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Reflection of P and SV waves at the free surface of a prestressed elastic half-space
Abstract
The propagation of plane waves in a prestressed elastic solid with incremental elastic coefficients possessing orthotropic symmetry is discussed. Two types of plane waves, called quasi-P and quasi-S waves, are shown to exist. An expression for the group velocity of these waves is obtained. The velocities of quasi-P and quasi-S waves are found to depend on the angle of propagation. The reflection coefficients of these waves incident at the free surface of a prestressed elastic half-space are derived. Critical reflection of quasi-P and quasi-S waves has been discussed. It is found that while quasi-S waves always have critical reflection, quasi-P waves can have critical reflection only if incremental elastic coefficients satisfy a certain condition. -Authors
... Sidhu and Singh [16] commented on the paper of Chattopadhyay et al. [7] and said that their results are not acceptable because the method of potentials used in their paper is not applicable for pre-stressed media. Later, Sidhu and Singh [17] investigated the problem of plane wave propagation in a pre-stressed elastic solid with incremental elastic coefficients possessing orthotropic symmetry. They obtained two types of plane waves, the so-called quasi P-waves and quasi S V-waves, showed that their velocities depend on the angle of propagation and derived the reflection coefficients of the reflected P-and S V-waves. ...
... These coefficients are obtained numerically for a particular model, and the results depicted graphically. The results of Sidhu and Singh [17] are recovered as a particular case of the present problem. ...
... Consider a plane qS V-wave propagating through the half-space M with phase velocity c 2 incident to the plane interface making an angle θ 0 with the normal. Sidhu and Singh [17] showed that c 2 is given by ...
The effect of initial stresses on incident quasi SV-waves at a plane interface between two dissimilar pre-stressed elastic half-spaces is investigated. The reflection and refraction coefficients of the reflected and refracted qSV- and qP-waves are derived with the help of appropriate boundary conditions. The coefficients are found to be functions of the angle of incidence and the initial stresses and incremental elastic parameters of the pre-stressed elastic half-spaces.
doi:10.1017/S1446181111000757
... Sidhu and Singh [16] commented on the paper of Chattopadhyay et al. [7] and said that their results are not acceptable because the method of potentials used in their paper is not applicable for pre-stressed media. Later, Sidhu and Singh [17] investigated the problem of plane wave propagation in a pre-stressed elastic solid with incremental elastic coefficients possessing orthotropic symmetry. They obtained two types of plane waves, the so-called quasi P-waves and quasi S V-waves, showed that their velocities depend on the angle of propagation and derived the reflection coefficients of the reflected P-and S V-waves. ...
... These coefficients are obtained numerically for a particular model, and the results depicted graphically. The results of Sidhu and Singh [17] are recovered as a particular case of the present problem. ...
... Consider a plane qS V-wave propagating through the half-space M with phase velocity c 2 incident to the plane interface making an angle θ 0 with the normal. Sidhu and Singh [17] showed that c 2 is given by ...
The effect of initial stresses on incident quasi SV-waves at a plane interface between two dissimilar pre-stressed elastic half-spaces is investigated. The reflection and refraction coefficients of the reflected and refracted qSV- and qP-waves are derived with the help of appropriate boundary conditions. The coefficients are found to be functions of the angle of incidence and the initial stresses and incremental elastic parameters of the pre-stressed elastic half-spaces.
... The closed-form formulae of these coefficients for the first-order approximation are then presented for a particular type of interface (cosine law interface). Some more results including the results of Sidhu and Singh [21] have been deduced as particular cases from the present problem. ...
... Let a plane elastic wave propagating through M with phase velocity c be incident at the corrugated interface making an angle y 0 with the normal. Sidhu and Singh [21] have shown that there exist two types of plane waves propagating with velocities given by ...
... These reflection coefficients exactly match those obtained by Sidhu and Singh [21] for the corresponding problem. ...
