Phase velocity effects of the wave interaction with exponentially sheared current

The nonlinear interaction between the unidirectional bichromatic wave-train and exponentially sheared current in water of an infinite depth is investigated. The model is based on the vorticity transport equation and the exact free surface conditions, without any assumptions for the existence of small physical parameters. Earlier works of the wave–current interaction were mainly restricted to either current acted on the monochromatic wave or irregular waves limited to irrotational current. Different from these previous works, no constraint is made in our model for amplitudes of the primary wave, and the current owns an exponential type profile along the vertical line. To ensure that the effect of vorticity on the phase velocity is consistent with earlier derivation, the case of a small amplitude wave traveling on the exponentially sheared current is examined firstly. Then the effect of nonlinearity on the phase velocity of primary waves in a bichromatic wave-train is considered. Accurate high-order approximations of the phase velocity are obtained under consideration of both the nonlinear wave self–self and mutual interactions. Finally, the combined effect of vorticity and nonlinearity on the phase velocity is investigated through the case of a bichromatic wave-train propagating on an exponentially sheared current. It is found that the characteristic current slope determines the effect of vorticity on the phase velocity caused by nonlinear wave self–self and mutual interactions, and the surface current strength may amplify/reduce this effect.     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 2: Hybrid spatial-spectral implementations

In Part 1 (van Groesen and Andonowati [1]), we derived models for the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. In this paper, we will describe hybrid spatial-spectral implementations of the models that retain the basic variational formulation of irrotational surface waves that underlays the derivation of the continuous models. It will be shown that the numerical codes are robust and efficient from results of simulations of two test cases of waves above a 1:20 sloping bottom from 30 m to 15 m depth: one simulation of a bichromatic wave train, and one of irregular waves of JONSWAP type. Measurements of scaled experiments at MARIN hydrodynamic laboratory and simulations with two other numerical codes will be used to test the performance. At the end of the full time trace of 3.5 h details of the irregular waves that travelled over more than 5000 m are clearly resolved with a correlation of more than 90%, in calculation times of less than 5% of the physical time. Also freak-like waves that appear in the irregular wave are shown to be modelled to a high degree of accuracy.     

Stability of bichromatic gravity waves on deep water

The stability of bichromatic gravity waves with small but finite amplitudes propagating in two directions on deep water is considered. Starting from the Zakharov equation, elementary quartet interactions are isolated and stability criteria are formulated. Results are illustrated for various combinations of bichromatic wave trains, from long-crested to standing waves. Two generic mechanisms operate: the first one is a modulational instability of one of the two components of the bichromatic wave train; the second mechanism is a modulation which couples both components of the wave train. However a third mechanism eventually comes into play: the resonant interaction of Phillips and Longuet-Higgins which leads initially to the linear growth of a third wave. When this latter is active, in particular for wave trains with wave vectors close together, it is shown by numerical integration that the long-time recurrence is destroyed.     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient waves in composite materials

A continuum theory for transient wave propagation in three-dimensional composite materials is given. The derived model provides a set of governing equations for the prediction of dynamic response of elastic composites to impulsive loadings. Pulse propagation normal to the direction of layering in periodically bilaminated media, and normal to the fiber direction in unidirectional long-fiber composites are obtained as special cases. The dynamic response of the composite is determined solely from the materials properties of the constituents (assumed in general to be orthotropic) and their geometrical dimensions. The predicted propagating transient waves are checked with exact solutions for impacted laminated composites, and with measured data for a fiber-reinforced material. Applications are given for pulse propagation in particulate composites and in tri-othogonally fiber-reinforced materials.     

具有精确色散性的非线性波浪数学模型

以完全非线性的自由表面边界条件为基础,以波面升高\eta和自由表面速度势\phi _\eta为待求变量,建立了新的波浪方程．方程在色散性上是完全精确的,非线性近似至三阶．与缓坡方程相比较,两者都具有精确的色散性,但该方程属于非线性模型,可模拟波浪的非线性效应,且适用于不规则波．方程的特点是属于微分-积分方程,对如何处理方程中积分项进行了讨论,并数值模拟了不同周期的线性波和二阶Stokes波,也模拟了波群的非线性演化,以对模型进行验证．      

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

粘弹性Hopkinson压杆中波的衰减和弥散

研究了线性粘弹性Hopkinson压杆中由于粘性效应和横向惯性效应引起的就力波衰减和弥散。导出计及横向惯性效应的线性粘弹性杆中纵波控制方程和应力解应变解，进而导出表征波衰竭和弥散性质的纵波传播系数的修正公式。这一修正公式计入粘性效应和几何效应，与Bacon公式相比，其形式简单，更便于在实验数据处理中应用。最后利用实验方法测定了有机玻璃杆的传播系数。     

EXACT ANALYSIS OF WAVE PROPAGATION IN AN INFINITE RECTANGULAR BEAM

The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam. Initially, by solving the three-dimensional elastodynamic equations a general analytic solution was derived for wave motion within the beam. And then for the beam with stress-free boundaries, the propagation characteristics of elastic waves were presented. This accurate wave propagation model lays a solid foundation of simultaneous control of coupled waves in the beam.     

