scholarly journals The effect of arbitrarily small rigidity on the free oscillations of the Earth

1997 ◽  
Vol 40 (5) ◽  
Author(s):  
F. Gilbert
Keyword(s):  
Shear Modulus ◽  
Singular Perturbation ◽  
Rayleigh Waves ◽  
Elastic Solid ◽  
Free Oscillations ◽  
Approximate Methods ◽  
Fluid State ◽  
The Earth ◽  
Elastic Interface ◽  
Small Shear

The system of propagator equations for an elastic solid becomes singular as the shear modulus becomes vanishingly small. In computational applications there is severe loss of precision as the limit of zero shear modulus is approached. The use of perturbation theory to address the effect of very small shear modulus, using the fluid state as a basis, is unsatisfactory because certain phenomena, e.g., Rayleigh waves, cannot be represented. Two approximate methods are presented to account for the singular perturbation. Since most of the Earth is nearly neutrally stratified, in which case the motion is nearly irrotational, one can impose the irrotational constraint and obtain a modified and reduced system of propagator equations. This system does not have the singular perturbation. In the second method the transition zone between a fluid and a solid is represented as an infinitesimally thin, Massive, Elastic Interface (MEI). The boundary conditions across the MEI are dispersive and algebraic. The limit of zero shear modulus is non-singular.

The response of the horizontal pendulum seismometer to Rayleigh and Love waves, tilt, and free oscillations of the earth

1968 ◽  
Vol 58 (5) ◽  
pp. 1385-1406
Author(s):  
P. W. Rodgers
Keyword(s):  
Correction Factor ◽  
Rayleigh Waves ◽  
Complete Response ◽  
Love Waves ◽  
Free Oscillations ◽  
Long Period ◽  
The Earth ◽  
Circular Particle ◽  
Rayleigh And Love Waves ◽  
Horizontal Pendulum

Abstract The horizontal pendulum seismometer is sensitive not only to acceleration along its sensitive axis but also to tilt, variations in the angle of inclination, and along-the-boom acceleration. The complete steady-state response of this type of seismometer to Rayleigh and Love waves, tilt, and free oscillations of the Earth is treated. An equation of motion is developed which includes the effects of tilt, variation in the angle of inclination, and along-the-boom acceleration. An approximate solution to this equation is obtained which separates out the response due to each effect. The response, including these effects, is developed for Rayleigh and Love waves and the conditions under which along-the-boom acceleration and variations in the angle of inclination are important are stated. The question “How much of the seismogram is due to tilt?” is answered in detail for long period Rayleigh waves and free oscillations. It is shown that the seismograms resulting from such waves can require sizable corrections depending on the wave parameters. A correction factor for Rayleigh waves is developed which is universal in the sense that it is independent of the parameters of the particular seismometer and thus applies to all pendulous horizontal seismographs. For Rayleigh waves it is a function only of ellipticity, phase velocity, and period. Correction factor curves for long-period retrograde Rayleigh waves are presented. For circular particle motions a ten per cent correction is required for a three hundred second Rayleigh wave. The problem of obtaining the horizontal ground motion is treated. The response of the horizontal seismometer as a tilt meter is examined; a conversion factor between displacement and tilt magnification is developed. The complete response to simultaneous spheroidal and torsional free oscillations of the Earth is developed. It is shown that the principal response to the low-order spheroidal modes is as a tilt meter. The relationship between the horizontal and vertical seismogram is developed.

Free oscillations of parameters of the Earth's motion in the Earth-Moon system

2001 ◽  
Vol 45 (11) ◽  
pp. 922-928 ◽  
Author(s):  
G. S. Kurbasova ◽  
L. V. Rykhlova
Keyword(s):  
Free Oscillations ◽  
Moon System ◽  
The Earth

use of apparent complex frequencies of free oscillations to estimate aspherical structure of the Earth

1991 ◽  
Vol 18 (5) ◽  
pp. 905-908 ◽  
Author(s):  
Naoki Shibata ◽  
Naoki Suda ◽  
Yoshio Fukao
Keyword(s):  
Free Oscillations ◽  
The Earth

Comparison of theoretical and observational expressions for elastic strains of the earth

1965 ◽  
Vol 55 (1) ◽  
pp. 153-163
Author(s):  
H. Takeuchi ◽  
L. E. Alsop
Keyword(s):  
Theoretical Studies ◽  
Free Oscillations ◽  
The Earth ◽  
Tidal Deformation ◽  
Elastic Strains

Abstract Transitional equations are provided between the quantities obtained in theoretical studies of tidal deformation, loading, and free oscillations of the earth and the empirical quantities obtained from observations of these phenomena. Tables of theoretical quantities are provided so that estimates may be made of the values to be expected observationally. Several examples are discussed.

