Permanent Wave Structures and Resonant Triads in a Layered Fluid

A closed three layer fluid with small density differences between the layers has two closely related modes of gravity wave propagation. The nonlinear interactions between the wave modes are investigated, particularly the nearly resonant or significant interactions. Permanent wave solutions are calculated, and it is shown that a permanent wave of the slower mode can generate resonantly a wave harmonic of the faster mode. The equations governing resonant triads of the two modes are derived, and solutions having a permanent structure are calculated from them. It is found that some resonant triad solutions vanish when the triad is embedded in the set of all harmonics with wavenumbers in its neighborhood     

Propagation of weakly nonlinear waves in fluid-filled thick viscoelastic tubes

In the present work, we studied the propagation of small-but-finite-amplitude waves in a prestressed thick walled viscoelastic tube filled with an incompressible inviscid fluid. In order to include the dispersion, the wall's inertial and shear effects are taken into account in determining the inner pressure–inner cross-sectional area relation. Using the reductive perturbation method, the propagation of weakly nonlinear waves in the long-wave approximation is investigated. After obtaining the general evolution equation in the long-wave approximation, by a proper scaling, it is shown that this general equation reduces to the well-known evolution equations such as the Burgers, Korteweg-de Vries (KdV), Koteweg-de Vries–Burgers (KdVB) and the generalized Burgers' equations. By proper re-scaling of the perturbation parameter, the modified form of the evolution equations is also obtained. The variations of the travelling wave profile with initial deformation and the viscosity coefficients are numerically evaluated and the results are illustrated in some figures.     

Numerical study of wave propagation in porous media with the use of the grid-characteristic method

Elastic wave propagation in a porous medium is numerically studied by applying the grid-characteristic method. On the basis of direct measurements of reflected and transmitted wave amplitudes, the reflection and decay coefficients are investigated as depending on the degree of porosity (percentage of the pore volume) and on the type of the filling substance (solid, liquid, or nothing). The reflection and decay coefficients are shown to be closely related to the porosity of the medium, which can be used in geological applications (estimation of porosity) and engineering applications (acoustic response attenuation).     

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

An energy‐preserving scheme is proposed for the three‐coupled nonlinear Schrödinger (T‐CNLS) equation. The T‐CNLS equation is rewritten into the classical Hamiltonian form. Then the spatial variable is discretized by using high‐order compact method to convert it into a finite‐dimensional Hamiltonian system. Next, a second‐order averaged vector field (AVF) method is employed in time which results in an energy‐preserving scheme. Some theoretical results such as convergence are investigated. In addition, it provides some numerical examples to illustrate the robustness and reliability of the theoretical results. It also explores the role of the parameters in the model and initial condition on the wave propagation.     

长波在弯曲管道中的传播——Ⅰ变截面弯管中长波传播的基本分析

本文研究长波在三维变截面弯管中的传播问题.通过建立正交曲线坐标系,以波数k和管道横截面的特征半径a的乘积ka作为小参数,对波动方程进行无量纲处理,用正则摄动法,把三维的Helmholtz方程化为二维的Laplace(或Poisson)方程和一维的Webster方程.并分析了管道的几何参数(横截面面积、管道中心线的曲率和挠度)对复速度势渐近展开的各阶项的影响.文中指出,横截面面积的变化首先影响浙近解的零阶项.在横截面的形状具有某种对称性时,管道中心线的曲率首先影响渐近解的二阶项,而挠度首先影响渐近解的三阶项.最后,给出了长波在弯曲圆管中传播的实例.     

Exact Solutions for Large Amplitude Waves in Dispersive and Dissipative Systems

This paper describes some applications of a transformation which takes a system of two, first order, quasilinear, partial differential equations in two dependent and two independent variables into a similar system. Typically, the equations govern wave propagation in dispersive and dissipative systems. It is shown that certain nonlinear equations which are of current practical interest can be transformed into linear equations.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Models of unidirectional propagation in heterogeneous excitable media

Two differential equation models of excitable media (threshold and recovery kinetics) with solutions that exhibit unidirectional propagation are presented. It is shown that unidirectional propagation in heterogeneous excitable media with non-oscillatory kinetics can be initiated from homogeneous initial data. Simulations on a reaction-diffusion model with FitzHugh-Nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one-dimensional circular spatial domain. An ordinary differential equation model with four semi-coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur.     

