The pattern of eigenfrequencies of radial overtones which is predicted for a specified Earth-model
Main Article Content
Abstract
In 1974 Anderssen and Cleary examined the distribution of eigenfrequencies
of radial overtones in torsional oscillations of Earth-models.
They pointed out that according to Sturm-Liouville theory this distribution
should approach asymptotically, for large overtone number m,
the value nnz/y, where y is the time taken by a shear-wave to travel
along a radius from the core-mantle interface to the surface, provided
elastic parameters vary continuously along the radius. They found that,
for all the models which they considered, the distributions of eigenfrequencies
deviated from the asymptote by amounts which depended on
the existence and size of internal discontinuities. Lapwood (1975) showed
that such deviations were to be expected from Sturm-Liouville theory,
and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouville
theory to apply to differential equations with discontinuous coefficients.
Anderssen (1977) used their results to show how to predict the pattern
of deviations —called by McNabb et al. the solotone effect— for a
given discontinuity in an Earth-model.
Recently Sato and Lapwood (1977), in a series of papers which will
be referred to here simply as I, II, III, have explored the solotone effect
for layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how the
form of the pattern of eigenfrequencies, which is the graph of
S — YMUJI/N — m against m, where ,„CJI is the frequency of the m"'
overtone in the I"' (Legendre) mode of torsional oscillation, is determined
as to periodicity (or quasi-periodicity) by the thicknesses and velocities
of the layers, and as to amplitude by the amounts of the discontinuities
(III). The pattern of eigenfrequencies proves to be extremely sensitive
to small changes in layer-thicknesses in a model.
In this paper we examine a proposed Earth-model with six surfaces
of discontinuity between core boundary and surface, and predict its
pattern of eigenfrequencies. When seismological observations become
precise enough, and can be subjected to numerical analysis refined
enough, to identify the radial overtones for large m, this pattern of
eigenfrequencies will prove to be a severe test for any proposed model,
including he one which we discuss below.
of radial overtones in torsional oscillations of Earth-models.
They pointed out that according to Sturm-Liouville theory this distribution
should approach asymptotically, for large overtone number m,
the value nnz/y, where y is the time taken by a shear-wave to travel
along a radius from the core-mantle interface to the surface, provided
elastic parameters vary continuously along the radius. They found that,
for all the models which they considered, the distributions of eigenfrequencies
deviated from the asymptote by amounts which depended on
the existence and size of internal discontinuities. Lapwood (1975) showed
that such deviations were to be expected from Sturm-Liouville theory,
and McNabb, Anderssen and Lapwood (1976) extended Sturm-Liouville
theory to apply to differential equations with discontinuous coefficients.
Anderssen (1977) used their results to show how to predict the pattern
of deviations —called by McNabb et al. the solotone effect— for a
given discontinuity in an Earth-model.
Recently Sato and Lapwood (1977), in a series of papers which will
be referred to here simply as I, II, III, have explored the solotone effect
for layered spherical shells, using approximations derived from an exacttheory which holds for uniform layering. They have shown how the
form of the pattern of eigenfrequencies, which is the graph of
S — YMUJI/N — m against m, where ,„CJI is the frequency of the m"'
overtone in the I"' (Legendre) mode of torsional oscillation, is determined
as to periodicity (or quasi-periodicity) by the thicknesses and velocities
of the layers, and as to amplitude by the amounts of the discontinuities
(III). The pattern of eigenfrequencies proves to be extremely sensitive
to small changes in layer-thicknesses in a model.
In this paper we examine a proposed Earth-model with six surfaces
of discontinuity between core boundary and surface, and predict its
pattern of eigenfrequencies. When seismological observations become
precise enough, and can be subjected to numerical analysis refined
enough, to identify the radial overtones for large m, this pattern of
eigenfrequencies will prove to be a severe test for any proposed model,
including he one which we discuss below.
Article Details
How to Cite
LAPWOOD, E. R. and SATO, R. (1977) “The pattern of eigenfrequencies of radial overtones which is predicted for a specified Earth-model”, Annals of Geophysics, 30(3-4), pp. 459–469. doi: 10.4401/ag-4832.
Issue
Section
OLD
Open-Access License
No Permission Required
Istituto Nazionale di Geofisica e Vulcanologia applies the Creative Commons Attribution License (CCAL) to all works we publish.
Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, so long as the original authors and source are cited. No permission is required from the authors or the publishers.
In most cases, appropriate attribution can be provided by simply citing the original article.
If the item you plan to reuse is not part of a published article (e.g., a featured issue image), then please indicate the originator of the work, and the volume, issue, and date of the journal in which the item appeared. For any reuse or redistribution of a work, you must also make clear the license terms under which the work was published.
This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your own work will ensure your right to make your work freely and openly available. For queries about the license, please contact ann.geophys@ingv.it.