一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

Evolution equation for nonlinear Scholte waves

The evolution equations for nonlinear Scholte waves (finite amplitude elastic waves propagating along liquid/solid interface), which account for the second order nonlinearity of a liquid, are derived for the first time. For mathematical simplicity the nonlinearity of the solid, which influence is expected to be weak in the case of weak localization of the Scholte wave, is not taken into consideration. The analysis of these equations demonstrates that the nonlinear processes contributing to the evolution of the Scholte wave can be divided into two groups. The first group includes nonlinear processes leading to wave spectrum broadening which are common to bulk pressure waves in liquids and gases. The second group includes the nonlinear processes which are active only in the frequency down-conversion (leading to wave spectrum conservation or narrowing), which are specific to the confined nature of the interface wave. It is demonstrated that the nonlinear parameters, which characterize the efficiency of various nonlinear processes in the interface wave, strongly depend on the relative properties of the contacting liquid and solid (or, in other words, on the deviation of the Scholte wave velocity from the velocities of sound in liquid and in solid). In particular, the sign of the nonlinear parameter responsible for the second harmonic generation can differ from the sign of the nonlinear acoustic parameter of the liquid. It is also verified that there are particular liquid/solid combinations where the nonlinear processes, which are inactive in the frequency up-conversion, dominate in the evolution of the Scholte wave. In this case distortionless propagation of the finite amplitude harmonic interface wave is possible. The proposed theory should find applications in nonlinear acoustics, geophysics, and nondestructive testing.     

脉冲反转技术在金属疲劳损伤非线性超声检测中的应用

材料性能退化总是伴随着某种形式的材料非线性力学行为,从而引起超声波传播的非线性,即高频谐波的产生。研究了利用脉冲反转技术测量金属材料超声学非线性系数的实验方法和信号处理算法,发展了一套可靠的测试实验系统,在相同条件下测量了同一试样在不同输入电压下的二次谐波和基波幅度,二次谐波幅度和基波幅度的平方近似成线性关系,表明实验系统是可靠的。利用该系统进行了一组LY12铝合金疲劳试样非线性超声检测实验。实验结果表明,超声非线性系数可以表征镁合金的疲劳早期退化,脉冲反转技术能够有效提取二次谐波时域信号,增强二次谐波的幅度,抑制主要由实验系统所产生的奇次谐波分量,为材料和结构早期力学性能退化的无损检测和评价提供一种有效的方法。     

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO/sub 3/) crystal, and its third-order elastic constant c/sub 444/ is determined.     

Evaluation of Fatigue Crack Orientation Using Non-collinear Shear Wave Mixing Method

In this paper, a non-collinear shear wave mixing technique is proposed for evaluation of fatigue crack orientation. Numerical analysis of the nonlinear interaction of two shear waves with crack is performed using two-dimensional finite-element simulations. The simulation results show that the nonlinear interaction of the two shears waves with cracks leads to the generation of transmitted and reflected sum-frequency longitudinal waves (SFLW), moreover the propagation direction of reflected SFLW is correlated with the orientation of crack, which can be used for crack orientation evaluation. Non-collinear wave-mixing experiments were conducted on specimens with fatigue crack. The experimental results show that the directivity of the generated SFLW agrees well with the simulation results, and non-collinear shear wave mixing has potential use in fatigue crack orientation evaluation.     

Parametric array technique for microbubble excitation

This study investigates the use of an acoustic parametric array as a means for microbubble excitation. The excitation wave is generated during propagation in a nonlinear medium of two high-frequency carrier waves, whereby the frequency of the excitation wave is the difference frequency of the carrier waves. Carrier waves of around 10 and 25 MHz are used to generate low-frequency waves between 0.5 and 3.5 MHz at amplitudes in the range of 25 to 80 kPa in water. We demonstrate with high-speed camera observations that it is possible to induce microbubble oscillations with the low frequency signal arising from the nonlinear propagation process. As an application, we determined the resonance frequency of Definity contrast agent microbubbles with radius ranging from 1.5 to 5 μm by sweeping the difference frequency in the range from 0.5 to 3.5 MHz.     

