Nonlinear wave propagation in optical fibres

A generalised mathematical theory leading to the evolution equation of nonlinear pulses propagating in an optical fibre waveguide is presented. The magnitudes of the optical self-steepening and third-order dispersion coefficients are calculated and their relative importance is assessed. The possibility of self-steepening is carefully examined and represented on a pulse-displacement-distance parameter diagram. It is concluded, for specimen single mode fibre data, that, even for a 1% deviation from zero group dispersion, the shock term interacts with the group dispersion to produce a pulse distortion in the form of a velocity change limited by the group dispersion rather than selfsteepening. Also, for such a fibre, self-steepening is so small that vast runs down the fibre are required for it to be seen. Any observation of self-steepening in fibres will require much closer tuning to zero group dispersion and will need the suppression of the third-order term.     

Derivative artificial compression method for conservation laws with multi-dimensional extension

A new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.     

多铁性扇环截面柱体中的波传播

论文基于线性磁电弹性理论，研究了具有扇环形截面的多铁性柱形波导中的弹性波传播问题.利用波动势函数法，解析推导获得波动特征方程，进而得到弥散关系.通过算例研究了波传播的关键特性，深入分析了弥散曲线、相速度曲线和截止频率变化情况.结果显示，波的相速度和截止频率非常依赖于波导结构的扇环截面半顶角、内外径比和层合界面的弱界面系数，对于给定材料的波导结构，这些参数也是控制其弥散特性的重要影响因素.值得指出的是，在柱面应力自由的边界条件下相速度曲线中存在独特的频率带隙，而这通常是在周期结构中才会出现.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

Self-interrupted regenerative metal cutting in turning

A new approach is used to study the global dynamics of regenerative metal cutting in turning. The cut surface is modeled using a partial differential equation (PDE) coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool. This approach automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Taylor's 3/4 power law model for the cutting force is adopted. Lower dimensional ODE approximations are obtained for the combined tool-workpiece model using Galerkin projections, and a bifurcation diagram computed. The unstable solution branch off the subcritical Hopf bifurcation meets the stable branch involving self-interrupted dynamics in a turning point bifurcation. The tool displacement at that turning point is estimated, which helps identify cutting parameter ranges where loss of stability leads to much larger self-interrupted motions than in some other ranges. Numerical bounds are also obtained on the parameter values which guarantee global stability of steady-state cutting, i.e., parameter values for which there exist neither unstable periodic motions nor self-interrupted motions about the stable equilibrium.     

A fast Godunov method for the water‐hammer problem

An efficient Godunov‐type numerical method with second‐order accuracy was developed to simulate the water‐hammer problem in piping. The exact solutions of the Riemann problem were analysed and illustrated on the intriguing solution diagram by properly introducing dimensionless variables within reasonably practical ranges. Based on the solution diagram, an efficient fast Riemann solver was also developed. Moreover, small perturbation analysis was performed to demonstrate the relations between the primitive variables, velocity and pressure, for the Riemann problem. The typical shock‐tube problem and the water‐hammer problem were implemented as sample ones to test the numerical method. It was shown that the present numerical method incorporated with Van Leer's flux limiter is a promising one to simulate fluid transient problem for piping in the present study. Copyright © 2002 John Wiley & Sons, Ltd.     

固定网格下的特征线法求解溃坝问题

溃坝问题是典型的非线性双曲方程的Riemann问题，其数值求解的难点在于对间断面的捕捉以及避免间断面处在数值计算过程中产生数值色散，因而为求解此问题所产生的各种数值计算方法的优劣也体现在这两个方面。本文针对溃坝问题提出一种新的计算方法。该方法基于对偶变量推导的浅水波方程，根据方程的特点，从方程的特征值和黎曼不变量出发，采用高精度的激波捕捉方法计算黎曼不变量的位置随时间的变化，然后映射至不随时间变化的固定网格。根据黎曼不变量的位置，采用保形分段三次Hermite插值将物理量映射至网格节点。计算结果显示，该方法不仅操作简单，计算量小，而且结果准确。     

Singularities and spectral asymptotics of a random nonlinear wave in a nondispersive system