波浪破碎的一种混合湍流模拟模式

近岸波浪的变形与破碎,一方面影响水体标识码运输,另一方面对消波护岸具有指导意义.本文提出一种三维混合湍流模拟模式方法,将求解区域分为造波区、波浪传播区和消波区. 造波区采用层流模式,通过基于 Fluent 的二次开发的 UDF 方法在边界进行速度造波.这种方法在给定入口速度的条件标识码据已知波高进行 精准插值,从而控制水的体积分数. 在波浪传播区域,采用大涡标识码行模拟研究,在消波区,采用 RANS 模型并利用多孔介质消波法进行消波. 模标识码 VOF 方法捕捉波浪破碎过程中的自由面变化.本文对波高为 5.5cm 的规则波 (M1)、波高为 13.5cm 的 规则波 (M3)、有效标识码 7.75cm 的 TMA 谱单向不规则波 (U1) 和有效波高为 19cm 的 TMA 谱多向不规则波 (B5) 展开标识码研究,并与前人的相关标识码果作比较,各条件下模拟结果与实验结果吻合.模拟结果说明本文提出的模型能够准确模拟出波浪传播过程中的折射和标识码象,并且能够捕捉的波浪破碎过程中的自由面变化,为三维波浪的传播与破碎的数值模拟提供一种模拟方法.      

Nonlinear dispersive waves in a relaxing medium

The dispersion of nonlinear waves in a relaxing medium is analysed by making use of the evolution equations for longitudinal waves. The dispersion relations are obtained and the behaviour of the waves compared to those that arise when they are governed by the well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations that describe unidirectional motion and also by the time regularized long wave (TRLW) equation that describes bi-directional motion. The nonlinear steady wave solutions are obtained. The general mathematical model used throughout this paper is obtained by the theory of nonlinear elasticity with weak relaxation effects (standard viscoelasticity). A further generalization using a four-element model is also discussed briefly.     

Stress waves in pyramids by photoelasticity

The propagation of stress waves in pyramids was studied photoelastically with the application of a laser-photomultiplier tube system and an internal polariscope for recording moving fringes. Dispersion and attenuation of stress waves were considered in a straight bar, a 5-deg pyramid, and a 20-deg pyramid made of Hysol 4290 epoxy plastic. In the straight bar and 5-deg pyramid, longitudinal waves propagate without any dispersion even though the waves attenuate as they progress down the models; in the 20-deg pyramid, however, the dispersion of the stress waves is quite significant. The distributions of the axial and radial stresses and the photoelastic fringe patterns obtained on the 20-deg pyramid show that the stress wave front is spherical with the maximum stress along the central axis of the pyramid. A one-dimensional theory of wave propagation without correction factors in a small-angle infinite cone compares well with the experimental results.     

Weakly nonlinear magnetohydrodynamic wave motion

One-dimensional magnetchydrodynamic (mhd) waves are treated including nonlinearity up to the second order in the wave amplitude; particular emphasis is on the effects of the diffusivity of the electromagnetic field on plane wave propagation. Unidirectional propagation is either governed by an equation of the Varley-Rogers type in the high frequency limit or by the Burgers equation in the low frequency limit. External resonant excitation in a finite layer leads to equations well known from resonance in various other physical systems.     

Resonance in rarefied gases

Dispersion and damping of ultrasound waves are a standard test for mathematical models of rarefied gas flows. Normally, one considers waves in semi-infinite systems in relatively large distance of the source. For a more complete picture, ultrasound propagation in finite closed systems of length L is studied by means of several models for rarefied gas flows: the Navier-Stokes-Fourier equations, Grad’s 13 moment equations, the regularized 13 moment equations, and the Burnett equations. All systems of equations are considered in simple 1-D geometry with their appropriate jump and slip boundary conditions. Damping and resonance are studied in dependence of frequency and length. For small L, all wave modes contribute to the solution.     

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     

USING LAPLACE TRANSFORM TO SOLVE THE VISCOELASTIC WAVE PROBLEMS IN THE SHPB EXPERIMENTS

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.     

粘塑性变截面杆中冲击波的演化

讨论了粘塑性变截面杆中的一维应力波传播规律。在粘塑性本构理论框架的基础上,利用特征关系和冲击波阵面上的突跃条件,得出了冲击波在传播过程中的演化规律,包括其微分方程和解析表达式,并对Bodner和Johnson-Cook材料和不同收缩形式的杆中冲击波的演化规律进行了讨论,同时计算和讨论了杆中冲击波后方应力波传播规律的特点。     

Analysis of micro-structural effects on phononic waves in layered elastic media with periodic nonsmooth coordinates

Propagation of elastic phononic waves in layered composite materials is analyzed by introducing nonsmooth periodic coordinates associated with structural specifics of the materials. Spatial scales of the original (smooth) coordinates are estimated by the wave lengths. In terms of the new coordinates, the homogenization procedure occurs naturally from the continuity conditions imposed on elastic displacements and forces at layer interfaces. As a result, higher-order asymptotic approximations describing spatiotemporal ‘macro’- and ‘micro’-effects of wave propagation are obtained in closed form. Such solutions provide visualizations for the wave shapes illustrating their structure induced local details. In particular, beat-wise mode shapes and effective anisotropy of acoustic wave propagation are revealed. The subharmonic beating in wave modes occur when wave lengths orthogonal to layers is about to ‘resonate’ with layer’ thickness. If the wave speed has a non-zero projection along the layers, then phase shifts between the beats are observed in different cross sections perpendicular to the layers.     

Laplace变换法研究SHPB实验中试件的黏弹性波传播问题

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.