Reflection and transmission of elastic waves by a periodic array of cracks: Oblique incidence

The interaction of elastic waves with a planar array of periodically spaced cracks of equal length is investigated. The angle of incidence is arbitrary. By the use of Fourier series techniques, the mixed-boundary value problem for a typical strip is reduced to a singular integral equation, which is solved numerically. It is shown that the reflected and transmitted displacement fields are the superposition of an infinite number of homogeneous and inhomogeneous plane body-wave modes. The reflection and transmission coefficients, which correspond to the modes of order zero, are plotted versus the frequency for three angles of incidence. Sharp resonance and antiresonance effects are observed. A check of the accuracy of the computations is provided by the balance of rates of energies.     

Reflection and transmission of antiplane surface waves by a surface-breaking crack in a layered elastic solid

The reflection and transmission of antiplane surface waves (Love waves) by a surface-breaking crack in a layered elastic solid is investigated. The crack is normal to the free surface, and breaks into the lower half-space solid. The formulation of the problem is reduced to a singular integral equation of the Cauchy type. In this equation, the unknown function, which is the slope of the crack-face displacement, is discontinuous at the interface between the two solids. It is shown that the magnitude of the discontinuity is related to the ratio of the shear moduli. A Gaussian numerical method is used to obtain the solution of the singular integral equation. At some distance from the plane of the crack, the wave motion is the superposition of a finite number of Love-wave modes. The amplitudes of these modes are readily evaluated in terms of the slope of the crack-face displacement. Curves are presented for the reflection coefficients corresponding to the first three modes and for the transmission coefficient as functions of the dimensionless frequency.     

Reflection and transmission of elastic waves by a moving slab

Scattering of elastic waves by a moving slab is considered. Two cases corresponding to strong contact and good lubrication are treated. It is shown that the motion introduces new effective compressional and shear wave velocities in the moving slab. The amplitude of the reflection and transmission coefficients are given for various angles of incidence, frequencies and velocities of motion.     

Magnetoelastic Rayleigh surface waves for a layer of periodic structure on an elastic half-space

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 7, pp. 48–52, July, 1988.     

Reflection and transmission of plane waves by layered periodic media

The reflection and transmission coefficients (at normal incidence) for plane parallel slabs consisting of N inhomogeneous and exactly identical parallel layers are expressed in terms of the corresponding coefficients for a single layer. The analysis is effected by two different methods, one of which stems from the so called invariant imbedding viewpoint while the other relies on a more conventional matrix argument.     

Diffraction of surface waves by floating elastic plates

Based on the dynamical theories of water waves and Mindlin thick plates, the diffraction of surface waves by a floating elastic plate is presented by using the Wiener–Hopf technique. Firstly, the problem is related to a wave guide in water of finite depth, which is analysed to determine the poles. The resulting hybrid boundary value problem is reduced to solving an infinite system of linear algebraic equations. The results obtained are compared with those calculated by an alternative analysis, and with experimental data. Finally, the effects of the geometric and physical parameters on the distribution of deflection and bending moments in plates are analysed and discussed.     

Scattering of elastic waves by three-dimensional surface topographies

Diffraction of plane harmonic waves by three-dimensional surface irregularities is investigated through the use of an indirect boundary integral equation method. The irregularity of an arbitrary shape is embedded in an elastic half-space and subjected to incident P, SV, SH, and Rayleigh waves. The material of the half-space is assumed to be linear, weakly anelastic, homogeneous and isotropic.

The accuracy of the method is demonstrated through comparison of the results with existing axisymmetric solutions. Several numerical examples for non-axisymmetric canyons are presented. The resulting amplification patterns exhibit strong sensitivity on type and angle of the incident waves and on the location of the observation point. Systematic comparisons of three-dimensional and corresponding two-dimensional models demonstrate similarity of the amplification pattern. The amplification is larger in some three-dimensional models than in two-dimensional ones. Strong coupling between SH and P-SV modes is observed for off-azimuthal incident waves. This phenomenon is specially pronounced for incident SH waves and it is intrinsic to three-dimensional scattering.     

