Generation of the Third Harmonics by Plane Waves in Murnaghan Materials

We demonstrate that it is expedient to use the complete expansion of the potential in terms of strain gradients for materials whose deformation is described by Murnaghan's potential. The cubic terms are retained in the constitutive equations, in addition to the classical quadratic terms. An analysis of the nonlinear system of wave equations reveals that the third harmonics can be generated. As an example, the nonlinear interaction of plane waves is analyzed for the following three cases of waves entering a medium: (i) a longitudinal wave, (ii) a vertically polarized transverse wave, and (iii) vertically and horizontally polarized transverse waves     

Cubically non-linear effects of plane waves in isotropic soft solid materials

In this paper we analyze mathematical properties of an isotropic soft solid model which is characterized by three elastic constants. The model was proposed to interpret measurements of weakly non-linear shear waves in gel-like and tissue-like media. In our analysis we are particularly interested in third order non-linear terms. We present for the first time the full equations of elastodynamics, as well as the equations for plane waves for this model, with cubically non-linear terms. Next, the interaction coefficients for non-linear interactions of three plane waves to produce the fourth wave are explicitly calculated. These coefficients show which of the three waves interact with each other and determine how strong the effect of interaction is on the produced fourth wave. It turns out that these coefficients are expressed in terms of some combinations of three elastic constants. The obtained results can be helpful in experimental determination of elastic constants which describe the model.     

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave     

Self-Switching of Waves in Materials

The problem is formulated on self-switching of acoustic waves in materials when the wavelength does not exceed considerably the representative dimension of the microstructure. Use is made of the microstructural theory of two-component mixtures, which is adapted well to study waves in composites. Truncated and evolutionary equations describing the interaction of two longitudinal plane waves — a strong pumping wave and a weak signal wave — are derived. A procedure of finding the exact solution to the evolutionary equations is described for the case where the wave numbers of the pumping and signal waves are equal. The solution is expressed in terms of the elliptic Jacobi function     

A new system of equations which predicts the evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption

The problem of the capillary-gravity waves which may arise at an interface between two stratified fluids of different densities is investigated. Particular attention is paid to the case when two different wave modes move at the same speed and to the wave train produced by the ensuing interaction. In contrast to most previous studies, the wave steepness and the wave bandwidth are not taken to be of the same order of magnitude, but the latter is of one order smaller. This leads to a system of nonlinear evolution equations which can be used to predict the subsequent progression of the wave field. These equations may be compared with the more usual nonlinear Schrödinger set which are valid under the equal bandwidth assumption and also a recently derived set which describe broader bandwidth waves. A large class of solutions to the equations is found and the corresponding wave profiles are presented.     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

Plane Waves in Cubically Nonlinear Elastic Media

The theory of plane waves in nonlinear materials described by the Murnaghan potential is proposed. The theory takes into account both the classical quadratic nonlinearity and the cubic nonlinearity of the basic wave equations. Some new opportunities for the wave interaction analysis are commented on: in addition to the second harmonics, a longitudinal plane wave generates the third one, a transverse plane wave generates the third harmonics, and horizontally and vertically polarized transverse plane waves jointly generate new waves     

Multiple scales analysis of wave�Cwave interactions in?a?cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass?Cspring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave?Cwave interaction problems.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

孤立波与多孔介质结构物的非线性相互作用

基于精确至Ｏ（εμ＾２,μ＾４）的多孔介质无压渗流模型方程和均匀流体质波动的Ｂｏｕｓｓｉｎｅｓｑ方程,本文对孤立波与多孔介质结构物的相互作用了较系统的数值实验。控制方程采用基于有限差分方程离散,在时域上采用了预估－校正方法进行了时间积分。在求解演化方程的过程中,引入“内迭代”过程实现流体域和多孔介质交界面的连接条件。结果表明孤立波在多孔介质上的反射与在不可渗透的界面上的反射类似,形成反向的孤立波但     

正向爆轰驱动高焓激波风洞的数值模拟

对充满氢氧可燃气体、带扩容腔的正向爆轰驱动的激波风洞进行了数值模拟。计算采用了欧拉方程,频散可控耗散差分格式（DCD）和改进的二阶段化学反应模型。在扩容腔附近采用二维轴对称计算模型,而在驱动段和被驱动段的直管道部分则采用一维计算模型。本文分析了爆轰波在管道中的传播、反射和绕射过程。计算结果表明扩容腔的尺寸对爆轰波的传播、反射、汇聚等起着决定性的作用;带扩容腔的正向爆轰驱动的激波风洞能够得到平稳的持续时间较长的气流,提高了实验的精确度和可重复性。     