In this paper, we study the propagation of high-intensity acoustic noise in free space and in waveguide systems. A mathematical model generalizing the Burgers equation is used. It describes the nonlinear wave evolution inside tubes of variable cross-section, as well as in ray tubes, if the geometric approximation for heterogeneous media is used. The generalized equation transforms to the common Burgers equation with a dissipative parameter, known as the “Reynolds–Goldberg number”. In our model, this number depends on the distance travelled by the wave. With a zero “viscous” dissipative term, the model reduces to the Riemann (or Hopf) equation. Its solution presents the field by an implicit function. The spectral form of this solution makes it possible to derive explicit expressions for both dynamic and statistical characteristics of intense waves. The use of a spectral approach allowed us to describe the high-intensity noise in media with zero and finite viscosity. Applicability conditions of these solutions are defined. Since the phase matching is fulfilled for any triplet of interacting spectral components, there is an avalanche-like increase in the number of harmonics and the formation of shocks. The relationship between these discontinuities and other singularities and the high-frequency asymptotic of intense noise is studied. The possibility is shown to enhance nonlinear effects in waveguide systems during the evolution of noise.     

Founding editor Dr Philip M. Gresho

A numerical technique is presented for the approximation of vertical gradient of the non‐hydrostatic pressure arising in the Reynolds‐averaged Navier–Stokes equations for simulating non‐hydrostatic free‐surface flows. It is based on the Keller‐box method that take into account the effect of non‐hydrostatic pressure with a very small number of vertical grid points. As a result, the proposed technique is capable of simulating relatively short wave propagation, where both frequency dispersion and non‐linear effects play an important role, in an accurate and efficient manner. Numerical examples are provided to illustrate this; accurate wave characteristics are already achieved with only two layers. Copyright © 2003 John Wiley & Sons, Ltd.     

Bloch–Floquet waves and localisation within a heterogeneous waveguide with long cracks

A bi-material waveguide is assumed to have an array of sufficiently long cracks parallel to the boundaries. The Bloch–Floquet waves propagating along such a waveguide are dispersive, and the band gaps are clearly identified. Slow waves are supported by a system of long cracks, and such modes are represented by the flat dispersion surfaces. Asymptotic analysis combines a lower-dimensional approximation together with the boundary layers occurring near the crack tips. Stress intensity factors are evaluated via the boundary layer analysis, which is matched with the outer fields corresponding to the lower-dimensional model. Evolution of such an elastic system is discussed as the cracks grow as a consequence of the stress concentration, which occurs for some slow waves leading to the crack opening. The asymptotic analysis is supplied with numerical simulations and physical examples.     

Propagation,Dispersion, and Localization of Elastic Waves in Low-Symmetric Anisotropic Waveguides

The results of a theoretical investigation into ultrasonic linear normal wavefields in anisotropic three-dimensional bodies with mechanical orthorhombic symmetry are presented. A detailed study is made of the dispersion dependencies for higher normal wave modes in a single-layered orthorhombic plate-type waveguide. Moreover, the distribution of complex branches corresponding to the edge standing wave modes is studied and their role in the transformation of the entire spectrum upon a change in the travel direction in the waveguide plane is analyzed. A new type of localization of the higher modes of high-frequency short-range normal waves in a crystal layer is described. An efficient method is developed for studying the spectrum of ultrasonic normal waves in a circular cylindrical waveguide made of an orthorhombic monocrystal     

Summation of Gaussian beams in a surface waveguide

The Gaussian beams summation method is applied to a surface waveguide problem in the parabolic approximation. It is shown by means of numerical comparison with an exact solution that the method is effective for computations. Its accuracy does not depend on the complication of the ray field, but the method fails for long distances from the source because of the beam's spreading if a large parameter of the problem is fixed.     

Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.     

