On Discontinuity Waves in Linear Piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Propagation and Evolution of Wave Fronts in Two-Phase Porous Media

An investigation is made into the propagation and evolution of wave fronts in a porous medium which is intended to contain two phases: the porous solid, referred to as the skeleton, and the fluid within the interconnected pores formed by the skeleton. In particular, the microscopic density of each real material is assumed to be unchangeable, while the macroscopic density of each phase may change, associated with the volume fractions. A two-phase porous medium model is concisely introduced based on the work by de Boer. Propagation conditions and amplitude evolution of the discontinuity waves are presented by use of the idea of surfaces of discontinuity, where the wave front is treated as a surface of discontinuity. It is demonstrated that the saturation condition entails certain restrictions between the amplitudes of the longitudinal waves in the solid and fluid phases. Two propagation velocities are attained upon examining the existence of the discontinuity waves. It is found that a completely coupled longitudinal wave and a pure transverse wave are realizable in the two-phase porous medium. The discontinuity strength of the pore-pressure may be determined by the amplitude of the coupled longitudinal wave. In the case of homogeneous weak discontinuities, explicit evolution equations of the amplitudes for two types of discontinuity waves are derived.     

Interaction of a weak discontinuity wave with a characteristic shock in a dusty gas

In this paper, the evolution of a characteristic shock in a dusty gas is investigated and its interaction with a weak discontinuity wave is studied. The transport equation for the amplitude of the weak discontinuity wave, which is of Bernoulli type, is obtained. The amplitudes of the reflected and transmitted waves after interaction of the weak discontinuity with the characteristic shock are evaluated by using the results of the general theory of wave interaction.      

Counter-intuitive results in acousto-elasticity

We present examples of body wave and surface wave propagation in deformed solids where the slowest and the fastest waves do not travel along the directions of least and greatest stretch, respectively. These results run counter to commonly accepted theory, practice, and implementation of the principles of acousto-elasticity in initially isotropic solids. For instance, we find that in nickel and steel the fastest waves are along the direction of greatest compression, not greatest extension (and vice-versa for the slowest waves), as soon as those solids are deformed. Further, we find that when some materials are subject to a small-but-finite deformation, other extrema of wave speeds appear in non-principal directions. Examples include nickel, steel, polystyrene, and a certain hydrogel. The existence of these “oblique”, non-principal extremal waves complicates the protocols for the non-destructive determination of the directions of extreme strains.     

One dimensional steepening of waves in non-ideal relaxing gas

In this paper, we studied the behavior of different modes of wave propagation and breaking of wave front by employing the theory of singular surfaces in a plane and radially symmetric flow of a non-ideal relaxing gas. The one dimensional steepening of waves is considered and the transport equation for the jump discontinuity of velocity gradient is obtained. The effects of relaxation and van der Waals excluded volume of the medium on the jump discontinuity of velocity gradient are analyzed.     

A numerical method for the determination of weakly non‐hydrostatic non‐linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non‐hydrostatic non‐linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non‐hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non‐linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near‐shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off‐shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non‐hydrostatic non‐linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.     

Validity ranges of interfacial wave theories in a two-layer fluid system

In the present paper, we endeavor to accomplish a diagram, which demarcates the validity ranges for interfacial wave theories in a two-layer system, to meet the needs of design in ocean engineering. On the basis of the available solutions of periodic and solitary waves, we propose a guideline as principle to identify the validity regions of the interfacial wave theories in terms of wave period T, wave height H, upper layer thickness d ₁, and lower layer thickness d ₂, instead of only one parameter–water depth d as in the water surface wave circumstance. The diagram proposed here happens to be Le Méhauté’s plot for free surface waves if water depth ratio r = d ₁/d ₂ approaches to infinity and the upper layer water density ρ ₁ to zero. On the contrary, the diagram for water surface waves can be used for two-layer interfacial waves if gravity acceleration g in it is replaced by the reduced gravity defined in this study under the condition of σ = (ρ ₂ − ρ ₁)/ρ ₂ → 1.0 and r > 1.0. In the end, several figures of the validity ranges for various interfacial wave theories in the two-layer fluid are given and compared with the results for surface waves. The project supported by the Knowledge Innovation Project of CAS (KJCX-YW-L02), the National 863 Project of China (2006AA09A103-4), China National Oil Corporation in Beijing (CNOOC), and the National Natural Science Foundation of China (10672056).     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonlinear wave propagation in binary mixtures of Euler fluids