A plane qP-wave (quasi-P-wave) is assumed to be incident at a corrugated interface between two dissimilar pre-stressed elastic solid half-spaces. Using Rayleigh's method of approximation, the reflection and transmission coefficients have been presented for the first-order approximation of the corrugation. These coefficients are obtained in closed form for a corrugated interface of periodic shape. We found that these coefficients depend on the angle of incidence, frequency of the incident wave, initial stresses and incremental elastic properties of the half-spaces. The coefficients corresponding to irregularly reflected and transmitted waves are found to be proportional to the amplitude of the corrugated interface and are also influenced significantly by the initial stresses of the half-spaces. Some more results including the results of Sidhu and Singh [Reflection of P and SV-waves at the free surface of a prestressed elastic half-space, Journal of the Acoustical Society of America 76(2) (1984) 594–598] have been deduced as particular cases from the present problem.
... Jeffrey's [1] and Gutenberg [2] considered the reflection of elastic plane waves at the surface of a solid halfspace. Sidhu and Singh [3] investigated the propagation of plane waves in a prestressed elastic solid possessing orthotropic symmetry and showed that the velocities of qL and qSV waves depend upon the angle of propagation. Rayleigh [4] considered the reflection and transmission of waves from an undulated boundary surface of an elastic solid. ...
... However, in general, such a decomposition of displacement vector is not possible in case of anisotropic materials [3]. ...
The paper concentrates on the study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor half-space underlying an inviscid liquid. The reflection and transmission coefficients varying with the incident angles are examined. Calculated results are verified by considering the quasilongitudinal ( q P ) and quasitransverse ( q S V ) waves. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection and transmission coefficients are carried out with the help of Gauss elimination method by using MATLAB programming software for silicon (Si) and germanium (Ge) semiconductors. The computer simulated-results have been plotted graphically for Si and presented in tabular form in case of Ge semiconductors. The study may be useful in semiconductors, geology, and seismology in addition to surface acoustic wave (SAW) devices.
... They demonstrated how the amplitude ratios of reflected waves depend on the angle of incidence, the characteristics of the rotating medium, and different levels of initial stresses. Sidhu and Singh [14] examined the reflection of plane waves at the free surface of an initially stressed elastic medium directly without using potentials. In this research, they demonstrated that the velocities of quasi-P and quasi-S waves vary based on the propagation angle. ...
We use an effective medium model to study the problem of reflection of plane waves from the free surface of a half-space occupied by an elastic particulate metacomposite. This problem has received little attention in the recent literature despite its significance from both practical and theoretical points of view. Classical formulas for the reflection angles and amplitudes of the reflected waves for a homogeneous elastic half-space with no wave attenuation are extended to a particulate metacomposite half-space with wave attenuation. We also include a detailed discussion concerning the reflected plane shear wave and surface compressional wave in the case of an incident shear wave propagating at an incident angle smaller than the critical angle. The efficiency and accuracy of the model are demonstrated via detailed comparisons between the predicted phase velocity and attenuation coefficient of plane waves in an (infinite) entire space and the corresponding results available in the literature. The implications of our results on the reflection of plane waves from the free surface of a hard sphere-filled elastic metacomposite are discussed. We mention that a quantitative validation of our results cannot be made here as a result of the lack of availability of established data in the existing literature.
... An analogous study was completed by Chattopadhyay et al. (1982) to additionally analyse the effect of initial stress and sandiness on the surface wave. The effect of stress on the critical reflection of longitudinal and shear waves on the free surface were described by Sidhu & Singh (1984). They also stated that the stressed isotropic media possessed orthotropic symmetry. ...