Effect of a viscous liquid loading on Love wave propagation

This paper describes a theory of surface Love waves propagating in a layered elastic waveguide loaded on its surface by a viscous (Newtonian) liquid. An analytical expression for the complex dispersion equation of Love waves has been established. The real and imaginary parts of the complex dispersion equation were separated and resulting system of nonlinear algebraic equations was solved numerically. The influence of the viscosity of liquid on the dispersion curves of phase velocity, the wave attenuation and the distribution of the Love wave amplitude is analyzed numerically. The propagation loss is produced only by the viscosity of liquids. Elastic layered waveguide is assumed to be loss-less. The numerical solutions show the dependence of the phase velocity change, the wave attenuation and the wave amplitude distribution in terms of the liquid viscosity and the wave frequency. The results of the investigations are fundamental and can be applied in the design and development of liquid viscosity sensors and biosensors, in Non-Destructive Testing (NDT) of materials, in geophysics and seismology.     

An integration method for detecting chaotic responses of nonlinear systems subjected to double excitations

A methodology designed for identifying chaos of the nonlinear systems subjected to double excitations is proposed. Based on simulations in this study, it is shown by bifurcation diagram that method of Poincaré sections, the conventional chaos-observing method, fails to pinpoint the onset of chaotic motions with the nonlinear systems subjected to double excitations. To remedy this problem, “K_s integration method” is proposed, which integrates the distance between trajectories and origin in phase plane over an excitation period and designates the obtained integration values as K_s's to take the roles of the sampling points derived by Poincaré sections in constructing bifurcation diagram. This “K_s integration method” is shown capable of providing valuable information in bifurcation diagram such that the parameter range leading to chaos can be easily decided and the number of distinguishable time-domain responses can be determined.     

Approximate method of designing a two-element airfoil

An approximate method is proposed for designing a two-element airfoil. The method is based on reducing an inverse boundary-value problem in a doubly connected domain to a problem in a singly connected domain located on a multisheet Riemann surface. The essence of the method is replacement of channels between the airfoil elements by channels of flow suction and blowing. The shape of these channels asymptotically tends to the annular shape of channels passing to infinity on the second sheet of the Riemann surface. The proposed method can be extended to designing multielement airfoils.     

Torsional wave in a circular micro-tube with clogging attached to the inner surface

In the present paper, we study the torsional wave propagation along a micro-tube with clogging attached to its inner surface. The clogging accumulated on the inner surface of the tube is modeled as an "elastic membrane" which is described by the so-called surface elasticity.A power-series solution is particularly developed for the lowest order of wave propagation.The dispersion diagram of the lowest-order wave is numerically presented with the surface(clogging) effect.     

Existence of backward propagating acoustic waves in supported layers

Conditions for the existence of acoustic waveguide modes with the direction of the group velocity opposite to that of the phase velocity in supported layers are investigated. We begin with a study of a clamped-free layer and show that the occurrence of the negative slope in the dispersion of the second and higher order modes leading to backward propagation is a commonly encountered phenomenon related to accidental degeneracies between longitudinal and transverse thickness resonances. For a layer on an elastic substrate, the negative dispersion slope exists only when the transverse velocity of the layer is very small compared to that of the substrate, which makes backward propagation a rarely occurring phenomenon in real structures. Finally, we explain how mode-crossing in certain bi-layer structures results in the negative slope in the dispersion of the fundamental mode.     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

An Improved Weighting Method for Inversely Determining Elastic Constants of Coating Layers from Dispersion Curves of Surface Acoustic Waves

This paper investigates the inversion of elastic constants of a coating layer from measured dispersion curves of a layered half-space sample. A systematic analysis on the sensitivity of dispersion curves is performed and the results provide important information in constructing a better inversion process. Numerical simulation has been given to demonstrate several possibilities for obtaining accurate elastic constants from the measured dispersion data over different frequency domains. Experiments have also been carried out using broadband Poly(vinylidene fluoride-trifluoroethylene) [P(VDF-TrFE)] focusing transducers and their surface wave measurement system over a wide frequency range of 4?~?120 MHz. Nickel coating layers electroplated on thick glass substrates are tested for the inversion. It is observed that the proposed inversion method, which carefully chooses the dispersion data at specific frequency ranges as well as adding some weightings to them, is indeed significantly improve the accuracy and convergence of the obtained elastic constants of the coated nickel layers. Young’s modulus of the coated nickel layers is also measured by a nano-indentation system and the results show good agreement with the data determined by the proposed inversion method.