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first-order degenerate perturbation theory

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Journal Article
Published: 01 August 1982
Bulletin of the Seismological Society of America (1982) 72 (4): 1155–1166.
...Sandra P. Morris; Robert J. Geller abstract Two methods which have been proposed for calculating the normal modes of laterally heterogeneous models of the Earth are first-order degenerate perturbation theory and the Rayleigh-Ritz variational procedure. The purpose of this paper is to compare...
Journal Article
Published: 01 February 1978
Bulletin of the Seismological Society of America (1978) 68 (1): 103–116.
... the normal modes of a laterally heterogeneous earth model has been considered by Madariaga (1972), Saito (1971), and Luh (1974) using first- order degenerate perturbation theory. Luh (1973) considered the effect of quasi- degeneracy. Usami (1971) used variational methods to find some very low-order tor...
Journal Article
Journal: Geophysics
Published: 01 June 2006
Geophysics (2006) 71 (3): D45–D58.
... perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas...
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Journal Article
Published: 25 February 2014
Bulletin of the Seismological Society of America (2014) 104 (2): 962–971.
... with azimuthal orders − l ≤ m ≤ l and degenerate, complex eigenfrequencies. The singlets as defined in (3) form a complete vector space of a dissipative, spherical, nonrotating, and isotropic earth model. With a perturbative rescaling into the (approximately) elliptical coordinates of the real Earth...
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Journal Article
Journal: Geophysics
Published: 01 January 2004
Geophysics (2004) 69 (4): 1053–1070.
... to inserting S   =   I and R   =   I in (equation 26 ). Based on paraxial ray theory ( Červený, 2001 ) and recently developed theory for first-order perturbation of isochron points ( Iversen, 2001a ), I have derived two systems of ODEs that can be used for numerical integration...
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Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (6): 2063–2072.
... of the Christoffel matrix Γ ik = c ijkl P j P l of the incident wave. Here we refer to the eigenvalues as almost identical if their difference is small, i.e., on the order of Δ c ijkl and Δρ. For a degenerate incident wave, we choose polarization vector e 6 i in such a way...
FIGURES
Journal Article
Published: 01 October 1972
Bulletin of the Seismological Society of America (1972) 62 (5): 1183–1193.
... in ~o For S waves the perturbation theory is degenerate. Let ~1 and ~2 be any two mutually orthogonal unit vectors in the plane normal to the wave vector ~. Then, again to first order in Zo 6c21(f0 and 6c22 are the two eigenvalues of the 2 x 2 symmetric matrix [~1 "b1 sl "b12 ] ~ b ~2 ~2 b S2_l. (23...
Journal Article
Journal: Geophysics
Published: 23 July 2008
Geophysics (2008) 73 (5): D63–D73.
... is simplified if perturbation theory is applied. The elastic medium is viewed as a background medium, and the attenuation effects are incorporated as perturbations. In the first-order approximation, the phase and ray velocities and their directions remain unchanged, being the same as those in the background...
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Book Chapter

Series: European Mineralogical Union Notes in Mineralogy
Publisher: European Mineralogical Union
Published: 01 January 2012
DOI: 10.1180/EMU-notes.12.4
EISBN: 9780903056366
... and the third terms on the left side of equation 19 vanish. The first-order term for energy becomes: Because of the nature of the external perturbation this is equivalent to: and its corresponding electronic wavefunctions and density are determined self-consistently, step by step. All...
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Journal Article
Published: 01 June 2003
Bulletin of the Seismological Society of America (2003) 93 (3): 1370–1380.
...: the multiplets are split. Within the framework of first-order perturbation theory this splitting can be linearly related to aspherical structure. Consider the k th multiplet, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle...
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Journal Article
Journal: Geophysics
Published: 28 September 2009
Geophysics (2009) 74 (5): WB203–WB213.
... for weak elastic anisotropy. The WAA parameters generalize WA parameters by incorporating attenuation effects. They can be represented alternatively by one set of complex values or by two sets of real values. Assuming high-frequency waves and using the first-order perturbation theory, all basic wave...
FIGURES
Journal Article
Published: 01 February 1989
Bulletin of the Seismological Society of America (1989) 79 (1): 85–100.
... that, for any ~I, ~ will lie in the orange-slice shaped sectors defined by the great circles that pass through 1 and ] and through l ' and ~. This region is shaded in Figure ld. ANALYSIS OF UNCERTAINTIES Our error analysis is based on a simple application of first-order perturbation theory. The notation...
Journal Article
Published: 01 June 1973
Bulletin of the Seismological Society of America (1973) 63 (3): 1175–1176.
...), equation (21)] may be readily superposed. The absence of a first-order direct effect on P-wave velocities is the only important conclusion to emerge from my work. The perturbation-theoretic formulation for S waves is degenerate, and this complicates the treatment of S waves for the general case...
Journal Article
Journal: Geophysics
Published: 16 May 2017
Geophysics (2017) 82 (4): WA11–WA21.
.... Similar observations are made for the S1-wave traveltime parameters as in the first model. However, due to the existence of the hyperbolic region on the slowness sheet of the fast S-wave (S2), the traveltime parameters are degenerated for both the wave modes. Upon closer inspection of Figures  13 – 15...
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Journal Article
Published: 01 August 2010
American Mineralogist (2010) 95 (8-9): 1257–1266.
... the frequency changes induced by D/H isotopic substitution ( Berglund et al. 1978 ). Perturbation theory applied to OH-stretching modes evidences that the large frequency shift mostly results from the small effective mass of the stretching modes related to the small mass of hydrogen; whereas the potential...
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Journal Article
Journal: Geophysics
Published: 28 September 2009
Geophysics (2009) 74 (5): WB147–WB152.
... . The S-wave polarization vectors are not unique because their only condition is to be orthogonal to each other and to lie in the plane perpendicular to n . This degeneration of the S-wave polarization, however, can be removed with first-order perturbation theory. Consider the two vectors e...
FIGURES
Journal Article
Published: 01 February 2004
Mineralogical Magazine (2004) 68 (1): 177–189.
... in the bands. However, this perturbation is documented in the IR spectra by a substantial increase in the FWHH as well as the occurrence of shoulders on the broadened bands. In contrast, at lower temperatures, the observed band splittings in the former triply degenerated states of ν 3 and ν 4 could...
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Journal Article
Journal: Geophysics
Published: 15 May 2017
Geophysics (2017) 82 (4): MR105–MR109.
... in pores associated with P-waves in the rock matrix. The basic idea is to approximate classification of waves in saturated rocks into two orders. The first order is a dominant P-wave in the matrix, and the second order is a companion fluid wave induced by the first-order wave. This approach is innovative...
FIGURES
Journal Article
Published: 01 June 2017
European Journal of Mineralogy (2017) 29 (3): 397–408.
... ): Phonons and related crystal properties from density-functional perturbation theory . Rev. Modern Phys ., 73 , 515 – 561 . Belousov , M.V. , Pogarev , D.E. , Shultin , A.A. ( 1970 ): Band intensity of extraplanar n 2 vibration of CO 3 2− and NO 3 − ions in crystals having...
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Journal Article
Published: 01 October 2002
American Mineralogist (2002) 87 (10): 1286–1290.
...Etienne Balan; A. Marco Saitta; Francesco Mauri; Céline Lemaire; François Guyot Abstract The theoretical infrared spectrum of lizardite [Mg 3 Si 2 O 5 (OH) 4 ] was computed using first-principles quantum mechanical calculations. Density functional perturbation theory allowed us to derive the low...
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