Solitary wave interactions and reshaping in coupled systems

The interaction of the components of composite solitary waves governed by nonlinear coupled equations is studied numerically. It is shown how predictions of the known exact traveling wave solutions may help in understanding and explaining the process of reshaping seen as head-on and take-over collisions of individual solitary waves. The most interesting results concern the switch in the sign or the periodic modulation of the amplitude of the solitary wave and the direction of its propagation due to collisions.     

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

爆轰波与金属的斜相互作用

应用二种截然不同的炸药PBX 95 0 1和T2 ,计算了爆轰波在不同入射角下与金属平板的斜相互作用。在正规反射区 ,计算结果与激波极线理论基本一致 ;应用燃烧模型 ,分别计算了Fe球壳装药JB90 0 3(HE)及JB90 14 (IHE)散心爆轰波的绕射传播 ,计算结果与实验很好地符合。在非正规反射区 ,二维拉氏程序计算结果明显地不同于经典理论结果 ,计算中没有出现Mach反射。计算结果显示 ,毗邻介质影响节点附近的爆轰波阵面形状及爆轰波速度 ;不同的反应率函数计算的节点图像不同。     

A two-way model for nonlinear acoustic waves in a non-uniform lattice of Helmholtz resonators

Propagation of high amplitude acoustic pulses is studied in a 1D waveguide connected to a lattice of Helmholtz resonators. An homogenized model has been proposed by Sugimoto (1992), taking into account both the nonlinear wave propagation and various mechanisms of dissipation. This model is extended here to take into account two important features: resonators of different strengths and back-scattering effects. An energy balance is obtained, and a numerical method is developed. A closer agreement is reached between numerical and experimental results. Numerical experiments are also proposed to highlight the effect of defects and of disorder.     

Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves

This paper studies nonlinear waves in a prestretched cylinder composed of a Blatz-Ko material. Starting from the three-dimensional field equations, two coupled PDEs for modeling weakly nonlinear long waves are derived by using the method of coupled series and asymptotic expansions. Comparing with some other existing models in literature, an important feature of these equations is that they are consistent with traction-free surface conditions asymptotically. Also, the material nonlinearity is kept to the third order. As these two PDEs are quite complicated, the attention is focused on traveling waves, for which a first-order system of ODEs are obtained. We use the technique of dynamical systems to carry out the analysis. It turns out that the system is three parameters (the prestretch, the propagating speed and an integration constant) dependent and there are totally seven types of phase planes which contain trajectories representing bounded traveling waves. The parametric conditions for each phase plane are established. A variety of solitary and periodic waves are found. An important finding is that kink waves can propagate in a Blatz-Ko cylinder. We also find that one type of periodic waves has an interesting feature in the profile slope. Analytical expressions for all bounded traveling waves are obtained.     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical computations of two-dimensional solitary waves generated by moving disturbances

Two-dimensional solitary waves generated by disturbances moving near the critical speed in shallow water are computed by a time-stepping procedure combined with a desingularized boundary integral method for irrotational flow. The fully non-linear kinematic and dynamic free-surface boundary conditions and the exact rigid body surface condition are employed. Three types of moving disturbances are considered: a pressure on the free surface, a change in bottom topography and a submerged cylinder. The results for the free surface pressure are compared to the results computed using a lower-dimensional model, i.e. the forced Korteweg–de Vries (fKdV) equation. The fully non-linear model predicts the upstream runaway solitons for all three types of disturbances moving near the critical speed. The predictions agree with those by the fKdV equation for a weak pressure disturbance. For a strong disturbance, the fully non-linear model predicts larger solitons than the fKdV equation. The fully non-linear calculations show that a free surface pressure generates significantly larger waves than that for a bottom bump with an identical non-dimensional forcing function in the fKdV equation. These waves can be very steep and break either upstream or downstream of the disturbance.     

