Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids

The present paper concerns with the linear theory of micropolar thermoelasticity for materials with voids. Some basic properties of wave numbers of the longitudinal and transverse plane harmonic waves are treated. The existence theorems of non-trivial solutions and eigenfrequencies of the interior homogeneous boundary value problems of steady vibrations are proved. The connection between plane harmonic waves and eigenfrequencies of the aforementioned problems is established.     

Water surface waves in a horizontally oscillating container

This paper presents the results of experimental verification of Sretenskii’s linear theory of gravity waves in a container partially filled with water and oscillating horizontally according to the harmonic law. It has been shown that this theory predicts the existence of an infinite ordered countable set of generation modes of unstable waves. It has been experimentally confirmed that the waves are unstable if the container oscillation frequency is equal to the frequency of any odd standing-wave mode. At even eigenfrequencies of container oscillations, the theory predicts wave amplitudes up to a constant term. Experiment has shown that, in this case, the waves are stable and have minimum amplitudes.     

Decay of an initial discontinuity of water depth in a finite-length channel: Experiment

This paper presents the results of an experimental study of water wave motion occurring in a finite-length channel after rapid removal of a shield which produced an initial difference in depth between the headwater and tailwater regions. It is shown that the spectra of depth oscillation frequencies contain eigenfrequencies characteristic of seiches. Examples of direct and reflected waves and data on water depth variation after the reflection of a bore type wave from a vertical wall are given.     

Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.     

Interaction of surface and flexural-gravity waves in ice cover with a vertical wall

The problem of the interaction of surface and flexural-gravity waves with a vertical barrier is solved in a two-dimensional formulation. It is assumed that the fluid is ideal and incompressible, has infinite depth, and is partially covered with ice. The ice cover is modeled by an elastic plate of constant thickness. The eigenfrequencies and eigenmodes of oscillation of the floating elastic ice plate, the deflection and deformation of ice, and the forces acting on the wall are determined.     

Resonant interactions of Tollmien-Schlichting waves in the boundary layer on a flat plate

Resonant interaction phenomena of Tollmien-Schlichting waves (T.-S. waves) are examined experimentally by spectral analysis method. Results demonstrate that, in the spectra measured in the vicinity of the critical layer of the unstable boundary layer, the energy of T.-S. waves concentrates in a narrow band of frequency with one to three peaks in the power spectra corresponding to the eigenfrequencies of T.-S. waves and the frequency with maximum growth rate located between the upper and lower branches of the neutral curve. The resonant interactions transfer the energy of these eigencomponents to their subsequent subharmonics in a range ofR _δ* where their growth rates increase from zero to a maximum value, and the boundary layer becomes turbulent after the third resonant interaction.     

三维轴对称基桩动力特性研究

针对目前基桩动力特性研究中把桩视为一维杆件的缺陷,采用三维轴对称模型,对基桩的动 力响应进行了更深入研究,并与一维杆件模型结果进行了对比. 结果表明桩的长径比对桩振 动的固有频率有影响,只在不考虑桩的高阶固有频率时,把桩视为一维杆件进行计算分析是 可行的;若只考虑前四阶固有频率,则只要桩的长径比大于3即可;桩-土刚度对桩振动的 第一、二阶固有频率有较大的影响,提出了由第一、二阶固有频率反算求出桩-土刚度,进 而求出桩的承载力的方法.     

Harmonic waves in one-, two- and three-dimensional composites: Bounds for eigenfrequencies

Harmonic waves in one-, two- and three-dimensional elastic composites with periodic structure are considered. Based on a new quotient recently proposed by Nemat-Nasser, lower and upper bounds for the eigenfrequencies are developed. For illustration waves propagating normal to the layers in layered composites, and normal to the fibers in fiber-reinforced composites, are considered. These examples show that the new quotient is very effective and yields very accurate results for the considered class of problems. While these results are of interest in their own right, they can be used to check the effectiveness of various approximate theories which recently have been proposed by various authors.     

Nonstationary axisymmetric problem of electroelasticity for a piezoceramic cylinder with circular polarization

This paper considers the problem of the propagation of forced electroelastic axisymmetric torsional waves in a hollow piezoceramic cylinder of finite dimensions, whose curvilinear electroded surfaces are subjected to shear stresses or an electrical potential. A new closed solution is constructed by expansion in vector eigenfunctions using the structural algorithm of finite integral transforms, which makes it possible to determine vibration eigenfrequencies, the stress-strain state of the element, and the potential and intensity of the induced electrical field. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 12–21, January–February, 2009.     

