限幅值法测量材料在基波和非线性引起的二次谐波作用下的衰减系数:利用准线性下的KZK方程推导基波和二次谐波的声压分布,并提取波束修正系数;采用短纯音信号进行非线性实验,对检测得到的基波和二次谐波声压进行衍射修正处理,有效抑制衍射对衰减系数测量的不利影响,继而通过线性拟合的方法计算得到更精确的基波和二次谐波的衰减系数。以水为例进行实验,研究了实验测量所得衰减系数的频率依赖关系,结果表明在非线性条件下水的衰减系数与频率间存在较强的线性关系,而线性条件下衰减系数随频率呈现二次方增长的特性则不适用于非线性条件。该研究提出了准确测量非线性声波衰减系数的方法,为更有效地应用非线性超声检测提供理论依据。      

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.     

The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid

The method of multiple scales is used to derive the fourth-order nonlinear Schrödinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer. The new terms of this equation describe the third-order linear dispersion effect and the nonlinearity dispersion effects. As the nonlinearity and the dispersion decrease, the equation uniformly transforms into the nonlinear Schrödinger equation for Stokes waves on the surface of a finite-depth fluid that was first derived by Hasimoto and Ono. The coefficients of the derived equation are given in an explicit form as functions of kh (h is the fluid depth, and k is the wave number). As kh tends to infinity, these coefficients transform into the coefficients of the NSEIV that was first derived by Dysthe for an infinite depth.     

Testing for inhomogeneity of the nonlinear parameter in an acoustic medium by ultrasonic phase conjugation

The principle of applying a selective phase conjugation of the second harmonic of a focused ultrasonic wave to diagnosing inhomogeneity of the nonlinear parameter in an acoustic medium is considered. A solution to the three-dimensional problem of harmonic generation by phase-conjugated waves in a nonlinear medium with a localized isoechogenous inclusion is obtained. The signal amplitudes detected by a transmitting-receiving transducer at the second and forth harmonics of a probe wave are calculated for varying position of the inclusion relative to the focus.     

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.     

表面张力对Rayleigh-Taylor不稳定性谐波的影响

Using the method of the parameter expansion up to the third order, explicitly investigates surface tension effect on harmonics at weakly nonlinear stage in Rayleigh-Taylor instability (RTI) for arbitrary Atwood numbers and compares the results with those of classical RTI within the framework of the third-order weakly nonlinear theory. It is found that surface tension strongly reduces the linear growth rate of time, resulting in mild growth of the amplitude of the fundamental mode, and changes amplitudes of the second and third harmonics, as is expressed as a tension factor coupling in amplitudes of the harmonics. On the one hand, surface tension can either decrease or increase the space amplitude; on the other hand, surface tension can also change their phases for some conditions which are explicitly determined.