Axisymmetric waves in electrohydrodynamic flows

The formation of nonlinear axisymmetric waves on inviscid irrotational liquid jets in the presence of radial electric fields is considered. Gravity is neglected but surface tension is considered. Electrohydrodynamic waves of arbitrary amplitude and wavelength are computed using finite-difference methods. Particular attention is paid to nonlinear traveling waves. In the first class of problems, an electric field generated by placing the liquid jet inside a hollow cylindrical electrode held at constant voltage, its axis coinciding with that of the jet, is studied. The jet is assumed to be a perfect conductor whose free surface is stressed by the electric field acting in the hydrodynamically passive annulus. In the second class of problems, the annular gas is a perfect conductor that transmits a constant voltage onto the liquid/gas surface. The liquid axisymmetrically wets a constant-radius cylindrical rod electrode placed coaxially with respect to the hollow outer electrode, and held at a different constant voltage. The fluid dynamics and electrostatics need to be addressed simultaneously in the inner region. Axisymmetric interfacial waves influenced by surface tension and electrical stresses are computed in both cases. The computations are capable of following highly nonlinear solutions and predict, for certain parameter values, the onset of interface pinching accompanied with the formation of toroidal bubbles. For given wave amplitudes, the results suggest that, for the former case, the electric field delays bubble formation and reduces wave steepness, while for the latter case the electric field promotes bubble formation, all other parameters being equal.     

超声导波混频表征和定位早期局部损伤的研究进展

超声导波具有远距离传输的特性，能够快速、有效地大范围检出薄板中的损伤或缺陷。非线性超声导波相较于传统超声导波，主要研究基波与材料中微观组织演化相互作用而产生的高阶谐波，对尺寸远小于基波波长的损伤或缺陷比较敏感。其中，超声导波的二次谐波相对容易激发，已被用于定量评估早期损伤。但是，超声导波的二次谐波容易受到测量系统非线性的干扰，并且无法定位材料中的局部损伤。超声导波混频在频率、模式、传播方向的选择上具有一定的灵活性，克服了二次谐波的缺点。目前，超声导波混频在理论、模拟和实验上取得了一定的进展，已被用于表征和定位金属材料中处于早期阶段的疲劳、热老化、微裂纹、冲击损伤和局部塑性变形等。高频段超声导波混频、兰姆波相向混频和非共线混频中差频谐波或和频谐波的传播性，以及更多类型损伤的定位和表征仍有待进一步研究。     

Fully nonlinear wave-current interactions and kinematics by a BEM-based numerical wave tank

A numerical wave tank (NWT) with fully nonlinear free-surface boundary conditions is developed to investigate nonlinear wave–wave and wave–current interactions and the resulting kinematics. In the present paper, the variation of wave amplitude and wave length of a monochromatic wave under several different speeds of steady uniform currents is studied through direct numerical simulations in the time domain. The nonlinear wave–current interactions are solved using a boundary integral equation method (BIEM) and a Mixed Eulerian–Lagrangian (MEL) time marching scheme. Both a semi-Lagrangian approach and Lagrangian (material-node) approach are employed and their performance is compared. A regridding algorithm based on cubic spline fitting is devised for updating the free-surface moving boundary in a stable and accurate manner. The incident waves are generated by feeding prescribed analytical waves on the input boundary. An efficient artificial numerical beach is devised and applied to dissipate wave energy and minimize wave reflections from the downstream wall. Nonlinear wave kinematics as a result of nonlinear wave–current interactions is calculated and the results are compared with a multi-layer Boussinesq model. The spatial variation of nonlinear wave profiles and kinematics affected by currents are also addressed and discussed.     