Computation of normal modes of three-dimensional resonators by semiclassical and variational methods: seismological implications

The problem investigated in this paper is that of calculating the eigenfrequencies of the bouncing-ball modes of oscillation of scalar waves in a three-dimensiol, arbitrarily shaped enclosure or resonator. With a view to future applications in seismology, we consider wave resonators limited above by a flat reflector—the earth's surface—and below by a curved one—the sediment/rock interface. The surface of the curved reflector is smooth but otherwise of arbitrary shape. The EBK (Einstein-Brillouin-Keller) method of semiclassical quantization is used to compute the eigenfrequencies. Conventional ray tracing is used to determine the trajectories of the rays in configuration space as well as in the Poincaré surfaces of section needed for the application of the method. For a prescribed set of mode numbers the EBK conditions produce a set of algebraic equations whose solution gives the required eigenfrequency and the corresponding eigentrajectory. Additionally, a system of four nonlinearly coupled ODE's is derived that simulates accurately the trajectories of the bouncing points of rays and can be used as a fast alternative to ray tracing. A variational method based upon the Rayleigh-Ritz formulation is used to obtain bounds on the eigenfrequencies and to determine the resonator's eigenfunctions. For separable and smooth nonseparable domains both approaches give consistent results. If the curvature of the reflector is convex towards the approaching ray, repeated reflections will result in highly unstable ray trajectories which ultimately becomne chaotic. In this case the EBK method fails. A brief discussion of the inherent tendency to chaotic behavior of the multiply reflecting ray system inside arbitrarily shaped enclosures is presented.     

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.     

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.     

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane （with respect to the lattice plane） and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave prop- agation are highlighted in high frequency domains. One im- portant result presented herein is the comparison between the first Bloch wave modes to the membrane and bend- ing/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homog- enized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retro- propagating Bloch wave modes with a negative group veloc- ity, and of the corresponding ＂retro-propagating＂ frequency bands.     

Generalized Rayleigh surface waves in a piezoelectric semiconductor half space

Meccanica - In this paper, plane strain surface waves, also named generalized Rayleigh surface waves, in a transversely isotropic piezoelectric semiconductor half space are investigated. The...     

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.     

Thermoviscoelastic surface waves of an assigned wavelength

This paper deals with the propagation of surface waves of an assigned wavelength on a thermoviscoelastic half-space. It is shown that a unique surface wave of an assigned wavelength, which satisfies the adopted criteria for behaviour at infinity, always exists. This wave is interpreted as a superposition of three dispersive inhomogeneous plane waves. The superposed waves have different directions of propagation and different phase velocities. Their directions of propagation are not parallel to the stress-free surface. The plane of constant amplitude that corresponds to each of these superposed waves is parallel to the stress-free surface and moves to it with a constant velocity, which is different for each of the superposed waves. The numerical computations refer to some typical values of the material and thermal constants at different values of the wavelength when the half-space is thermally insulated.     

Reflection and refraction of plane waves under initial stresses at an interface

The paper deals with the phenomena of reflection and refraction of plane elastic waves at a plane interface between two semi-infinite elastic solid media in contact, when both the media are initially stressed. It has been shown analytically that both reflected and refracted P and SV waves depend on initial stresses present in the media. The numerical values of reflection and refraction coefficients for different initial stresses and the angle of incidence have been calculated by computer and the results are given in the form of graphs. Many results are found in the paper which are not seen in initially stress-free media.     

On the types and number of plane waves in hypoelastic materials

General principles are formulated for modeling the elastic deformation of materials and analyzing plane waves in nonlinearly elastic materials such as hyperelastic, hypoelastic, and those governed by the general law of elasticity. The results of studying the propagation of plane waves in hypoelastic materials are further outlined. The influence of initial stresses and initial velocities on the types and number of plane waves is studied. Wave effects characteristic of hypoelastic materials are predicted theoretically. One of such effects is blocking of certain types of plane waves by initial stresses __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 96–107, November 2005.     

Cubically non-linear effects of plane waves in isotropic soft solid materials

In this paper we analyze mathematical properties of an isotropic soft solid model which is characterized by three elastic constants. The model was proposed to interpret measurements of weakly non-linear shear waves in gel-like and tissue-like media. In our analysis we are particularly interested in third order non-linear terms. We present for the first time the full equations of elastodynamics, as well as the equations for plane waves for this model, with cubically non-linear terms. Next, the interaction coefficients for non-linear interactions of three plane waves to produce the fourth wave are explicitly calculated. These coefficients show which of the three waves interact with each other and determine how strong the effect of interaction is on the produced fourth wave. It turns out that these coefficients are expressed in terms of some combinations of three elastic constants. The obtained results can be helpful in experimental determination of elastic constants which describe the model.