BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN VARIANT BOUSSINESQ EQUATIONS

The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.     

有限谱ENO格式在爆轰波数值模拟中的应用

使用有限谱ENO(EssentiallyNon Oscillatory)格式 ,采用V .P .Korobeinikov二步化学反应模型 ,对稀释的化学当量的氢气和氧气混合物中非定常自维持爆轰波进行了二维数值模拟 ,研究了波的产生和演化机理 ,分析了爆轰波的三波结构和传播过程。计算得到的爆轰波参数和结构与以前的计算和实验结果一致。研究表明 :有限谱ENO格式可以成功地模拟非定常自维持爆轰波。     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Gravity-capillary waves in the presence of constant vorticity

Periodic and solitary gravity-capillary waves propagating at a constant velocity at the surface of a fluid of finite depth are considered. The vorticity in the fluid is assumed to be constant. Analytical solutions are presented for waves of small amplitude. For waves of large amplitude, numerical solutions are computed by boundary integral equation methods. The results unify previous findings for irrotational gravity capillary waves and gravity waves with constant vorticity. In particular solitary waves with oscillatory tails and branches of solutions which exist only for waves of large amplitude are found.     

Bragg scattering of sound by surface waves in shallow water

We present a theory of Bragg scattering of sound by surface gravity waves which are nearly periodic with a slowly varying envelope. Through several examples we illustrate how the resonant scattering of sound from a CW source in shallow water can reveal the characteristics of weak surface waves. Only plane sound waves of a guided mode are studied in this paper. Specific examples include the scattering by surface waves which are modulated in amplitude and/or phase. The incident sound waves are sinusoidal but detuned. Applications to short surface waves riding on long surface waves are examined.     

On waves of general type propagating at the interface between an elastic half-space and a liquid

Free Boundary waves of general type at a plane interface between an elastic half-space and a compressible, non-viscous liquid are considered. It is found that the waves are not necessarily plane waves and the motion is not necessarily confined to a plane perpendicular to the interface. All physical quantities associated with the waves are derivable from a single scalar function and the characteristic equation for the waves is identical with the corresponding equation for Stoneley-type (plane) waves.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.     

Diffraction of ship-type waves

The family of progressive waves propagating along the surface of a heavy liquid contains not only the simple waves which are usually studied but also waves of very complex structure, which in form are reminiscent of the crests of ship-generated waves. These waves are defined by formulas which contain an arbitrary function of a single argument. This paper studies the diffraction of such waves caused by the presence of an immersed vertical half-plane which impedes wave motion.     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

Some recent developments in elastic waves in solids

Three topics on recent developments in elastic waves which are of special interest to researchers in experimental mechanics are discussed. They are: (1) the elastic waves in anisotropic media with particular reference to waves in fiber-reinforced composite materials; (2) the coupled thermoelastic waves in isotropic or anisotropic media; and (3) the interaction of elastic waves with magnetic fields in nonferrous metals, polycrystalline ferromagnetic alloys and saturated ferrimagnetic crystals.     

超声柱面导波技术及其应用研究进展

综述无损检测中的超声柱面导波技术及其应用研究进展。给出导波的频散及多模式特征,着重评述超声导波的模式和频率选择、导波的激励和接收方法、导波与缺陷的相互作用、信号处理与特征提取以及导波技术在无损检测中的应用前景。      

Distortion of Digital Image Correlation (DIC) Displacements and Strains from Heat Waves

“Heat waves” is a colloquial term used to describe convective currents in air formed when different objects in an area are at different temperatures. In the context of Digital Image Correlation (DIC) and other optical-based image processing techniques, imaging an object of interest through heat waves can significantly distort the apparent location and shape of the object. There are many potential heat sources in DIC experiments, including but not limited to lights, cameras, hot ovens, and sunlight, yet error caused by heat waves is often overlooked. This paper first briefly presents three practical situations in which heat waves contributed significant error to DIC measurements to motivate the investigation of heat waves in more detail. Then the theoretical background of how light is refracted through heat waves is presented, and the effects of heat waves on displacements and strains computed from DIC are characterized in detail. Finally, different filtering methods are investigated to reduce the displacement and strain errors caused by imaging through heat waves. The overarching conclusions from this work are that errors caused by heat waves are significantly higher than typical noise floors for DIC measurements, and that the errors are difficult to filter because the temporal and spatial frequencies of the errors are in the same range as those of typical signals of interest. Therefore, eliminating or mitigating the effects of heat sources in a DIC experiment is the best solution to minimizing errors caused by heat waves.     