Corner problems and global accuracy in the boundary element solution of nonlinear wave flows

The numerical model for nonlinear wave propagation in the physical space, developed by Grilli, et al.^12,13, uses a higher-order BEM for solving Laplace's equation, and a higher-order Taylor expansion for integrating in time the two nonlinear free surface boundary conditions. The corners of the fluid domain were modelled by double-nodes with imposition of potential continuity. Nonlinear wave generation, propagation and runup on slopes were successfully studied with this model. In some applications, however, the solution was found to be somewhat inaccurate in the corners and this sometimes led to wave instability after propagation in time.

In this paper, global and local accuracy of the model are improved by using a more stable free surface representation based on quasi-spline elements and an improved corner solution combining the enforcement of compatibility relationships in the double-nodes with an adaptive integration which provides almost arbitrary accuracy in the BEM numerical integrations. These improvements of the model are systematically checked on simple examples with analytical solutions. Effects of accuracy of the numerical integrations, convergence with refined discretization, domain aspect ratio in relation with horizontal and vertical grid steps, are separately assessed. Global accuracy of the computations with the new corner solution is studied by solving nonlinear water wave flows in a two-dimensional numerical wavetank. The optimum relationship between space and time discretization in the model is derived from these computations and expressed as an optimum Courant number of 0.5. Applications with both exact constant shape waves (solitary waves) and overturning waves generated by a piston wavemaker are presented in detail.     

Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.     

Nonlinear waves in a prestressed elastic tube filled with a layered fluid

In this work, we studied the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an incompressible layered fluid, where the outer layer is assumed to be inviscid whereas the cylindrical core is considered to be viscous. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the longwave approximation is studied. The governing equation is shown to be the Korteweg-de Vries-Burgers' (KdV-B) equation. A travelling wave type of solution to this evolution equation is sought and it is observed that the formation of shock wave becomes evident with increasing core radius parameter.     

Weak or strong nonlinearity: the vital issue

Marine hydrodynamics is characterised by both weak nonlinearities, as seen for example in drift forces, and strong nonlinearities, as seen for example in wave breaking. In many cases their relative importance is still a controversial matter. The phenomenon of particle escape, seen in linear theory, appears to offer a guide to when strongly nonlinear effects will start to become important, and what will happen when they do. In the case of the “ringing” of vertical cylinders in steep waves, particle escape is shown to correspond approximately to local wave breaking, which leads to the cavitation responsible for “ringing”. Another example is rogue waves, where recent results from weakly nonlinear theory are disappointing, and also fail to explain the rogue waves seen in relatively shallow water, as in the data from the Draupner and Gorm platforms. Recent laboratory experiments, too, show wave crests continuing to grow in height after all frequency components have come into phase, which is inconsistent with weakly nonlinear theory. Particle escape, which is more frequent in shallow water, offers a simple alternative explanation for these observations, as well as for the violent motion at the wave crests, which often confuses rogue-wave data. Extreme wave crests have long been known to be strongly nonlinear, so it appears possible that rogue waves are primarily a strongly nonlinear phenomenon. Fully nonlinear computations of two interacting regular waves are presented, to explore further the connection between particle escape and wave breaking. They are combined with Monte-Carlo simulations of particle escape in hurricane conditions, and the very few measurements of large breaking waves during hurricanes. It is concluded that large breaking waves will have occurred about once per hour, and once per 100 h, respectively, in the recent hurricanes LILI and IVAN. These findings call into question the use of non-breaking wave models in the design codes for fixed steel offshore structures.     

Exact nonlinear travelling hydromagnetic wave solutions

Summary Exact travelling wave solutions for hydromagnetic waves in an exponentially stratified incompressible medium are obtained. With the help of two integrals it becomes possible to reduce the system of seven nonlinear PDE's to a second order nonlinear ODE which describes an one dimensional harmonic oscillator with a nonlinear friction term. This equation is studied in detail in the phase plane. The travelling waves are periodic only when they propagate either horizontally or vertically. The reduced second order nonlinear differential equation describing the travelling waves in inhomogeneous conducting media has rather ubiquitous nature in that it also appears in other geophysical systems such as internal waves, Rossby waves and topographic Rossby waves in the ocean.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