In the present paper, we study the propagation of acceleration and shock waves in a binary mixture of ideal Euler fluids, assuming that the difference between the atomic masses of the constituents is negligible. We evaluate the characteristic speeds, proving that they can be separated into two groups: one is related to the case of a single Euler fluid, provided that an average ratio of specific heats is introduced; the other is new and related to the propagation speed due to diffusion. We evaluate the critical time for sound acceleration waves and compare its value to that of a single fluid. We then study shock waves, showing that three types of shock waves appear: sonic and contact shocks, which have counterparts in the single fluid case, and the diffusive shock, which is peculiar to the mixture. We discuss the admissibility of the shock waves using the Lax-Liu conditions and the entropy growth criterion. It is proved that the sonic and the characteristic shock obey the same properties as in the single fluid case, while for the diffusive shock there exists a locally exceptional case that is determined by a particular value of the concentration of the constituents, for which the genuine nonlinearity is lost and no shocks are admissible. For other values of the unperturbed concentration, the diffusive shock is stable in a bounded interval of admissibility.Received: 15 December 2002, Accepted: 28 June 2003 Correspondence to: T. RuggeriS. Simi: On leave from the Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Serbia     

On wave propagation in inextensible elastic bodies

In this paper we study the propagation of acceleration waves in inextensible elastic bodies. ¹ While the computations are but an exercise, the results are interesting and quite unlike the corresponding results for unconstrained bodies. Indeed, a wave travelling in the direction of inextensibility must necessarily be transverse, and, when the reaction stress is compressive and sufficiently large, the corresponding speed of propagation becomes non-real, so that even transverse waves fail to exist.We also study (infinitesimal) progressive waves and we find the corresponding propagation condition to be the same as that for acceleration waves. Here, however, non-real speeds of propagation have a definite physical meaning: they imply exponential growth of the wave. Thus, in particular, when the reaction stress is compressive and sufficiently large, a transverse progressive wave travelling in the direction of inextensibility grows without bound. We conjecture that this indicates the presence of local buckling. ²     

Theory of non-linear wave propagation with application to the interaction and inverse problems

The propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical models are derived from the theories of thermoclasticity or viscoclasticity taking into account the geometric and physical non-linearities and dispersion. On the basis of eikonal equations for the associated linear problem the transport equations of the nth order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are analyzed. Conditions are also studied when the interaction of non-linear waves does not take place. The inverse problem of determining materials properties according to pulse shape changes is discussed.     

粘塑性靶板中冲击波的演化

讨论了粘塑性靶板中的一维应变波的传播规律。利用广义特征理论导出了应力波传播的特征线和特征关系。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的规律，并以Ｂｏｄｎｅｒ－Ｐａｒｔｏｍ幂函数型粘塑性材料为例，计算和讨论了板中应力波传播规律的特点     

Piezoelectric wafer active sensor embedded ultrasonics in beams and plates

In this paper we present the results of a systematic theoretical and experimental investigation of the fundamental aspects of using piezoelectric wafe active sensors (PWASs) to achieve embedded ultrasonics in thin-gage beam and plate structures. This investigation opens the path for systematic application of PWASs forin situ health monitoring. After a comprehensive review of the literature, we present the principles of embedded PWASs and their interaction with the host structure. We give a brief review of the Lamb wave principles with emphasis on the understanding the particle motion wave speed/group velocity dispersion. Finite element modeling and experiments on thin-gage beam and plate specimens are presented and analyzed. The axial (S ₀) and flexural (A ₀) wave propagation patterns are simulated and experimentally measured. The group-velocity dispersion curves are validated. The use of the pulse-echo ultrasonic technique with embedded PWASs is illustrated using both finite element simulation and experiments. The importance of using high-frequency waves optimally tuned to the sensor-structure interaction is demonstrated. In conclusion, we discuss the extension of these results toin situ structural health monitoring using embedded ultrasonics.     