Diffraction of cnoidal waves by vertical cylinders in shallow water

Diffraction of nonlinear waves by single or multiple in-line vertical cylinders in shallow water is studied by use of different nonlinear, shallow-water wave theories. The fixed, in-line, vertical circular cylinders extend from the free surface to the seafloor and are located in a row parallel to the incident wave direction. The wave–structure interaction problem is studied by use of the nonlinear generalized Boussinesq equations, the Green–Naghdi shallow-water wave equations, and the linearized version of the shallow-water wave equations. The wave-induced force and moment of the Green–Naghdi and the Boussinesq equations are presented when the incoming waves are cnoidal, and the forces are compared with the experimental data when available. Results of the linearized equations are compared with the nonlinear results. It is observed that nonlinearity is very important in the calculation of the wave loads on circular cylinders in shallow water. The variation of wave loads with wave height, wavelength and the spacing between cylinders is studied. Effect of the neighboring cylinders, and the shielding effect of upwave cylinders on the wave-induced loads on downwave cylinders are discussed.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

The cumulative interaction of finite amplitude waves in strings

A “two time scale” asymptotic expansion procedure describing the modulation of a propagating simple wave governed by a system of non-linear partial differential equations is applied to the deflection waves of non-linear elastic strings. Rapid deflection signals propagating into a general slowly varying disturbance are modulated. In addition, they themselves affect the equations for that disturbance. The two effects are separated naturally when, to prevent the cumulative growth inherent in most “high frequency” procedures, an averaging technique is introduced. The interaction of two deflection waves is given as a specific example.     

A mixture theory analysis for the surface-wave propagation in an unsaturated porous medium

The mixture theory is employed to the analysis of surface-wave propagation in a porous medium saturated by two compressible and viscous fluids (liquid and gas). A linear isothermal dynamic model is implemented which takes into account the interaction between the pore fluids and the solid phase of the porous material through viscous dissipation. In such unsaturated cases, the dispersion equations of Rayleigh and Love waves are derived respectively. Two situations for the Love waves are discussed in detail: (a) an elastic layer lying over an unsaturated porous half-space and (b) an unsaturated porous layer lying over an elastic half-space. The wave analysis indicates that, to the three compressional waves discovered in the unsaturated porous medium, there also correspond three Rayleigh wave modes (R1, R2, and R3 waves) propagating along its free surface. The numerical results demonstrate a significant dependence of wave velocities and attenuation coefficients of the Rayleigh and Love waves on the saturation degree, excitation frequency and intrinsic permeability. The cut-off frequency of the high order mode of Love waves is also found to be dependent on the saturation degree.     

Linear interactions affecting the propogation of waves in a linear elastic fluid

Small linear interactions affecting the propogation of waves in a linear elastic fluid are investigated. These linear interactions may occur as a result of impurities on the surface of a linear elastic fluid. These interactions are imposed on the linear wave equations which were investigated in Momoniat (Propogation of waves in a linear elastic fluid, submitted for publication) using the non-classical contact symmetry method. The occurrence of a small parameter in the wave equations under consideration in this paper makes the problem ideal for analysis using an approximate non-classical contact symmetry method. Approximate contact symmetries and approximate solutions are determined and discussed for the problems under consideration. Comparisons are made with the case of no interaction.     

Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Self-switching of displacement waves in elastic nonlinearly deformed materialsSelf-commutation d'ondes de déplacement dans les milieux élastiques non-linéaires

The problem of self-switching plane waves in elastic nonlinearly deformed materials is formulated. Reduced and evolution equations, which describe the interaction of two waves the power pumping wave and the faint signal wave are obtained. For the case of wave numbers matching the pumping and signal waves, a procedure of finding the exact solution of evolution equations is described. The solution is expressed by elliptic Jacobi functions. The existence of the power wave self-switching is shown and commented. To cite this article: J. Rushchitsky, C. R. Mecanique 330 (2002) 175–180.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 40–51, May 2005.