Influence of dislocations on magnon-phonon couplings: A phenomenological approach

On the basis of a nonlinear theory of finitely deformable elastoviscoplastic ferromagnetic crystals developed in a companion paper, the present work presents an attempt at a phenomenological study of the influence of dislocations and viscoplastic flow on the behavior of spin waves (the collective modes of oscillations typical of ferromagnetism). This is achieved by linearizing the above mentioned nonlinear theory about a fundamental ferromagnetic phase with a practically vanishing viscoplastic threshold. The main results obtained after a study of wave modes and asymptotic evaluations in terms of a piezomagnetic coupling parameter are the evidence of a magnetoacoustic resonance between spin waves and left circularly polarized transverse elastoviscoplastic disturbances, a slight shift towards higher wave numbers of the corresponding critical wave number as compared to the perfectly elastic-crystal case and the fact that spin waves suffer a damping which is directly proportional to the piezomagnetic coupling parameter and to the reciprocal primary relaxation time (the relaxation time associated with the viscosity processes inherent in viscoplasticity, in the absence of restoring effects.     

Nonlinear oscillations in a thin ring — I. Three-wave resonant interactions

Summary Nonlinear resonant interactions between planar waves in a thin circular ring are investigated. It is found that a high-frequency azimuthal wave is unstable against a pair of secondary low-frequency waves. The secondary waves are of two types; either two bending or azimuthal and bending. These are in phase with the primary wave. All three together compose a resonant triad. Such kind of instability causes the stress amplification in the ring. The stress growth constant and the period of energy exchange between the waves are estimated based on analytical solutions to the evolution equations driving the triad. The lowest-order nonlinear approximation analysis predicts stability for bending waves. A good qualitative agreement of the obtained results with some known experimental data is observed.     

Modulation of stochastic diffusion by wave motion

Pollutants that are chemically inert flow with the carrier fluid passively while diffuse at the same time. In this study, the stochastic diffusion behavior of the passive pollutant in a progressive or standing wave field is examined with analytical means. Our focus is on the nonlinear interactions between the stochastic diffusion and the deterministic wave motions, and we limit the scope to cases whereby a small parameter, ε, exists between the advective and diffusive displacements, which then allows a perturbation analysis to be performed. With a sinusoidal progressive wave, the results show that the deterministic wave motion can either increase or decrease the embedded stochastic diffusion depending on the wave characteristics. Longer wave lengths and shorter wave periods tend to promote diffusion significantly, while shorter wave lengths and longer wave periods act in the opposite manner but with a much smaller effect. An analysis of the standing wave motion, represented by a combination of left and right moving progressive waves, shows that the effects due to two opposing waves to the stochastic diffusion can be superimposed.     

Progressive Cross Waves due to the Horizontal Oscillations of a Vertical Cylinder in Water. Evolution Equations

The paper deals with the theoretical analysis of progressive cross waves excited due to the horizontal oscillations of a vertical, surface-piercing circular cylinder in water of constant depth. Although cross waves are a phenomenon well known in laboratory wave tanks, it seems that they have not been observed around horizontally oscillating structures in fluid up to now. Such observations have recently been carried out by the authors on various models of offshore gravity platforms subjected to earthquake-like horizontal excitation in a water tank. The theoretical analysis of the problem is based on a method developed by Becker and Miles (1992) for the radial cross waves due to the motion of an axisymmetric cylindrical wavemaker. Whitham's average-Lagrangian approach is applied. It is shown that the energy transfer to the cross wave is described by the functional which is quadratic, both in the forced basic wave and in the cross wave. Therefore, the solution to second-order problems is necessary for the derivation of the evolution equations. The evolution of the cross wave is found to be described by two complex nonlinear partial differential equations with coefficients depending on a slow radial variable both in linear and nonlinear terms. The evolution equations are coupled through the nonlinear terms and through the boundary conditions as well.