Effect of Depth of a Point Source on the Motion of the Surface of an Elastic Solid Sphere

Alterman, Z.; Abramovici, F.

doi:10.1111/j.1365-246x.1966.tb03501.x

Cited by 17 publications

(9 citation statements)

References 7 publications

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“…As shown in [9], the R-wave length is close to the length of the P -wave generated by the source. From formula (1.1) it follows that the dimensionless length of the P -wave is λ ≈ 10.…”

Section: Calculations Of Motion In the R-wavementioning

confidence: 87%

Movement of the ground in Rayleigh waves produced by underground explosions

Симоненко

Shishkin

Шишкина

2006

J Appl Mech Tech Phys

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Analytic representations are obtained for the displacement and stress fields in the Rayleigh surface wave (R-wave) generated in an elastic half-space by an internal source that produces the same seismic P -wave as an underground explosion. Oscillograms, particle trajectories, and stresses in the halfspace and on its surface are calculated. Relations for the energy flux in the R-wave are obtained. For rock salt, the fraction of the explosion energy transferred to the R-wave is estimated. It is established that this fraction can reach values of about 1% of the total explosion energy if the explosion is a contained one. As the charge depth is increased, the energy of the R-wave decreases in approximately inverse proportion to the depth.Introduction. Elastic surface Rayleigh waves (R-waves) [1] result from dynamic actions on the surface of elastic bodies. In structures of small dimensions, they are used as ultrasonic waves. Rayleigh waves are also observed in large structures and engineering constructions. In addition, R-waves are produced by explosions, earthquakes, and impacts of cosmic bodies on the planets. Seismic R-waves are used to probe the Earth's crust and to study its structure, and long R-waves are used to study the Earth's mantle. Rayleigh waves produced by explosions contain a significant fraction of the explosion energy, and at a certain distance from the epicenter, they dominate the other seismic waves. They contain information on the energy source and the properties of the medium. For example, records of R-waves from some underground nuclear explosions suggest that spall fracture of the medium occurs at the epicenters of the explosions [2]. In [3], it is shown that during impacts of cosmic bodies on the Earth, the focusing of R-waves in the antipode region (the region opposite to the site of impact) can lead to the formation of unusual geological structures such as explosion pipes or diatremes.Rayleigh waves produced by a point source in an elastic half-space were considered in [4][5][6]. Petrashen' [7] studied the Lamb problem for the case of an isotropic elastic sphere and obtained expressions for Rayleigh waves on the surface of an elastic sphere. Onis'ko and Shemyakin [8] studied the movement of the ground surface for an explosion in a half-space, and Alterman and Abramovici [9] explored the movement of the surface of an elastic sphere for a contained explosion. Brekhovskikh [10] investigated Rayleigh waves produced by a harmonious source and propagating along the curved surface of an elastic body. The present paper gives more detailed results for Rayleigh waves on both the surface of an elastic half-space and inside it for explosions at a great depth. The energy flux transferred by the Rayleigh wave is considered, and the fraction of the explosion energy converted into the R-wave energy is estimated. Such data are required to obtain more exact estimates of the damage to various engineering constructions from R-waves and describe the dynamic geological processes occurring in the regions that are ...

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“…As shown in [9], the R-wave length is close to the length of the P -wave generated by the source. From formula (1.1) it follows that the dimensionless length of the P -wave is λ ≈ 10.…”

Section: Calculations Of Motion In the R-wavementioning

confidence: 87%

Movement of the ground in Rayleigh waves produced by underground explosions

Симоненко

Shishkin

Шишкина

2006

J Appl Mech Tech Phys

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“…Its radial component is 13 times the radial component of the direct P pulse. The diffracted pulses and the ' logarithmic ' pulses were discussed previously (Alterman & Kornfeld 1963, 1965. Figs.…”

Section: I37mentioning

confidence: 99%

The Motion of a Sphere Caused by an Impulsive Force and by an Explosive Point-Source

Alterman

Abramovici

1967

Geophys J Int

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An exact solution is obtained for the displacement of the surface of a uniform elastic solid sphere of radius a due to an impulsive force applied in radial direction at a point situated at a depth d below the surface.The theoretical seismograms, which were computed for several epicentral distances 9 ( O k 9 < 180"), show several modes of surface waves in addition to various reflected and diffracted pulses. The higher modes of surface waves are a result of sphericity only and are compared with the observed higher modes and with the higher modes in a plane layered model. A comparison of the present SV-P source with an explosive P-source shows that the Rayleigh-waves R I i from the SV-P source have a larger high-frequency content than R from the P-source, show less dispersion and have larger amplitudes when caused by shallow and medium-depth sources. Higher mode surface waves R 2 i have larger angular than radial component. They are larger in the results from the P-source, and smaller from the SV-P source. The mode R S i has larger radial than angular component and is found equally from both sources. The GVv surface-waves are the largest guided waves to arrive from the deep SV-P source, they are largest except for the Rayleigh-wave from the shallow and medium depth SV-P sources. The radial component of CV. is 2-3 times larger than its angular component and is similar to the G, guided waves from an SH-torque source. GVv is absent in results from the P-source.The connection between the higher modes of surface waves and groups of certain reflected pulses is given. ' Refracted' pulses, found in results from the shallow SV-P source, have a larger angular than radial component. In several cases the angular component is larger than that of the direct pulse.

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“…Following the considerable development of electronic computers, it has become possible to draw theoretical seismograms (Sat8 & Usami 1962Sat8 et al 1962Sat8 et al , 1963aSat8 et al , 1967aSat8 et al , 1967bSat8 et al , 1963bUsami & Sat6 1964Alterman & Kornfeld 1964Usami et al 1965;Alterman & Abramovici 1966) and to compare amplitudes of body waves on the seismograms with those obtained by the ray theory. But by this approach it is difficult to find the period range for which the ray theory 527 is applicable, in other words, to see for what period range amplitudes and travel times of body waves can be explained by the ray theory.…”

Section: Introductionmentioning

confidence: 99%

Amplitude of Body Waves in a Heterogeneous Sphere Comparison of Wave Theory and Ray Theory

Satô

1969

Geophys J Int

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According to ray theory, the path of propagation of body waves is curved due to hetogeneity of the medium, and the amplitude is obtained by calculating the geometrical spreading effect. The ray theory is valid when the variation of physical constants such as velocity and density is slight within a wavelength. It is frequently required, however, to know the behaviour of waves in rather long period range where it is uncertain whether the ray theory is applicable or not.In this paper, the expression for the disturbance due to a point SHsource when the medium consists of many heterogeneous spherical shells with second order discontinuities is obtained in an exact form. The solution is computed numerically and compared with that expected from the ray theory.When there is no singular point on the travel-time curve, the ray theory seems to give a fairly good approximation in a wide frequency range, although the existence of the second order discontinuity effects observed amplitudes with long period if the observing point is close to the boundary.At the caustic or the cusp point of the travel-time. curve, where the amplitude cannot be calculated from simple ray theory, the amplitude is nearly proportional to (period)-) and is not extremely large. The phase shift is nearly equal to n/4.

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Effect of Depth of a Point Source on the Motion of the Surface of an Elastic Solid Sphere

Cited by 17 publications

References 7 publications

Movement of the ground in Rayleigh waves produced by underground explosions

Movement of the ground in Rayleigh waves produced by underground explosions

The Motion of a Sphere Caused by an Impulsive Force and by an Explosive Point-Source

Amplitude of Body Waves in a Heterogeneous Sphere Comparison of Wave Theory and Ray Theory

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