Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.     

The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.     

Reflection and transmission of electromagnetic waves from a nonlinear anisotropic slab between two linear isotropic media

Summary In this paper, the author has studied the reflection from and transmission through a homogenous nonlinear anisotropic slab (anisotropy being due to an external magnetic field) bounded by two linear isotropic media. Nonlinear equations describing the growth of the two modes of propagation of an electromagnetic wave in the direction of the magnetic field, in an anisotropic nonlinear medium, have been set up and solved; the solutions have been used to obtain expressions for the reflected and transmitted components of the incident wave. The simpler problem of the reflection and transmission from an isotropic nonlinear slab has also been discussed as a special case.     

The interaction of surface waves with fixed semi immersed cylinders having symmetric cross sections

The reflection of water waves by a semi immersed cylinder having a symmetric cross section is studied for both Dirichlet and Neumann boundary conditions on the cylinder. The method of conformal transformations as utilized by Ursell and by Tasai for the radiation problem is adapted to the present diffraction problem. The problem is solved by expansions of the reflected wave potential using nonorthogonal functions (wave free potentials). These functions are not complete, and an additional source and a dipole are required. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown strengths of the source and the dipole terms. Numerical results are obtained for the reflection coefficient, transmission coefficient, horizontal force on cylinder, vertical force on cylinder. In the long wave region analytical approximations are obtained for these functions when the cross section is circular. The reflection and transmission coefficients are very different for the two boundary conditions in the long wave region, the Dirichlet reflection coefficient being much larger than the corresponding Neumann coefficient. This behavior is similar to acoustic and electromagnetic diffraction problems in two dimensions. On leave of absence from Itek Corporation, Lexington (Mass.), U.S.A.     

Unsteady shock generated vorticity

Abstract. A two-dimensional theory is developed for the vorticity just downstream of a curved, unsteady shock wave. By utilizing Crocco's equation, an explicit formula is obtained for the vorticity that does not require a perfect gas and that holds for arbitrary conditions upstream of the shock wave. The analysis is applied to the flow just downstream of the reflected shock that occurs in a single-Mach reflection pattern. Flow conditions are based on an interferometric photograph of Ben-Dor and Glass (1978). In this case, the reflected shock is weak everywhere from its upstream intersection with the wall to the triple point. The vorticity has a singularity and a change of sign near the triple point that indicates the presence of a weak shear layer downstream of this location. Received 20 December 1999 / Accepted 19 March 2000     

Non-dispersive waves in nonlinear anisotropic dielectrics

The propagation and interaction of transverse electro-magnetic disturbances in a half space of anisotropic non-dispersive dielectric and the associated reflection problem are discussed. For certain types of material symmetry, solutions are obtained in the nearly linear approximation. It is shown that an incident pulse produces a reflected disturbance during a time interval larger than the duration of the pulse. In the case of a sinusoidal incident signal the spectrum of the reflected wave contains non-harmonic frequencies.     

Propagation,reflection, and transmission of SH-waves in slightly compressible,finitely deformed elastic media

The propagation, reflection, and transmission of SH waves in slightly compressible, finitely deformed elastic media are considered in this paper. The dispersion relation for SH-wave propagation in slightly compressible, finitely deformed layer overlying a slightly compressible, finitely deformed half-space is derived. The present paper also deals with the reflection and refraction (transmission) phenomena due to the SH wave incident at the plane interface between two distinct slightly compressible, finitely deformed elastic media. The closed form expressions for the amplitude ratios of reflection and refraction coefficients of the reflected and refracted SH waves are obtained from suitable boundary conditions. For the numerical discussions, we consider the Neo-Hookean form of a strain energy function. The phase speed curves, the variations of reflection, and transmission coefficients with the angle of incidence, and the plots of the slowness sections are presented by means of graphs.     

Structure of compression shock waves in porous elastoplastic materials

The shock-wave structure in a porous elastoplastic material is studied. In a certain range of parameters, the existence of a four-wave structure of a compression shock wave is possible. Regimes in which a reflected shock wave does not appear at all have been found in the problem of shock-wave reflection from a rigid wall. In this case, the entire energy of the incident shock wave transforms to thermal energy due to dissipation induced by the viscous collapse of the pores. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 27–32, November–December, 1998.     