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.     

The thickness of detonation waves visualised by slight obstacles

The reflection of detonation waves from slight obstacles, which hardly disturb the wave propagation, is observed by time-resolved schlieren photography. The following stoichiometric mixtures are used: pure and argon-diluted hydrogen-oxygen, hydrogen-air, acetylene-oxygen, and acetylene-air. Initial pressures are varied such that cell widths range from 1.4 up to 108 mm, which is twice the side length of the square cross-section of the tube. The trajectories of the incident and the reflected waves in the x,t-plane are used to determine lower limit values for the wave thickness. The considerable influences of the obstacle shape and of the evaluation method on the results are discussed in detail, and error sources are analyzed. The method has been improved since a previous publication by the authors. The ratio of the lower limit values to the cell width spreads from 0.4 to 0.8 in the medium cell size range. It decreases with increasing marginality and seems to increase at small scale. A unique correlation between the lower limit value and the tube diameter, both referred to the cell size, that was proposed earlier in the literature has to be refused. The velocities of the reflected waves are presented as additional information on the post-detonation wave state. The sonic transition is discussed theoretically, enhancing the stream tube model, and practically, based on detailed observations for marginal detonations.Received: 29 April 2003, Accepted: 20 October 2003, Published online: 3 February 2004PACS: 47.40.-xCorrespondence to: M. Weber     

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.     

Mechanism of shear attenuation near the interface under combined compression and shear impact loading

We investigate the compressional/shear coupling plastic wave propagation characteristics analytically for ideal elastic–plastic materials in both stress and particle velocity spaces, focusing on the shear wave attenuation near the interface occurring in pressure–shear plate impact experiments. The results show that the shear attenuation is strongly associated with the wave propagation characteristics of the coupling waves. In the stress space, as the shear stress increases, an adjustment of the stress components is observed and the final stress state along the wave path is a combined pure shear- and hydrostatic pressure-state. In the particle velocity space, the wave structures with different loading and maximal transverse particle velocity are obtained. The maximal transverse particle velocity varies with the longitudinal velocity and forms a boundary line. Once the loading transverse velocity exceeds this line, a transverse particle velocity discontinuity occurs at the impact interface. If the bonding strength is sufficiently high, there will be a shear band in the target in the extreme vicinity of the interface.     

多孔材料中声波的传播与演化

采用两相多孔介质的拉格朗日模型来描述一种理论流体充填的多孔弹性固体材料,其中孔隙度的变化满足一个附加的平衡方程。     

A σ‐coordinate non‐hydrostatic model with embedded Boussinesq‐type‐like equations for modeling deep‐water waves

A σ‐coordinate non‐hydrostatic model, combined with the embedded Boussinesq‐type‐like equations, a reference velocity, and an adapted top‐layer control, is developed to study the evolution of deep‐water waves. The advantage of using the Boussinesq‐type‐like equations with the reference velocity is to provide an analytical‐based non‐hydrostatic pressure distribution at the top‐layer and to optimize wave dispersion property. The σ‐based non‐hydrostatic model naturally tackles the so‐called overshooting issue in the case of non‐linear steep waves. Efficiency and accuracy of this non‐hydrostatic model in terms of wave dispersion and nonlinearity are critically examined. Overall results show that the newly developed model using a few layers is capable of resolving the evolution of non‐linear deep‐water wave groups. Copyright © 2009 John Wiley & Sons, Ltd.