The stress-dependent wave reflection and transmission (R/T) coefficients for the layered transversely isotropic media with a vertical symmetry axis (VTI) are seldom investigated in the published literature. To fill this gap, we propose the exact formulas for the plane wave R/T coefficients on the welded and nonwelded boundaries between two distinct VTI half-spaces under the effect of initial stress. The theory of acoustoelasticity is employed to describe the influence of initial applied stress on the overall elastic properties of VTI media and to represent two different boundary conditions. The normal and tangential compliances are used to characterize the discontinuity of nonwelded boundary based on the linear-slip model, and their stress dependences are ideally considered according to the effect of stress on fracture aperture. Then the plane-wave displacement equations are substituted into the boundary conditions to yield the analytic formulas for frequency-independent R/T coefficients for the welded interface and frequency-dependent R/T coefficients for the nonwelded interface. The stress-dependent wave slowness vector and polarization vector embedded in R/T coefficients can be directly computed with the Christoffel equation given by the acoustoelastic equation. Modeling results graphically show the effects of initial stress on the angle-dependent wave velocities, Thomsen elastic anisotropy parameters, fracture compliances, the R/T coefficients and seismic reflection responses for welded and nonwelded interfaces in detail. The R/T coefficients are more sensitive to initial stress at relatively large incidence angles for the designed two-layer model with welded or nonwelded boundary. The proposed R/T coefficient formulas and modeling results are relevant to in-situ stress detection, fracture characterization, and exploration for oil and gas in shale reservoirs in high-stress fields.
... Sinha and Sinha 20 investigated the problems of reflection of thermoelastic waves at a solid half-space with thermal relaxation. Sidhu and Singh 21 investigated the reflection of P and SV waves in an initially stressed elastic half-space at a traction free surface. Chandrasekharaiah and Srinatiah 22 studied thermoelastic plane waves in a rotating isotropic solid. ...
This research article deals with the propagation of plane waves in a rotating magneto-thermoelastic half-space with diffusion in the context of Lord–Shulman theory of thermoelasticity. The governing equations are formulated for a specialized plane and solved to obtain the velocity equation, which indicates the existence of four coupled plane waves. The wave velocities are computed for a particular medium, and the effect of the diffusion parameter is shown graphically on these waves. Impedance boundary conditions are applied to study the reflection of plane waves from the stress-free and thermally insulated/isothermal surface. Reflection coefficients and energy ratios of the reflected waves are also obtained, and the variations of reflection coefficients and energy ratios against the angle of incidence are shown graphically.
... Some problems of interest in different elastic media are discussed by Deresiewicz (1960); Christie (1955); Parfitt. and Eringen (1969); Sinha and Sinha (1974); Haskell (1962); Dey and Addy (1977); Borcherdt (1982); Sidhu and Singh (1984); Rokhlin et al. (1986); Ogden and Sotiropoulos (1998) The use impedance boundary conditions in various fields of physics like acoustics and electromagnetism is common. Literature on wave propagation phenomenon with impedance boundary conditions is at an early stage. ...
... There is a number of works, see for examples [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], investigating the reflection and transmission of a qP-wave or a qSV-wave in prestressed elastic media. However, all of them considered either the reflection of these waves from the surface of a pre-stressed elastic half-space or the reflection and transmission of them at the interface between the two pre-stressed elastic half-spaces. ...
... The theory developed by Biot was later used by Dey and Addy [3,4] to study the effect of initial stress on plane waves. Later, the same theory was applied for various studies on the reflection and transmission of P and SV waves in isotropic materials [5][6][7][8] and for anisotropic materials [9]. More recently, various problems using classical linear theory with initial stress and other additional conditions have been studied [10][11][12][13]. ...
In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on plane waves in an incompressible material. For this problem, the initial stress is not associated with a finite elastic deformation and the material is assumed to be isotropic in the absence of the initial stress. The theory of superposition of infinitesimal deformations on finite deformation is applied to a problem of plane incremental motions in an initially stressed incompressible homogeneous elastic half-space. The general formulation of the problem is presented first and then specialized using a prototype strain energy function. Homogeneous plane waves are considered and the analysis is carried out for incompressible materials in both the deformed and the undeformed reference configurations. In addition to this, problems for the reflection of small amplitude homogeneous waves from the plane boundary of an initially stressed half-space are also considered and graphical results are included, which show the effect of initial stress on reflection. It is noted that the reflection coefficients in this case behave in a similar fashion when the initial stress is a pre-stress.