Theory of regular and mach reflection of shock waves in a two-phase gas-liquid medium

An equilibrium model of a two-phase gas-liquid medium, with allowance for the proportion, density, and compressibility of the components, and with a difference from [1] in that the adiabatic velocity of sound is introduced, has been used in order to study the regular and Mach (elementary theory) reflection of a shock wave of moderate intensity from a solid wall throughout the whole range of gas proportions. A complicated nonmonotonic variation has been found for the pressure on the wall behind the reflected wave, the angle of reflection, and the angle of departure of the triple point as functions of the gas proportion, the angle of incidence, and the intensity of the incident wave. In particular, it is shown that oblique reflection for moderate and low gas contents leads to the formation of a stronger reflected shock wave than does normal reflection. The effect of the gas proportion on the position of the boundary between the regions of regular and Mach reflection has already been studied in [2]. The results are described of serial calculations of the parameters of reflection for an air-water mixture, and these results agree fairly well for normal reflection with the known experimental data [3] for low and moderate gas contents. In the limiting case, the results agree with the known results for single-phase media [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–190, March–April, 1985.     

The phase shift for reflection of elastic waves in hexagonal piezoelectric crystals

The reflection of a transverse wave from the surface of a piezoelectric crystal of class 6 is considered for the case when the reflection plane coincides with or is close to the plane of transverse isotropy. The dependence of the phase shift of the reflected wave on crystal piezomoduli and other parameters is investigated. Reflection angles corresponding to maximum sensitivity of the phase shift φ to a change in piezomoduli are found. The influence of the piezoeffect on the reflection phase shift turns out to be most pronounced at grazing incidence along the transverse isotropic direction. Then the switching on of the (arbitrarily small) piezoeffect changes the phase from 0 to π, which gives annihilation between incident and reflected bulk waves.     

Propagation of quasishear waves in prestressed materials with unbonded layers

The propagation of quasi-shear waves along the layers of a prestressed composite material is described using the three-dimensional linearized theory of elasticity. The composite consists of two alternating layers that are free to slip relative each other. The dispersion equation is analyzed numerically. The effect of the prestresses on the wave velocity is studied     

The slowness surfaces of incompressible and nearly incompressible elastic materials

The slowness surface of a compressible elastic material has three sheets whilst that of an incompressible elastic material has only two sheets. The explanation for this qualitative difference is found to be that as the material approaches an incompressible limit the inmost sheet becomes a small sphere collapsing to the origin whilst the other two sheets tend to the two sheets of the limiting incompressible solid. The theory of nearly incompressible materials is developed here because of its important applications to rubberlike solids. Some results on the wave polarisations and on the convexity of the slowness surfaces are also given.     

Three-dimensional supersonic internal flows

In order to examine the transition between regular and Mach reflection in a three-dimensional flow, a range of special geometry test pieces, and inlets, were designed. The concept is to have a geometry consisting of two plane wedges which results in regular reflection between the incident waves off the top and bottom of the inlet capped by two curved end sections causing Mach reflection. The merging of these two reflection patterns and the resulting downstream flow are studied using laser vapor screen and shadowgraph imaging supported by numerical simulation. An angled Mach disc is formed which merges with the line of regular reflection. A complex wave pattern results with the generation of a bridging shock connecting the reflected wave from the Mach reflection with the reflected waves from the regular reflection. In order to experimentally access the flow within the duct, a number of tests were conducted with one end cap removed. This resulted in a modified flow due to the expansive flow at the open end the influence of which was also studied in more detail.     

Elastic waves in an electro-microelastic solid

The present study is concerned with the wave propagation in an electro-microelastic solid. The reflection phenomenon of plane elastic waves from a stress free plane boundary of an electro-microelastic solid half-space is studied. The condition and the range of frequency for the existence of elastic waves in an infinite electro-microelastic body are investigated. The constitutive relations and the field equations for an electro-microelastic solid are stemmed from the Eringen’s theory of microstretch elasticity with electromagnetic interactions. Amplitude ratios and energy ratios of various reflected waves are presented when an elastic wave is made incident obliquely at the stress free plane boundary of an electro-microelastic solid half-space. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Numerical computations are performed for a specific model to calculate the phase speeds, amplitude ratios and energy ratios, and the results obtained are depicted graphically. The effect of elastic parameter corresponding to micro-stretch is noticed on reflection coefficients, in particular. Results of Parfitt and Eringen [Parfitt, V.R., Eringen, A.C., 1969. Reflection of plane waves from a flat boundary of a micropolar elastic half-space. J. Acoust. Soc. Am. 45, 1258–1272] have also been reduced as a special case from the present formulation.     