Interaction of interfacial and internal waves *

Reflection of internal waves at an interface between the mixed layer and a stratified fluid under the layer is investigated experimentally. Reflection coefficients of the internal waves and transmission coefficients of interfacial waves are obtained for the case in which the interfacial waves propagate along the interface. In order to reveal the energy budget of the two kinds of waves, the above reflection and transmission coefficients are employed in consideration of dissipation of the waves. These coefficients are investigated for different values of a ratio of the frequencies of the interfacial and internal waves. When the ratio is , 1 or 2 at the same horizontal wavenumber, overreflection takes place (i.e., reflection coefficient > 1), while under the same condition the transmission coefficient of the interfacial wave is less than unity. This indicates that the energy source of the overreflection of the internal wave is the interfacial wave. Overreflection also takes place when the two waves have the same frequencies and are in phase. On the other hand, the interfacial wave overtransmits the energy (i.e., transmission coefficient > 1) when the two waves are out of phase.     

The influence of constraints on the properties of acceleration waves in isotropic thermoelastic media

The properties of acceleration waves are investigated for situations in which the waves propagate in isotropic heat-conducting elastic media subject to arbitrary sets of constraints. Conditions under which waves may exist in the presence of constraints are investigated for classes of constraints broad enough to encompass all those encountered in practice. Attention is focussed on principal waves, and results are presented for the growth of the amplitudes of such waves first for fronts of arbitrary curvature, and subsequently by specialisation for plane, cylindrical and spherical waves travelling in material which has undergone one-dimensional plane deformation, cylindrically symmetric and spherically symmetric deformation, respectively.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

Superposition of finite-amplitude transverse waves in deformed Mooney-Rivlin and neo-Hookean materials

In this paper, waves propagating in Mooney-Rivlin and neo-Hookean non-linear elastic materials subjected to a homogeneous pre-strain are considered. In a previous paper, Boulanger and Hayes [Finite-amplitude waves in deformed Mooney-Rivlin materials, Q. J. Mech. Appl. Math. 45 (1992) 575-593] showed, for deformed Mooney-Rivlin materials, that the superposition of two finite-amplitude shear waves polarized in different directions (orthogonal to each other) and propagating along the same direction is an exact solution of the equations of motion. The two waves do not interact. Here, we are interested in superpositions of waves propagating in different directions. Two types of superpositions are considered: superpositions of waves polarized in the same direction, and also superposition of waves polarized in different directions. It is shown that such superpositions are exact solutions of the equations of motion with appropriate choices of the propagation and polarization directions.     

Acceleration waves in rate type plastic material with general work-hardening

The acceleration waves propagating in the rate type plastic material are investigated, where the material has isotropic and kinematic work-hardening. The waves are classified into unloading and loading waves, and have, in general, different propagation velocities. The propagation velocities of the principal waves are obtained for the Prandtl-Reuss material and the T material.     

Observations on waveforms of capillary and gravity-capillary waves

Due to extreme conditions in the field, there has not been any observational report on three-dimensional waveforms of short ocean surface waves. Three-dimensional waveforms of short wind waves can be found from integrating surface gradient image data (Zhang 1996a). Ocean surface gradient images are captured by an optical surface gradient detector mounted on a raft operating in the water offshore California (Cox and Zhang 1997). Waveforms and spatial structures of short wind waves are compared with early laboratory wind wave data (Zhang 1994, 1995). Although the large-scale wind and wave conditions are quite different, the waveforms are resoundingly similar at the small scale. It is very common, among steep short wind waves, that waves in the capillary range feature sharp troughs and flat crests. The observations show that most short waves are far less steep than the limiting waveform under weak wind conditions. Waveforms that resemble capillary-gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989, Vanden-Broeck and Dias 1992). Capillaries are mainly found as parasitic capillaries on the forward face of short gravity waves. The maximum wavelength in a parasitic wave train is less than a centimeter. The profiles of parasitic wave trains and longitudinal variations are shown. The phenomenon of capillary blockage (Phillips 1981) on dispersive freely traveling short waves is observed in the tank but not at sea. The short waves seen at sea propagate in all directions while waves in the tank are much more unidirectional.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.