Abstract
We have seen that the complete seismic field induced in a radially heterogeneous sphere can be expressed as an infinite sum of standing waves, namely the normal modes. However, we know from seismogram analysis that most of the recorded earth motion can be explained in terms of propagating waves. There must exist a link, therefore, between these two seemingly different aspects of wave motion.
After having spent years trying to be accurate, we must spend as many more in discovering when and how to be inaccurate.
(Ambrose Bierce)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
Alsop E (1964) Spheroidal free periods of the earth observed at eight stations around the world. Bull. Seism. Soc. Am 54: 755–776.
Alterman Z, Jarosch H, Pekeris CL (1961) Propagation of Rayleigh waves in the earth. Geophys Jour Roy Astron Soc (London) 4: 219–241.
Alterman Z, Kornfeld P (1966) Effect of a fluid core on propagation of an SH-torque pulse from a point-source in a sphere. Geophys 31: 741–763.
Ansell JH (1978) On the scattering of SH waves from a point source by a sphere. Geophys Jour Roy Astron Soc (London) 54: 349–387.
Ben-Menahem A (1962) Radiation of seismic body waves from a finite moving source in the earth. Jour Geophys Res 67: 345–350.
Ben-Menahem A (1964) Mode-ray duality. Bull Seismol Soc Amer 54: 1315–1321.
Benndorf H (1905) Über die Art der Fortpflanzung der Erdbebenwellen in Erdinnern. Mitt Erdbebenkomm Wien 29: 1407.
Biswas NN, Knopoff L (1970) Exact earth-flattening calculations for Love waves. Bull Seismol Soc Amer 60: 1123–1137.
Bremmer H (1949) Terrestrial Radio Waves. Elsevier, New York, 343 pp.
Bromwich TJ I’A (1904) The caustic by reflection, of a circle. Amer Jour Math 26: 33–44.
Brune JN (1964) Travel times, body waves, and normal modes of the earth. Bull Seismol Soc Amer 54: 2099–2128.
Brune JN (1966) P and S wave travel times and spheroidal normal modes of a homogeneous sphere. Jour Geophys Res 71: 2959–2965.
Brune JN, Nafe JE, Alsop LE (1961) The polar phase shift of surface waves on a sphere. Bull Seismol Soc Amer 51: 247–257.
Cayley A (1857) A memoir upon caustics. Phil Trans Roy Soc (London) A147: 273–312.
Chapman CH (1969) Seismic wave diffraction theory. Ph.D. Thesis, Cambridge University.
Duwalo G, Jacobs JA (1959) Effects of a liquid core on the propagation of seismic waves. Canad Jour Phys 37: 109–128.
Engdahl ER (1968) Seismic waves within Earth’s outer core: multiple reflection. Science 161: 263–264.
Engdahl ER, Flinn EA, Romney CR (1970) Seismic waves reflected from the Earth’s inner core. Nature 228: 852–853.
Franz W (1954) Über die Greenschen Funktionen des Zylinders und der Kugel. Zeit Naturforschung 9a: 705–716.
Gans R (1915) Fortpflanzung des Lichtes durch ein inhomogenes Medium. Ann Phys 47: 709–736.
Gilbert F (1976) The representation of seismic displacements in terms of travelling waves. Geophys Jour Roy Astron Soc (London) 44: 275–280.
Gilbert F, Helmberger DV (1972) Generalized ray theory for a layered sphere. Geophys, Jour Roy Astron Soc (London) 27: 57–80.
Gilbert F, MacDonald GJF (1960) Free oscillations of the earth. I. Toroidal oscillations. Jour Geophys Res 65: 675–693.
Gutenberg B (1951) PKKP, P’P’ and the Earth’s core. Trans Amer Geophys Union 32: 373–390.
Herrin E (1968) Seismological tables for P phases. Bull Seismol Soc Amer 58: 1193–1241.
Imai I (1954) Die Beugung Electromagnetischer Wellen an einen Kreiszylinder. Zeit Phys 137: 31–48.
Jacobs JA (1975) The Earth’s Core. Academic Press, London, 253 pp.
Jeans JH (1923) The propagation of earthquake waves. Proc Roy Soc (London) A102: 554–574.
Jeffreys H, Lapwood ER (1957) The reflection of a pulse within a sphere. Proc Roy Soc (London) A241: 455–479.
Jobert N (1960) Calcul de la dispersion des ondes de Love de grand periode a la surface de la terre. Ann Geophys 16: 393–413.
Keller JB, Rubinow SI, Goldstein M (1963) Zeros of Hankel functions and poles of scattering amplitudes. Jour Math Phys 4: 829–832.
Kennett BLN (1972) Seismic wave scattering by obstacles on interfaces. Geophys Jour Roy Astron Soc (London) 28: 249–266.
Landisman M, Satô Y, Nafe J (1965) Free vibrations of the earth and the properties of its deep interior regions. Part I. Density. Geophys Jour Roy Astron Soc (London) 9: 439–502.
Langer RE (1931) On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order. Trans Amer Math Soc 33: 23–64.