Wave propagation in an orthotropic micropolar elastic solid

Two-dimensional plane wave propagation in an orthotropic micropolar elastic solid is studied. There exist three types of coupled waves in xy-plane, whose velocities depend upon the angle of propagation and material parameters. A problem on reflection of these plane waves from a stress-free boundary is considered. The reflection coefficients of various reflected waves are computed numerically for a particular model of the solid. The effects of anisotropy upon the velocities and reflection coefficients are depicted graphically for different angles of propagation.     

Gas content factor in the interaction of shock waves of different intensity in a two-phase gas-liquid medium

The transition from regular to Mach interaction is investigated in connection with the interaction of two plane weak or moderate shock waves of different intensity in a two-phase gas-liquid medium over the entire range of gas contents. A nonmonotonic dependence of the transition limit and the flow parameters on the gas content is detected. The investigation extends the results of [1] corresponding to the reflection of a shock wave from a wall. At intermediate gas contents in the case of opposing shock waves, analogous to the normal reflection of a shock wave from a solid wall, the results are in agreement with [2]. In the case of weak shock waves non-linear asymptotic expansions [3] are employed. In the extreme cases of single-phase media the results coincide with the findings of [3, 4]. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 172–174, November–December, 1986.     

Oblique reflection of shock waves from rigid porous materials

A study to determine the general gas dynamic behaviour associated with the impact of a shock wave on a porous wedge has been undertaken. A number of interesting features are noted. The pattern of wave reflection is shown to be significantly affected by the inflow of gas into the wedge. This has the effect of reducing the triple point trajectory angle for cases of Mach reflection and for strongly reducing the reflection angle in regular reflection. The permeability of the wedge has a significant effect on the strength of the reflected wave and in some cases this wave can be attenuated to the extent that it is almost eradicated. Pressure measurements taken under the wedge are characterized by oscillations which are of similar shape, for a given wedge, over a range of shock wave Mach numbers. It is shown that the wave transmitted into the wedge is attenuated to varying degrees depending on the material properties, and that for weak incident waves the mean propagation velocity can be less than the sound speed in the pore fluid. Photographs taken using a specially constructed wedge which allows the transmitted wave to be visualised, show that the transmitted wave is nearly plane.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

The reflected pressure field in the interaction of weak shock waves with a compressible foam

This paper deals with the waves that are reflected from slabs of porous compressible foam attached to a rigid wall when impacted by a weak shock wave. The interest is in establishing possible attenuation of the pressure field after a shock or blast wave has struck the surface. Foam densities from 14 to 38 kg/m³ were tested over a range of shock wave Mach numbers less than 1.4. It is shown that the initial reflected shock wave strength is accurately predicted by the pseudo-gas model of Gelfand et al. (1983), with a pressure ratio of approximately 80% of the value for reflection off a rigid wall. Evidence is presented of gas entering the foam during the early stages of the process. A second wave emerges from the foam at a later stage and is separated from the first by a region of constant velocity and pressure. This second wave is not a shock wave but a compression front of significant thickness, which emerges from the foam earlier than predicted by the pseudo-gas model. Analysis of the origin of this wave points to much more complex flows within the foam than previously assumed, particularly in an apparent decrease in average wave front speed as the foam is compressed. It is shown that the pressure ratio across both these waves taken together is slightly higher than that for reflection off a rigid wall. In some cases this compression wave train is followed by a weak expansion wave.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

A numerical simulation of reflection processes of a detonation wave on a wedge

Abstract. A two dimensional numerical simulation has been performed to study reflection processes of detonation waves on a wedge. The numerical scheme adopted is the flux corrected transport scheme and a two-step chemical reaction is assumed for a stoichiometric oxyhydrogen mixture diluted with argon. Transverse wave structures of the detonation are produced by artificial disturbances situated in front of a one-dimensional Chapman-Jouguet detonation wave. Numerical grids are generated by solving a Laplace equation. Results show that in the case where Mach reflection occurs, the cells in the Mach stem are smaller than those in the incident wave and are distorted in shape. There is also an initiating stage during which the cells in the Mach stem are created. The critical angle beyond which Mach reflection cannot occur is discussed. Received 15 October 1999 / Accepted 27 March 2000     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.