Original Article Analysis of acceleration waves in crystalline solids based on a continuum model incorporating microscopic thermal vibration

Acceleration waves in crystalline solids at finite temperatures are studied by applying the method of singular surface to a new continuum model derived from a nonequilibrium statistical-mechanical lattice model. The propagation velocities of the waves and the ratios of the mechanical and thermal amplitudes are determined. The differential equations which govern the variation of the amplitudes with time are also obtained. The analysis is valid in a wide temperature range including the melting point. The temperature dependence of the wave propagation is found to be singular at the melting point, and its physical implication is discussed. Received: August 1, 1996     

Weak shock waves in isotropic solids at finite temperatures up to the melting point

Propagation speeds and Rankine–Hugoniot relations for weak shock waves in isotropic solids are derived analytically in order to elucidate mechanical and thermal properties of the waves. In the analysis, we adopt a new continuum model for the solids, which takes into account explicitly microscopic thermal vibration of the constituent atoms. As the model is valid in a wide temperature range up to the melting point, we can discuss the relations at high temperatures even near the melting point. Typical numerical results are also shown and discussed as illustrations.      

Experimental Investigation of the Generation of Internal Waves by a Vertical Cylinder in a Near-Surface Pycnocline

A bench study of the amplitudes, mode composition, and phase structure of the internal waves generated by a vertical cylinder in the presence of a near-surface pycnocline has been performed; the pycnocline took the form of a stratified fluid layer located between two quasi-homogeneous layers of thicknesses h ₁ and h ₂=2h ₁. In the experiments, the cylinder traveled at velocities critical with respect to internal wave generation. Different cases of model submergence relative to the pycnocline are considered. The dependence of the mode structure and the amplitude-phase characteristics of the forced internal waves on the body velocity and its relative submergence is analyzed. The parameters of both steady and unsteady wave systems are studied.The data obtained make it possible to predict the forced wave parameters and the critical body velocities for given model dimensions and pycnocline parameters.     

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     

Plane waves in a hypoplastic half-space under periodic boundary conditions

The paper presents a numerical study of the propagation of plane waves in a half-space occupied by a granular material, with periodic boundary conditions for velocity or stresses prescribed at the boundary of the half-space. The constitutive behaviour of the material is described by a simplified hypoplastic equation which takes into account different values of the stiffness for different directions of deformation, and the coupling between shear and volumetric strains owing to dilatancy. These two features are responsible for a nonlinear character of longitudinal waves and for the generation of longitudinal motion by transverse disturbances. It is shown that longitudinal and transverse boundary disturbances produce qualitatively the same longitudinal waves at large distances from the boundary. As a longitudinal wave propagates, the amplitude of oscillations decreases and eventually vanishes, resulting in a single non-oscillating wave.Received: 10 September 2002, Accepted: 31 March 2003 Correspondence to: Y. A. Berezin     

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.     

Computation of interface wave motions by reciprocity considerations

The current theoretical study deals with computation of Stoneley waves along a solid–solid interface and Scholte waves (also called Scholte-Gogoladze) along a solid–liquid interface by reciprocity considerations. Closed-form solutions of the wave motions generated by a time-harmonic line load applied in two bonded elastic half-spaces of different material properties are derived in a simple manner. In order to perform direct applications of reciprocity theorems, we introduce in this article new expressions for the displacements of free interface waves. Reciprocity relations between an actual state, interface wave motion generated by a time-harmonic line load, and a virtual state, an appropriately chosen free wave traveling along the interface, are derived. Scattered amplitudes of Stoneley waves and Scholte waves due to the load are thus computed. To show application of the obtained results, scattering of Stoneley wave by a delamination at the interface is then studied.     