Cubic Nonlinearity in Elastic Materials: Theoretical Prediction and Computer Modelling of New Wave Effects

Our object of interest is nonlinear interaction of waves in elastic materials. The new model of a material is proposed that takes into account the mechanism of simultaneous quadratic and cubic nonlinear deformations. Introduction of cubic nonlinearity into the model makes the general wave picture more complicated and creates new possibilities for the wave analysis. We present four possibilities for the evolution of profiles of plane harmonic waves. It is noted that quadratic and cubic nonlinearities emerge first of all in the second and third harmonics generation, respectively. Further, we discuss the results of computer modelling of the wave profile evolution. The influence of the progress of second and third harmonics on the wave profile evolution is studied separately. We study separately how second and third harmonics influence the evolution of the wave profile. We also investigate how the progress of harmonics depends on the initial frequency and amplitude. We find two distinct schemes of the evolution progress: the scheme (in) with four stages for the second harmonics and the scheme with three stages for the third harmonics. As a result the influence of both harmonics could be observed simultaneously, and such a case is demonstrated in the paper. Nevertheless this phenomenon is not necessarily present in every material which explains the absence of experimental observations of the third harmonics by this time.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Wave Propagation in a Layer of Magnetic Liquid

A linearized equation for the propagation of surface gravitational waves in a layer of magnetized liquid of finite depth is examined. The liquid is assumed to be inviscid, incompressible, and to possess magnetization properties in the absence of electrical conductivity, while the motion is assumed to be irrotational. Travelling wave solutions are obtained. The dependences of the phase and group velocities of the magnetic liquid on the magnetic parameters are studied. It is shown that for some values of the magnetic parameters there is an interval of short wavelengths for which the group velocity is negative, which indicates that the wave energy propagates in the negative direction.     

Effect of interconvertibility of electromagnetic and gravitational waves in strong external electromagnetic fields and the propagation of waves in the field of a charged “black hole” : PMM vol. 38, n 6, 1974, pp. 1122–1129

It is shown that the behavior of an arbitrary wave propagating in the field of a nonrotating charged black hole is defined (with the use of quadratures) by four functions. Each of these functions obeys its second order equation of the wave kind. Short electromagnetic waves falling onto a black hole are reflected by its field in the form of gravitational and electromagnetic waves whose amplitude was explicitly determined. In the case of the wave carrying rays winding around the limit cycle the reflection and transmission coefficients were obtained in the form of analytic expressions.Various physical processes taking place inside, as well as outside a collapsing star, may induce perturbations of the gravitational, electromagnetic and other fields, and lead to the appearance in the surrounding space of waves of various kinds which propagate over a distorted background and are dissipated along its inhomogeneities.In the absence of rotation and charge in a star, the analysis of small perturbations of the gravitational fields is based on the system of Einstein equations linearized around the Schwarzschild solution. In [1, 2] this system of equations, after expansion of perturbations in spherical harmonics and Fourier transformation with respect to time, was reduced to two independent linear ordinary differential equations of second order of the form of the stationary Schrödinger equation for a particle in a potential force field. Each of these equations defines one of two possible independent perturbation kinds: “even” and “odd” (the different behavior of spherical tensor harmonics at coordinate inversion is the deciding factor in the determination of the kind of perturbation [1, 2]). Although these equations were derived with the superposition on the perturbations of the metric of specific coordinate conditions, they define, as shown in [4], the behavior of invariants of the perturbed gravitational field, which imparts to the potential barriers appearing in these equations an invariant meaning.The system of Maxwell equations on the background of Schwarzschild solution also reduces to similar equations, which differ from the above only by the form of potential barriers appearing in these [5].In the presence in the unperturbed solution of a strong electromagnetic field the gravitational and electromagnetic waves interact with each other, and transmutation takes place. The train of short periodic electromagnetic waves generates the accompanying train of gravitational waves. This phenomenon was first analyzed in [6] on and arbitrary background. It was shown in [7, 8] that dense stars surrounded by hot plasma may acquire a charge owing to splitting of charges by radiation pressure and the “sweeping out” of positrons nascent in vapors in strong electrostatic fields. The interaction of waves becomes particularly clearly evident in the neighborhood of black holes which may serve as “valves” by maintaining equilibrium between the relict electromagnetic and gravitational radiation in the Universe. Rotation of black holes intensifies this effect [6].If a nonrotating star possesses an electrostatic charge, the definition of perturbations of the electromagnetic and gravitational fields must be based on the complete system of Einstein-Maxwell equations linearized around the Nordström-Reissner solution. (Small perturbations of electromagnetic field outside a charged black hole were considered in [9, 10] on the basis of the system of Maxwell equations on a “rigid” background of the Nordström-Reissner solution, without taking into account the interconvertibility of gravitational and electromagnetic waves, which materially affects their behavior in the neighborhood of a charged black hole). Here this system of equations which define the interacting gravitational and electromagnetic perturbations are reduced to four independent second order differential equations, two for each kind of perturbations (an importsnt part is played here by the coordinate conditions imposed on the perturbations of the metric, proposed by the authors in [4]). Perturbation components of the metric and of the electromagnetic field are determined in quadratures by the solutions of these equations. If the charge of a star tends to vanish, two of the derived equations convert to equations for gravitational waves on the background of the Schwarzschild solution [1, 2], while the twoothers become equations which are equivalent to Maxwell solutions on the same background. The short-wave asymptotics of derived equations is determined throughout including the neighborhood of the limit cycle for the wave carrying rays. These solutions far away from the point of turn coincide with those obtained in [6] for any arbitrary background. Approximation of geometric optics does not provide correct asymptotics for impact parameters of rays which are close to critical for which the Isotropie and geodesic parameters wind around the limit cycle. This case is investigated below.A similar situation in the Schwarzschild field was analyzed in [11], where analytic expressions for the wave reflection and transmission coefficients were determined, and the integral radiation stream trapped by a black hole produced by another radiation component of the dual system was calculated.     