On the RR→MR transition of asymmetric shock waves in steady flows

In this paper, RR→MR transition of asymmetric shock waves has been theoretically studied. The transition can occur between the sonic-point and maximum-deflection criteria due to the the effects of expansion fans which are inherent flow structures. Comparison shows a better agreement among experiments and the analytical results. Some discrepancies reported in previous studies among experiments and theory have also been explained based on the threshold for RR→MR transition.      

Numerical study of formation of solitary strain waves in a nonlinear elastic layered composite material

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.     

Derivation of a higher order nonlinear Schrödinger equation for weakly nonlinear Rossby waves

In the paper, with the help of a perturbation expansion method a new higher order nonlinear Schrödinger (HNLS) equation is derived to describe nonlinear modulated Rossby waves in the geophysical fluid. Using this equation, the modulational wave trains are discussed. It is found that the higher order terms favor the instability growth of modulational disturbances superimposed on uniform Rossby wave trains, but the instability region becomes narrower. In addition, the latitude and uniform background basic flow are found to affect the instability growth rate and instability region of uniform Rossby wave train. However, for a geostrophic flow the background basic flow does not affect the modulational instability of uniform Rossby wave train.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves

A new nonlinear evolution equation is derived for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by a harmonic forcing. Instead of the traditional approach involving a base state of the long wave limit, a base state of harmonic waves is assumed for the perturbation analysis. This approach is considered to be more appropriate for channels of length just a few multiples of the depth. The dispersion relation found approaches the classical long wave limit. The weakly nonlinear dispersive waves satisfy a KdV-like nonlinear evolution equation with steeper nonlinearity.     

Modeling of long nonlinear waves on the interface in a horizontal two-layer viscous channel flow

The dynamics of two-dimensional waves of small but finite amplitude are theoretically studied for the case of a two-layer system bounded by a horizontal top and bottom. It is shown that for relatively large steady-state flow velocities and at certain fluid depth ratios the vertical velocity profile is nonlinear. An evolutionary equation governing the fluid interface disturbances and allowing for the long-wave contributions of the layer inertia and surface tension, the weak nonlinearity of the waves, and the unsteady friction on all the boundaries of the system is derived. Steady-state solutions of the cnoidal and solitary wave type for the disturbed flow are determined without regard for dissipation losses. It is found that the magnitude and the direction of the flow can alter not only the lengths of the waves but also their polarity.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 143–158. Original Russian Text Copyright © 2005 by Arkhipov and Khabakhpashev.     

激波在不同三角楔面内传播特性的数值研究

激波在收缩管内的反射与聚焦会形成高温高压区,点燃可燃混合气并诱导爆轰,因此对爆轰发动机的点火具有重要意义。本文基于二维N-S方程,结合五阶WENO格式,对马赫数为6的正激波在三角形楔面内的反射与聚焦现象进行了数值研究。结果表明,楔面顶角的变化对激波的反射类型以及聚焦均有明显的影响:随着顶角的增加,激波的反射类型从马赫反射向过渡马赫反射和双马赫反射转变,且壁面上的前向射流更加明显;三波点第一次碰撞产生的高温高压区足够满足可燃混合气体的点火条件,且其温度与压力值随顶角的增加而增大;当激波在楔面上发生临界双马赫反射时,温度与压力达到最大;当顶角增加到一定值时,激波在楔面反射转变为常规反射,不会产生激波对碰,因而没有高温高压区。     

Experiments on the interaction of a pair of cylindrical weak blast waves in air

This paper reports the results of experiments and computations on the interaction of a pair of cylindrical blast waves in air. The waves were generated by exploding wires, and both symmetrical and unsymmetrical interactions were observed. The experimental data includes schlieren photographs of the wave interactions, their radii, shock Mach number and pressure versus time, as well as various cross plots and data on the shock regular/irregular interaction transition condition. The flow fields were computed with the help of the Total Variation Diminishing (TVD) method, and appear to represent the experimental results reasonably well. Some attention is also given to the blast scaling laws of the type discussed by Sakurai (1965) and Oshima (1960).This article was processed using Springer-Verlag TEX Shock Waves macro package 1990.