... A significant number of problems on reflection phenomenon of elastic waves at free surface and interfaces of different elastic media may be found in literature. Some problems of interest in different elastic media are given in references list [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. ...
... These results are the results of Sidhu and Singh (1984) for the relevant problem. ...
The propagation of elastic waves due to incident qSV -wave (quasi SV -wave) at a corrugated interface between two dissimilar pre-stressed elastic half-spaces has been investigated. Adopt-ing the Rayleigh's method of approximation, the reflection and transmission coefficients are derived for the first order approximation of the corrugation. We have observed that these coeffi-cients are functions of initial stresses, incremental elastic properties and angle of propagation of the incident wave. The coefficients corresponding to irregularly reflected and refracted waves depend on frequency & the amplitude of corrugated interface.
... Some basic aspects of plane wave propagation and reflection in pre-stressed solids have been examined in the paper by Sidhu and Singh [13], while Norris [10] has pointed to errors in several earlier papers by a number of authors. Inhomogeneous 'longitudinal' plane waves in a deformed elastic solid is done by Destrade and Hayes [4], and recently Destrade and Ogden [5] studied surface waves in a stretched and sheared incompressible elastic materials. ...
This paper is concerned with the effect of pure shear on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material. For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave (with angle of reflection equal to the angle of incidence) although its amplitude may vanish at certain discrete angles but is independent of the state of deformation. Reflected wave amplitude is exactly equal to one in this case. For a second class of constitutive laws similar behavior is found for certain combinations of angle of incidence, material properties and de-formations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and a surface wave. Here surface wave amplitude depends upon the pure shear and the reflected wave amplitude is not equal to one in general. The dependence of the amplitudes of the reflected, and surface waves on the angle of incidence, the states of deformation is illustrated graphi-cally.
Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in-situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log datasets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.
The research article deals with the reflection of plane waves in a micropolar thermoelastic diffusion solid half-space with voids in the context of Lord-Shulman theory of thermoelasticity. The governing equations are formulated to obtain two velocity equations. The plane wave solution of these velocity equations indicate the existence of six plane waves, a set of four coupled longitudinal displacement waves and a set of two coupled transverse displacement wave namely coupled longitudinal displacement wave (LD−1), coupled thermal wave (LD−2), coupled mass diffusion wave (LD−3), coupled longitudinal void volume fraction wave (LD−4) and coupled transverse displacement wave (TD−1), coupled transverse micro-rotational wave (TD−2). A reflection of the plane wave from a stress-free thermally insulated surface is studied to obtain the reflection coefficients and energy ratios of reflected waves. The reflection coefficients and energy ratios of the reflected waves are computed for a particular material and the variations of the reflection coefficients and energy ratios against the angle of incidence are shown graphically.
The plane wave propagation in a homogeneous transversely isotropic thermally conducting elastic solid is studied with two thermal relaxation times. Three types of plane waves, quasi-P, quasi-S and thermal waves, are shown to exist. The analytical expressions for their velocities of propagation are obtained. The velocities of these waves are found to depend on the angle of propagation and frequency. This dependence of velocities on the direction of propagation and frequency is shown graphically. Effects of thermal parameters and anisotropy upon these velocities are observed.
We derive frequency equation for Love waves propagating in an elastic layer overlying a poro-elastic solid half-space. The interface between layer anf half-space is considered as loosely bonded; in general, the layer and substratum are considered under initial stress. The effects of looseness of interface and the initial stress on the phase velocity of Love waves have been studied. The results for welded contact at the layer-substratum interface are also discussed in the presence and absence of initial stresses in the media. It is observed that the imperfectness of the interface is effective for long wavelength waves and the phase velocity of Love waves propagating in crustal layer of the earth shows a considerable decrease with the increase in sandiness properties of the layer.