Numerical investigations of shock waves in gas-particle mixtures

The propagation of shock waves in gas-particle mixtures in one- and two-dimensional geometries is numerically investigated. Two schemes for approximating conservation laws for particles, which are collectively treated as a continuum medium, are compared and discussed. Different models of the drag coefficient and Nusselt number, directly affecting the interaction between the gas and particle phases, are used for obtaining shock profiles, and the results are compared. The oblique shock reflections at a solid wedge in a gas-particle mixture are simulated. The results demonstrate that the reflection pattern changes as the shock propagates along the wedge, revealing strong non-selfsimilarity of the phenomenon.Received: 22 May 2003, Accepted: 28 August 2003, Published online: 12 November 2003 Correspondence to:T. Saito     

Evolution Equations for Plane Cubically Nonlinear Elastic Waves

Consideration is given to the nonlinear theory of elastic waves with cubic nonlinearity. This nonlinearity is separated out, and the interaction of four harmonic waves is studied. The method of slowly varying amplitudes is used. The shortened and evolution equations, the first integrals of these equations (Manley–Rowe relations), and energy balance law for a set of four interacting waves (quadruplet) are derived. The interaction of waves is described using the wavefront reversal scheme     

Propagation of plane waves in thermoelastic cubic crystal material with two relaxation times

A problem concerned with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated. The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem. The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness, thermal contact conductance, slip and welded boundaries. Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence have been compared graphically for different incident waves. It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.     

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.     

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     

Structure of permanent waves in density-stratified media

Non-linear steady waves in density-stratified channels are discussed. Under the assumption of non-diffusive incompressible flow, a method to classify all waveforms is given for moderate wave amplitudes, when the densityq at infinity decreases with altitude (q<0). The classification is worked out explicitly for the two highest critical Froude numbers. Furthermore, special density distributions are constructed with the purpose of showing that, for example, solitary waves exist even ifq>0.Dedicated to Professor Gaetano Fichera on the occasion of his 70th birthday. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, September 1992.     

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.     

A Comparison of Nonlinear Water Wave Models

We compare the numerical evolution of one-dimensional gravity waves in response to a traveling surface pressure pulse using a highly accurate boundary integral method and two relatively efficient approximate models (West et al. and Benney–Luke). In both water of finite-depth and in the deep-water limit, the steady state effect of the decaying pressure ramp is to create a profile which approximates a Stokes wave. Moreover, the transient surface profile appears to evolve through a series of Stokes waves of time-varying amplitude. Results show all three models yield similar predictions for lower amplitude waves, while the West et al. and boundary integral predictions differ from the Benney–Luke model at higher amplitudes.     

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.     

Wave interaction in a nonequilibrium gas flow

By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of non-resonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(ε^-1), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived.     

Propagation of acceleration waves in incompressible saturated porous solids

Within the framework of the incompressible porous media model, the propagation properties of acceleration waves in liquid-filled porous solids is discussed. The incompressibility of the two constituents in the model forces the amplitudes of the longitudinal waves in the skeleton and in the liquid to satisfy a certain relation. The two propagation speeds are presented by examination for the existence of acceleration waves and only longitudinal and transverse waves are realizable in the incompressible two-phase porous materials.     

Shock waves at microscales

We present a model for the effects of scale, via molecular diffusion phenomena, on the generation and propagation of shock waves. A simple parametrization of the shear stresses and heat flux at the wall leads to the determination of new jump conditions, which show that, for a given wave Mach number at small scales, the resulting particle velocities are lower but the pressures are higher. Also, the model predicts that the flow at small scale is isothermal and that the minimum wave velocity can be subsonic. Experiments with a miniature shock tube using low pressures to simulate the effects of small scale have shown qualitative agreement with the proposed model. In fact, the effects of scale appear even more important than what has been incorporated in the model.PACS: 47.40.-xReceived: 14 November 2002, Accepted: 2 April 2003, Published online: 18 June 2003     

Oscillatory line source for water waves in shear flow

The linearized water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. The vorticity is uniform, with zero velocity at the free surface. Then there will be at most two emitted waves, and no Doppler effects. Exact far-field waves are derived, with radiation conditions applied at infinity. An upstream wave will always exist, whereas the downstream wave exists only when the angular frequency of oscillation exceeds the vorticity. The wave radiation problem is solved also for oscillating vortex and dipoles. The amplitudes and energy fluxes are calculated.