Reflection of harmonic elastic waves from a convex axisymmetric cavity

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 3, pp. 27–32, March, 1991.     

Scattering of elastic waves by a periodic array of cracks in 3-D space

The problem of scattering of normal incident time harmonic plane elastic waves by a co-planar periodic array of cracks in 3-D space is investigated. The scattered waves consist of a superposition of an infinite number of wave modes [M, N]^T and [M, N]^L,M. N=0, 1, 2, , but only a finite number of them are propagating wave modes. The numerical calculation has been made for rectangular cracks and P wave incidence. The reflection coefficient of [O, O] order,R ₀ ³ , has been studied in detail for various wave numbers and parameters of the geometry for the problem. The reliability of the numerical calculation has been checked by an application of the balance of rates of energies. For an elongated rectangular crack,R ₀ ³ in the corresponding 2-D problem in [2] is recovered. The dynamic stress intensity factors around the crack edge have been obtained. The results as the wave number goes to zero have been compared with those in the correspoding static case. Good agreement is observed.     

Reflection and transmission of elastic waves by the spatially periodic interface between two solids (Theory of the integral-equation method)

The linear theory of two-dimensional reflection and transmission of time-harmonic, elastic waves by the spatially periodic interface between two perfectly elastic media is developed. A given phase progression of the incident wave in the direction of periodicity induces a modal structure in the elastodynamic field and leads to the introduction of the so-called spectral orders. The main tools in the analysis are the elastodynamic Green-type integral relations. They follow from the two-dimensional form of the elastodynamic field reciprocity theorem, where in the latter a Green state adjusted to the periodicity of the structure at hand is used. One of these relations is a vectorial integral equation from which the elastodynamic field quantities can be determined.

The consequences of field reciprocity in the structure and of the conservation of energy are developed in view of their serving as a check on numercal results to be obtained from the relevant integral equations.

The formalism thus developed applies to profiles, if periodic, of arbitrary shape and size and can without too serious difficulties be implemented on a computer. The major difficulty in this respect is the relevant Green function, the series representation of it being slowly convergent. Its evaluation becomes tractable after an appropriate technique for accelerating the convergence. The only practical limitations are then put by the speed of the computer and its storage capacity.     

Interaction of elastic waves with a periodic array of collinear inplane cracks

A boundary integral equation method is applied to the study of the interaction of plane elastic waves with a periodic array of collinear inplane cracks. Numerical results are presented for the dynamic stress intensity factors. The effects of the wave type, wave frequency, wave incidence angle, and crack spacing on the dynamic stress intensity factors are analyzed in detail. The project supported by the Committee of Science and Technology of Shanghai and Tongji University     

Reflection of long waves from a “nonreflecting” bottom profile

The transformation of long surface waves in a zone of variable depth is investigated within the framework of shallow-water theory. In the particular case of a bottom profile containing a so-called “nonreflecting” relief segment adjacent to an even bottom, expressions for the reflected and transmitted pulse waves are obtained in explicit form. It is shown that waves are reflected from such a profile. The role of distributed and concentrated reflection of a wave propagating above an uneven bottom is discussed.     

Reflection and transmission of elastic waves through a couple-stress elastic slab sandwiched between two half-spaces

The reflection and transmission of elastic waves through a couple-stress elastic slab that is sandwiched between two couple-stress elastic half-spaces are studied in this paper. Because of the couple-stress effects, there are three types of elastic waves in the couple-stress elastic solid, two of which are dispersive. The interface conditions between two couple-stress solids involve the surface couple and rotation apart from the surface traction and displacement. The nontraditional interface conditions between the slab and two solid half-spaces are used to obtain the linear algebraic equation sets from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then, the energy fluxes carried by the various reflection and transmission waves are calculated numerically and the normal energy flux conservation is used to validate the numerical results. The special case, couple-stress elastic slab sandwiched by the classical elastic half-spaces, is also studied and compared with the situation that the classical elastic slab sandwiched by the classical elastic half-spaces. Incident longitudinal wave (P wave) and incident transverse wave (SV wave) are both considered. The influences of the couple-stress are mainly discussed based on the numerical results. It is found that the couple-stress mainly influences the transverse modes of elastic waves.     