The problem of plane SV-wave propagation at a plane interface between elastic homogeneous and non-homogeneous half-spaces has been attempted. The reflection and transmission coefficients corresponding to reflected and transmitted P-wave and SV-waves have been derived. It was found that these coefficients are functions of angle of incidence, elastic parameters and inhomogeneous parameters.
This paper illustrates the influence of pre-stress and finite strain on the reflection of homogeneous plane waves from a plane boundary in an incompressible isotropic elastic solid. For a special class of constitutive laws it is shown that an incident SV wave propagating in a principal plane of the finite deformation gives rise to a surface wave in addition to a reflected SV wave for every angle of incidence (except discrete angles for which the surface wave amplitude vanishes). For materials not in this special class and for certain states of deformation and stress, on the other hand, two SV waves are in general reflected when the angle of incidence (measured from parallel to the plane boundary) is less than a critical value. One reflected wave has angle of reflection equal to the angle of incidence. The angle of reflection of the second reflected wave may be greater than, equal to or less than the angle of incidence, but for the second reflected wave to have time-averaged energy flux directed into the material away from the boundary its wave normal is, for certain angles of incidence, necessarily directed out of the material. For angles of incidence greater than the critical angle the situation is similar to that for the special class of materials. This contrasts with the situation in the (compressible) classical theory in which an incident SV wave leads to a surface wave for angles of incidence less than some critical value. The dependence of the reflection coefficients on the angle of incidence depends markedly on the state of deformation and stress, and the range of behaviour is illustrated graphically for a specific material model. The surface wave speed increases indefinitely as the angle of incidence approaches normal incidence while its amplitude approaches zero. This is a consequence of the degeneracy associated with the incompressibility constraint that the P-wave speed is infinite.
3D solutions of the dynamical equations in the presence of external forces are derived for a homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves. Expressions for slowness surfaces and apparent velocities for these waves are derived analytically as well as numerically and represented graphically.
The effect of pre-stress on the propagation and reflection of plane waves in an incompressible isotropic elastic half-space has been examined recently by the authors (Ogden & Sotiropoulos, 1997). In the present paper the corresponding analysis for compressible materials is detailed. In the two-dimensional context considered for incompressible materials the (homogeneous) plane waves were necessarily shear waves. By contrast, in the compressible context pure shear waves can propagate only in specific directions in the considered principal plane and, in a general direction, a quasi-shear wave may be accompanied by a quasi-longitudinal wave, as is the case in the anisotropic linear theory. The dependence of the (in-plane) slowness section on the pre-stress (and finite deformation) and on the choice of constitutive law is elucidated. This information is used to determine the reflection coefficients for reflection of either a (quasi-) shear wave or a (quasi-) longitudinal wave from the boundary of the half-space and to characterize the different cases which arise depending on the geometry of the slowness section. The theoretical results are illustrated by numerical calculations for the range of possible types of behaviour with reference forms of strain-energy function and different states of finite deformation and to the question of stability of the half-space.
In the paper under discussion, Chattopadhyay e t a l. [J. Acoust. Soc. Am. 7 2, 255–263 (1982)] studied the problem of the reflection of elastic waves at the plane free boundary of an initially stressed sandy medium. The authors effected the decoupling of the P and S V motion for waves propagating in the horizontal and vertical directions which are the principal directions of the initial stress tensor considered. However, while discussing the reflection of P and S V waves, the authors use the above decoupling for waves propagating in an arbitrary direction. In fact, the expressions for the displacement potentials assumed by the authors do not satisfy the equations of motion. Consequently, most of the equations and results of the subject paper are either irrelevant or incorrect. Similarly, the results obtained by the authors for Rayleigh waves in an initially stressed sandy medium are also wrong.
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.