Langer RE (1937) On the connection formulas and the solutions of the wave equation. Phys Rev 51: 669–676.
Luneburg RK (1964) Mathematical Theory of Optics. University of California Press, Berkeley.
Nussenzveig HM (1965) High-frequency scattering by an impenetrable sphere. Ann Phys 34: 23–95.
Odaka T (1978) Derivation of asymptotic frequency equations in terms of ray and normal mode theory and some related problems. Jour Phys Earth 26: 105–121.
Pekeris CL (1965) Asymptotic theory of the free torsional oscillations of the earth. Proc Natl Acad Sci (US) 53: 1253–1261.
Richards PG (1973) Calculation of body waves, for caustics and tunnelling in core phases. Geophys Jour Roy Astron Soc (London) 35: 243–264.
Richards PG (1976) On the adequacy of plane-wave reflection/transmission coefficients in the analysis of seismic body waves. Bull Seismol Soc Amer 66: 701–717.
Sato R (1969) Body wave amplitude near shadow boundary: SH wave. Jour Phys Earth 17: 1–12.
Sato R, Lapwood ER (1977) The asymptotic distribution of torsional eigenfrequencies of a spherical shell. I. Jour Phys Earth 25: 257–282.
Satô Y (1961) Normal mode interpretation of the sound propagation in whispering galleries. Nature 189: 475–476.
Satô Y, Usami T, Landisman M, Ewing M (1963) Basic study of the oscillations of a sphere, Part V. Propagation of Torsional disturbances on a radially heterogeneous sphere. Case of a homogeneous mantle with a liquid core. Geophys Jour Roy Astron Soc (London) 8: 44–63.
Satô Y, Usami T, Landisman M (1968) Theoretical seismograms of torsional disturbances excited at a focus within a heterogeneous spherical earth—case of a Gutenberg— Bullen A’ earth model. Bull Seismol Soc Amer 58: 133–170.
Scholte JGJ (1956) On Seismic Waves in a Spherical Earth. Publ. 65, Roy. Nederland Meteorological Institute, The Hague, 55 pp.
Shimamura H, Sato R (1965) Model experiments on body waves—travel times, amplitudes, wave-forms and attentuation. Jour Phys Earth 13: 10–33.
Shimshoni M, Ben-Menahem A (1970) Computation of the divergence coefficient for seismic phases. Geophys Jour Roy Astron Soc (London) 21: 285–294.
Shimshoni M, Sylman Y, Ben-Menahem A (1973) Theoretical amplitudes of body waves from a dislocation source in the earth. II. Core phases. Phys Earth Planet Int 7: 59–91.
Sills LB (1978) Scattering of horizontally polarized shear waves by surface irregularities. Geophys Jour Roy Astron Soc (London) 54: 319–348.
Singh SJ, Ben-Menahem A (1969a). Decoupling of the vector wave equation of elasticity for radially heterogeneous media. Jour Acoust Soc Amer 46: 655–660.
Singh SJ, Ben-Menahem A (1969b) Asymptotic theory of body waves in a radially heterogeneous earth. Bull Seismol Soc Amer 59: 2039–2059.
Singh SJ, Ben-Menahem A, Shimshoni M (1972) Theoretical amplitudes of body waves from a dislocation source in the earth. I. Core reflections. Phys Earth Planet Int 5: 231–263.
Sommerfeld A (1964) Partial Differential Equations in Physics. Academic Press, New York, 335 pp.
Sommerfeld A (1964) Optics. Academic Press, New York, 383 pp.
Sylman Y, Shimshoni M, Ben-Menahem A (1974) Theoretical amplitudes of body waves from a dislocation source in the earth. III. P, S and surface reflections. Phys Earth Planet Int 8: 130–147.
Van der Pol B, Bremmer H (1937) The diffraction of electromagnetic waves from an electrical point-source round a finitely conducting sphere, with application to radiotelegraph, and the theory of the rainbow. Phil Mag (Ser 7) 24: 141–176, 825-864.
Vered M, Sylman Y, Ben-Menahem A (1974) Generalized reflection and transmission coefficients for seismic sources in a multi-layered spherical earth model. Pure Appl Geophys 112: 821–835.
Watson GN (1918) The diffraction of electric waves by the earth. Proc Roy Soc (London) A95: 83–99.
Woodhouse JH (1972) Diffraction by anomalous regions in the earth’s mantle. Geophys Jour Roy Astron Soc (London) 32: 295–324.
Yanovskaya TB (1958) The dispersion of Rayleigh waves in a spherical layer. Izv Geophys Ser 801-817 (English translation, 461-467).
Zoeppritz K, Geiger L, Gutenberg B (1912) Über Erdbeben wellen. V Nachr Kgl Ges Wiss Göttingen Math-Phys Kl, 121–206.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Ben-Menahem, A., Singh, S.J. (1981). Asymptotic Theory of the Earth’s Normal Modes. In: Seismic Waves and Sources. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5856-8_8
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5856-8_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-5858-2
Online ISBN: 978-1-4612-5856-8
eBook Packages: Springer Book Archive