Reflection of surface waves by a vertical step at the bottom of a basin

A study is made of the propagation of surface waves in a basin with a vertical step in the bottom. The problem is solved by fitting solutions on the horizontal boundary that continues the bottom of the shallow-water region of the basin; the factorization method is used. It is shown that the reflection of the waves depends on the wavelength of the incident waves and the difference between the two depths in the basin.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–185, January–February, 1984.     

Reflection and transmission of obliquely incident Rayleigh surface waves by a viscoelastic interphase

The reflection and transmission of obliquely incident Rayleight surface waves by an interphase between two quarter spaces of identical or different materials, have been investigated. The mechanical behavior of the interphase is represented by a thin viscoelastic layer. By using the full space Green's functions due to a spatially harmonic line load, the mathematical statement of the 3-dimension problem is reduced to a 2-dimension system of singular integral equations. The far-field behavior of the scattered waves leads to the definition of reflection and transmission coefficients,R andT. The system of the singular integral equations are solved forR andT with the boundary element method. The results are presented for selected values of the elastic constants of the joined quarter spaces, the parameters of the interphase and the incident angles of Rayleigh surface waves.     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

Reflection and transmission of elastic waves by the spatially periodic interface between two solids (Numerical results for the sinusoidal interface)

The integral-equation method for calculating the reflection and transmission of elastic waves by the spatially periodic interface between two solids, developed in a previous paper, is applied to a sinusoidal interface, and numerical results are presented. The computations have been carried out for four different heights of the profile (the plane interface included), a single frequency of operation, two combinations of elastic solids, and the four types of excitation. We have considered the interface between granite and slate, the interface between copper and flint glass, and P- as well as SV-wave incidence in either of the media.A peaked behaviour of the reflection and transmission factor occurs at angles of incidence where an elastodynamic spectral mode changes from propagating to evanescent and vice versa. An additional anomaly occurs in cases where the horizontal wave number of one of the spectral orders coincides with the horizontal wave number of a Stoneley wave along the corresponding plane interface. The latter phenomenon is the more pronounced, the shallower the corrugation of the interface is.     

Reflection and transmission of elastic waves at an elastic/porous solid saturated by two immiscible fluids

Wave propagation in a porous elastic medium saturated by two immiscible fluids is investigated. It is shown that there exist three dilatational waves and one transverse wave propagating with different velocities. It is found that the velocities of all the three longitudinal waves are influenced by the capillary pressure, while the velocity of transverse wave does not at all. The problem of reflection and refraction phenomena due to longitudinal and transverse wave incident obliquely at a plane interface between uniform elastic solid half-space and porous elastic half-space saturated by two immiscible fluids has been analyzed. The amplitude ratios of various reflected and refracted waves are found to be continuous functions of the angle of incidence. Expression of energy ratios of various reflected and refracted waves are derived in closed form. The amplitude ratios and energy ratios have been computed numerically for a particular model and the results obtained are depicted graphically. It is verified that during transmission there is no dissipation of energy at the interface. Some particular cases have also been reduced from the present formulation.     

Internal waves induced by moving periodic perturbations applied to the surface of a stratified fluid

Investigations of internal wave generation by moving perturbations are of considerable interest for submarine navigation, hydroacoustics, ocean seismology, etc. The main results for perturbations of constant intensity were published in [1–3]. In the present paper we continue the investigations and study moving perturbations whose intensity varies periodically in time. The perturbations are approximated by surface shape variations or an external pressure on the surface. The vertical displacement of the water particles relative to the equilibrium position is obtained in the form of a series in terms of waves modes for a given density stratification. A calculation algorithm and a program for computing each of the wave modes have been compiled. The boundaries of the wave regions and constant-phase lines are constructed and the displacement amplitudes are calculated. It is shown that there are resonance relations between the oscillation frequency and the perturbation velocity for which the displacement for a given mode becomes infinite (in the linear theory). Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–135, July–August, 1994.