Free oscillation observations through 1968

1969 ◽  
Vol 59 (5) ◽  
pp. 2079-2099
Author(s):  
John S. Derr
Keyword(s):  
Confidence Level ◽  
Free Oscillation ◽  
Statistical Tests ◽  
Standard Errors ◽  
Free Oscillations ◽  
The Earth ◽  
Earth Models ◽  
Earth Structures ◽  
Path Dependent ◽  
Consistent Basis

abstract All observations of the free oscillations of the Earth published through 1968 are weighted to produce a set of means and standard errors of the means. Fundamental orders 0 to 97 for spheroidal and 2 to 99 for torsional are treated, as well as many overtones up to order 49. Statistical tests indicate that some observations are path dependent at the 99 per cent confidence level. Comparison of these means and standard errors with published Earth models indicate that they form a consistent basis for inversion of free oscillation observations to infer Earth structures.

scholarly journals Mantle Rayleigh waves from the Kamchatka earthquake of November 4, 1952*

1954 ◽  
Vol 44 (3) ◽  
pp. 471-479
Author(s):  
Maurice Ewing ◽  
Frank Press
Keyword(s):  
Internal Friction ◽  
Rayleigh Waves ◽  
Shear Velocity ◽  
Velocity Curve ◽  
The Core ◽  
Long Period ◽  
A Value ◽  
The Earth ◽  
Dispersion Data ◽  
Kamchatka Earthquake

Abstract Mantle Rayleigh waves from the Kamchatka earthquake of November 4, 1952, are analyzed. The new Palisades long-period vertical seismograph recorded orders R6–R15, the corresponding paths involving up to seven complete passages around the earth. The dispersion data for periods below 400 sec. are in excellent agreement with earlier results and can be explained in terms of the known increase of shear velocity with depth in the mantle. Data for periods 400-480 sec. indicate a tendency for the group velocity curve to level off, suggesting that these long waves are influenced by a low or vanishing shear velocity in the core. Deduction of internal friction in the mantle from wave absorption gives a value 1/Q = 370 × 10−5 for periods 250-350 sec. This is a little over half the value reported earlier for periods 140-215 sec.

scholarly journals Rayleigh-Type Waves in a Rotating Fiber-Reinforced Half Space under the Action of Magnetic Field, Gravity and Surface Stress

2021 ◽  
pp. 1-7
Author(s):  
Narottam Maity ◽  
◽  
S P Barik Barik ◽  
P K Chaudhuri ◽  
◽  
...  
Keyword(s):  
Magnetic Field ◽  
Surface Stress ◽  
Rayleigh Waves ◽  
Elastic Solid ◽  
Frequency Equation ◽  
Complete Agreement ◽  
Fiber Reinforced ◽  
Electrically Conducting ◽  
Stress Rotation ◽  
Wave Velocity Equation

The aim of the present article is to analyze the propagation of Rayleigh waves in a rotating fiber-reinforced electrically conducting elastic solid medium under the influence of surface stress, magnetic field and gravity. The magnetic field is applied in such a direction that the problem can be considered as a two dimensional one. The wave velocity equation for Rayleigh waves has been obtained. In the absence of gravity field, surface stress, rotation and fiberreinforcement, the frequency equation is in complete agreement with the corresponding classical results. The effects on various subjects of interest are discussed and shown graphically. Comparisons are made with the corresponding results in absence of surface stress

Statistical properties of Rayleigh waves due to scattering by topography

1967 ◽  
Vol 57 (1) ◽  
pp. 83-90
Author(s):  
J. A. Hudson ◽  
L. Knopoff
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Seismic Noise ◽  
Forward Scattering ◽  
Statistical Properties ◽  
Body Waves ◽  
Simplified Model ◽  
Two Dimensional ◽  
Especial Reference ◽  
The Earth

abstract The two-dimensional problems of the scattering of harmonic body waves and Rayleigh waves by topographic irregularities in the surface of a simplified model of the earth are considered with especial reference to the processes of P-R, SV-R and R-R scattering. The topography is assumed to have certain statistical properties; the scattered surface waves also have describable statistical properties. The results obtained show that the maximum scattered seismic noise is in the range of wavelengths of the order of the lateral dimensions of the topography. The process SV-R is maximized over a broader band of wavelengths than the process P-R and thus the former may be more difficult to remove by selective filtering. An investigation of the process R-R shows that backscattering is much more important than forward scattering and hence topography beyond the array must be taken into account.

The Elastic Coefficients of the Theory of Consolidation

1957 ◽  
Vol 24 (4) ◽  
pp. 594-601
Author(s):  
M. A. Biot ◽  
D. G. Willis
Keyword(s):  
Nonlinear Systems ◽  
Shear Modulus ◽  
Physical Interpretation ◽  
Compressible Fluid ◽  
Elastic Solid ◽  
Fluid Content ◽  
Alternate Forms ◽  
Elastic Coefficients ◽  
Methods Of Measurement

Abstract The theory of the deformation of a porous elastic solid containing a compressible fluid has been established by Biot. In this paper, methods of measurement are described for the determination of the elastic coefficients of the theory. The physical interpretation of the coefficients in various alternate forms is also discussed. Any combination of measurements which is sufficient to fix the properties of the system may be used to determine the coefficients. For an isotropic system, in which there are four coefficients, the four measurements of shear modulus, jacketed and unjacketed compressibility, and coefficient of fluid content, together with a measurement of porosity appear to be the most convenient. The porosity is not required if the variables and coefficients are expressed in the proper way. The coefficient of fluid content is a measure of the volume of fluid entering the pores of a solid sample during an unjacketed compressibility test. The stress-strain relations may be expressed in terms of the stresses and strains produced during the various measurements, to give four expressions relating the measured coefficients to the original coefficients of the consolidation theory. The same method is easily extended to cases of anisotropy. The theory is directly applicable to linear systems but also may be applied to incremental variations in nonlinear systems provided the stresses are defined properly.

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