P. Borejko

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

Abstract 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.

Basic theory of small-amplitude waves in a constrained elastic body

The effects of one or more internal constraints on the properties of waves in elastic media have been considered by many authors with particular reference to incompressibility and inextensibility, the constraints most widely used in modelling the behaviour of solid materials. The idea of broadening the theoretical framework of such studies by assigning a fully general form to the constraints is seemingly due to Scott [1975] who investigated the propagation and growth of acceleration waves of arbitrary shape in an elastic body experiencing one or two constraints. Subsequently Borejko & Chadwick [1980] have discussed the energetics of acceleration waves in an arbitrarily constrained elastic body and, in the same context, an analysis of simple waves has been presented by Whitworth [1982].

Transient elastic waves in a half-space: comparison of the DBIE-method with the method of generalized ray

SummaryThe two-dimensional problem of wave propagation in an elastic half-space is studied by the DBIEM (Direct Boundary Integral Equation Method) combined with the finite difference procedure applied to the time variable. The present hybrid formulation employs the fundamental solution depending neither on the frequency nor on the time variable. Time records of surface responses of the half-space are computed and compared with those obtained by the numerical evaluation of exact analytical solutions to this problem.

Application of the generalized rays to transient waves in an elastic half-space due to a buried line source

SummaryThe aim of this investigation is to study by means of the generalized rays the effects of a free surface on the propagation of elastic transient waves emanating from a line source of explosion, which varies with time as the Heaviside step function with rounded shoulders. This is accomplished by the numerical evaluation and analysis of the exact solutions provided by this theory. Graphs are obtained which show displacements and stresses as functions of time at three different receivers in the interior of an infinite medium and at the free surface of a homogeneous half-space. The character of surface responses, for short observation times, is the same as in the infinite medium, with the difference that they are all amplified (surface receiver effect). The late time surface responses are characterized by the presence of a smooth pulse representing the Rayleigh surface wave. The Rayleigh pulse in the non-vanishing surface stress takes its ultimate shape already close to the epicentre, while the Rayleigh pulse in the surface displacements developes fully at the remote distance from the epicentre.

Seismic Waves in Layered Soil: The Generalized Ray Theory

Methods of structural dynamics find practical applications in the earthquake resistant design of major structures, such as sky-scrapers, nuclear power stations, hospitals and large dams and bridges located in seismically active regions. In the past, earthquake resistant design criteria for such structures usually employed the so-called design response spectrum (see Newmark and Rosenblueth [2.3–1] and Clough and Penzien [2.3–2]). This spectrum is based on estimated values of certain numerical indices of the expected strong ground motion at the structure site. The indices, constructed from empirical relations supplied by the analysis of available ground motion records and other historical data (see, e.g., Hays [2.3–3]), include peak values of the ground displacement, velocity and acceleration as well as their dominant periods. Since the significant amount of reliable data for the most important cases of severe events and small epicentral distances is still not available, the reliability of these empirical relations is doubtful. Consequently, in modern civil engineering practice, the design response spectrum is applicable at preliminary design stages; the ultimate proportioning of a structure requires an explicit description of the expected ground motion at the site (see Clough and Penzien [2.3–2]). (Actually, the design response spectrum is not applicable to structures with nonlinear responses or involving various types of structural interactions. For such structures, an actual time history record of ground motion is, as it was pointed out by Clough and Penzien [2.3–2], indispensable). More reliable criteria for structural design employ more detailed descriptions of the expected ground motion (see Scanlan [2.3–4]). The best information is provided by complete time histories of ground displacement, velocity and acceleration likely to occur at the site.

The Method of Generalized Ray-Revisited

In Section 2, ROTATION OF COORDINATES, the Authors derived the emittance functions in the Weyl-Sommerfeld representation of the wave potentials for a horizontal instantaneous single force from those known for a vertical force from conditions of invariance of the phase and amplitude of plane waves under coordinate rotation, Eqs. (10) ∼ (13) and (18) ∼ (20). That transformation implies the validity of the commonly applied identity for the (force) vector components when rotating the vector in the opposite sense to the coordinate rotation. Further, in the three-dimensional case, the vertical force poses an axisymmetric problem which is compatible with the Fourier transformation applied to the coordinates in the horizontal plane.

Stress waves in layered rocks

SummaryIn this paper two problems will be solved analytically: the reflection of plane stress waves at a free boundary and the reflection and refraction of plane stress waves at a welded interface separating two dissimilar materials. In particular, stress reflection and refraction coefficients will be derived and the superimposed state of stress in regions of wave interference will be calculated. The solutions are visualized in a series of examples by means of computer-generated isochromatic fringe patterns. They show the partition of stress between reflected and refracted waves, stress distribution in regions of wave interference and demonstrate stress wave propagation in laycred structures.

Seismic Waves in an Impacted Half-Space: Time Domain BEM Versus the Method of Generalized Ray

The general two-dimensional time-dependent BEM formulation accounting for the effect of initial conditions and body forces is applied to the problem of a transient line load acting normal to the surface of an elastic half-space. The complete time records of the horizontal and vertical displacements at two different receivers are numerically evaluated and compared with those obtained by the numerical evaluation of the exact solution to this problem provided by the method of generalized ray. The agreement is excellent showing the validity as well as the quality of the present numerical implementation of the time-dependent BEM.

Energy relations for acceleration waves in elastic materials

It is well known that, for a plane sinusoidal body wave of small amplitude in an elastic medium which is anisotropic in relation to a natural reference state, the kinetic and strain energy densities are universally equal and that, furthermore, the ray (or group) velocity of the wave coincides with the velocity of energy propagation and is co-directional with the normal to the slowness surface of the material at the point representing the wave. In this paper it is shown how an analogous set of relationships may be established for an acceleration wave of arbitrary form advancing into a stationary, stress-free region of a body composed of a non-heat conducting material. The counterpart of the equipartition of energy is found to hold regardless of the constitution of the body. For the remaining results the transmitting material is taken to be elastic, but subject, possibly, to one or more internal constraints. Some particular examples of constraints are discussed, namely incompressibility, inextensibility and combined incompressibility and inextensibility.

Amplitude Versus Offset for Refracted Waves

The analysis of amplitude versus offset (AVO) is widely used in reflection seismic processing and interpretation. By this technique lithological information (S-wave velocity and density) can be extracted from P-wave reflection coefficients.