The paper deals with the phenomenon of reflection of plane elastic waves in a free surface when the medium is initially stressed. It has been shown analytically that the reflected P and SV waves depend on initial stresses present in the medium. The numerical values of reflection coefficients for different initial stresses and the angle of incidence have been calculated by the Computer I.C.L. 1901-A and the results are given in the form of graphs. Many interesting results are found in the paper which are not seen in an initially stress-free medium.RésuméCet article traite du phénomène de la réflexion d'ondes élastiques planes sur une surface libre lorsque le milieu possède une contrainte initiale. On a montré analytiquement que les ondes réfléchies P et SV dépendent des contraintes initiales présentes dans le milieu. Les valeurs numériques des coefficients de réflexion pour différentes contraintes initiales et l'angle d'incidence ont été calculés sur un ordinateur I.C.L. 1901-A; on donne les résultats sous forme de graphes. Dans cet article, on trouve beaucoup de résultats intéressants qu'on ne rencontre pas dans un milieu libre de contraintes initiales.ZusammenfassungDiese Arbeit behandelt die Reflektion ebener elastischer Wellen an einer freien Oberflache wenn der wellenleitende Stoff unter Vorspannung steht. Analytisch wurde gezeigt, dass die reflektierten P- und SV-Wellen von der vorhandenen Vorspannung abhangen. Zahlenwerte fur die Reflektionskoe-ffizienten fur verschiedene Vorspannungen und Einfallwinkel wurden auf einem I.C.L. 1901-A Rechner berechnet und werden in graphischer Form dargestellt. In dieser Arbeit wird eine Anzahl interessanter Ergebnisse beschrieben, die in spannungsfreien Stoffen nicht zu beobachten sind.
Because of the similarity between the equations of motion governing infinitesimal vibrations due to a small perturbing force superimposed on an already existing state of finite strain and the equations of linear anisotropic elasticity, methods of analysis used in one may be extended to the second.
In particular, in this paper, the technique of ray expansions is considered. The analysis is found to be virtually identical in the two cases although there are differences in physical interpretation.
Simple explicit solutions of the dynamical equations for a prestressed solid under horizontal compression in a gravity field are derived to establish the magnitudes of two factors influencing seismic wave propagation velocities: (1) The overburden pressure which, at depth z, leads to wave velocity changes of order rhogz/mu, where p is rock density and mu is the rigidity. Compressional velocity increases with depth, whereas shear velocity decreases: at depths of the order of 150 km this effect is of the order of 5%. (2) The deviatoric stress, or stress difference P, which leads to anisotropic velocity changes of order P/mu these depend both on the direction of propagation and the polarization of the waves. This phenomenon is only likely to be important for seismic theory under conditions approaching internal instability and buckling.
The phenomenon of reflection of elastic waves at a free surface of an initially stressed sandy medium has been investigated. The reflection coefficients have been computed numerically for both P and SV motion. The variations of reflection coefficients for different values of eta , initial stress parameter, and angle of incidence have been represented graphically. The effects of initial stress and sandiness of the layer on surface waves have also been studied. To the best of our knowledge the idea of a sandy medium under initial stress is a new one which is physically possible. Taking into consideration of the theories given by W. H. Weiskopf and M. A. Biot the equations of motion have been deduced.
The linearized equations of motion and linearized boundary and continuity conditions governing small elastic-gravitational disturbances away from equilibrium of an arbitrary, uniformly rotating, self-gravitating, perfectly elastic Earth model with an arbitrary initial static stress field are derived. The appropriate form of Rayleigh's variational principle and of the Betti reciprocal theorem and the Volterra dislocation relation for such a configuration are given. The latter is then used to derive an explicit expression for the equivalent body forces to be applied in the absence of a seismic dislocation in order to produce a dynamical response of the Earth model equivalent to that produced by the dislocation. It is found that if the initial static stress in the vicinity of the dislocation is purely hydrostatic, then a point tangential displacement dislocation has as an exactly equivalent body force the familiar double couple of moment 0, A0s0. If however the hypocentral static stress field has a deviatoric part, then additional equivalent body forces must be used properly to model a seismic dislocation. The necessary additional equivalent forces are explicitly exhibited; theoretically their existence provides a method of estimating hypocentral stresses, but the application of any such method is probably premature.