Plasma waves reflection from a boundary with specular accommodative boundary conditions

In the present work the linearized problem of plasma wave reflection from a boundary of a half-space is solved analytically. Specular accommodative conditions of plasma wave reflection from plasma boundary are taken into consideration. Wave reflectance is found as function of the given parameters of the problem, and its dependence on the normal electron momentum accommodation coefficient is shown by the authors. The case of resonance when the frequency of self-consistent electric field oscillations is close to the electron (Langmuir) plasma oscillations frequency, namely, the case of long-wave limit is analyzed in the present paper.     

Numerical Prediction of Aerodynamic Noise Generated from an Aircraft in Low Mach Number Flight

The paper describes numerical prediction of aerodynamic noise generated from an Aircraft. Simulation of turbulent flow is done solving the incompressible Navier-Stokes equation, where turbulence is modeled applying the orthogonal subgrid scale (OSGS) method with dynamical subscales. Because of comparison, the same simulation is done using the LES (Large Eddy simulation). It is shown how simulation of turbulent flow affects the prediction of acoustic sources calculated using Lighthill's analogy. Translation from the time to frequency domain is done through DFT (Direct Fourier Transform), which gives smaller usage of memory. Acoustic sources are used in the inhomogeneous Helmholtz equation to simulate pressure wave propagation in the domain. It is shown that OSGS with dynamical subscales gives better representation of the spectrum. Overall, better prediction of energy transfer across large and small eddies will give better allocation and presentation of acoustics sources. These sources will change wave propagation of the pressure in the acoustic field. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

具有波阻抗不连续特性的粘弹性介质中的逆散射问题

在时间域内讨论了粘弹性介质的逆散射问题,其中粘弹性介质的波阻抗在远离入射波作用面一侧的交界面上是不连接的。介质的散射算子,传播算子所满足的微分积分方程可以用来反演未知的粘弹性介质的松弛模量,文中给出的反演过程只须利用介质层一侧的反射算子在一个走时来回的时间内的实验测量数据。最后,给出了数值算例,计算结果表明,利用方法可以较准确的反演得到材料松弛模量。     

Waves of polarized light in nonlinear birefringent fibre

A propagation of a short optical pulse in nonlinear birefringent fibre is described by a system of two coupled Schrödinger equations. By means of variational Anderson method this system reduces to the system of ordinary differential equations for spatial evolution of pulse parameters. In two ultimate cases the analytical solutions of the equations are managed to be found. It is shown that at some critical power of the input pulse W_c the regime of propagation changes. For the power exceeding W_c the radiation concentrates in one channel. The numerical investigation of the intermediate cases was done when by the variation of the input pulse power one can achieve the comparable effectiveness of the competing processes of dispersion broadening and nonlinear pulse compression. The numerical simulations show that in the range of critical values of the nonlinear coupling coefficient the transition takes place to the chaotic phase and amplitude behavior of the coupled waves of different polarizations. The research is important to understand the processes of ultra short digital pulses propagation in optical fibre links.     

Influence of induced magnetic field on peristaltic flow in an annulus

This paper looks at the influence of the induced magnetic field on peristaltic transport through a uniform infinite annulus filled with an incompressible viscous and Newtonian fluid. The present theoretical model may be considered as mathematical representation to the movement of conductive physiological fluids in the presence of the endoscope tube (or catheter tube). The inner tube is uniform, rigid, while the outer tube has a sinusoidal wave traveling down its wall. The flow analysis has been developed for low Reynolds number and long wave length approximation. Exact solutions have been established for the axial velocity, stream function, axial induced magnetic field, current distribution and the magnetic force function. The effects of pertinent parameters on the pressure rise and frictional forces on the inner and outer tubes are investigated by means of numerical integrations, also we study the effect of these parameters on the pressure gradient, axial induced magnetic field and current distribution. The phenomena of trapping is further discussed.     

Convection-induced anisotropy in excitable media subject to solenoidal advective flow fields

The effects of solenoidal velocity fields on the propagation of spiral waves in excitable media is studied numerically by means of a time-linearized method. It is shown that the advective field distorts the spiral wave at moderate frequencies, whereas, at large frequencies, the average shape of the spiral wave is nearly identical to that in the absence of convection, although its inner and outer parts exhibit spatial oscillations whose frequency increases as that of the velocity field is increased. At low frequencies and high amplitudes of the velocity field, the concentration of the activator and the wave propagation are controlled by the symmetry of the velocity and the number and location of the stagnation points, and the concentration of the activator may exhibit either counter-rotating regions or a layered structure.