Normal Mode Theory and Free Oscillation of the Earth

Contents

    1. Introduction
    2. Vibration of 1D media
    3. Normal mode theory
    4. Vibration of 2D and 3D media
    5. Free oscillation of an elastic sphere
    6. SNREI Earth model with self-gravity
    7. Normal mode seismogram
    8. Realistic Earth models
    9. Recent developments

Schedule

  1. organization meeting, overview of the course.
  2. a brief history of theoretical seismology and seismic observation; Earth's interior structure.
  3. wave EQ in 1-D media, general solution, variable separation method.
  4. boundary conditions, fixed end string, eigenmodes and eigenfrequencies, fundamental mode and overtones, traveling wave.
  5. a fixed string initially with a impulse shape, reflection; attenuation.
  6. normal mode theory, operators and abstract vector space, symmetric and Hermitian operators.
  7. normal mode theory, positive definite and bounded below operator, orthogonality and completeness.
  8. normal mode summation.
  9. vibration of a rectangular membrane, degeneration
  10. vibration of a circular membrane, cylindrical coordinate, Bessel's function, mode shapes as a function of depth for different n and m.
  11. vibration of a circular membrane, horizontal wavenumber, dispersion diagram, phase velocity and group velocities, surface wave.
  12. vibration of a liquid sphere, Legendre function, spherical harmonics, spherical Bessel's function.
  13. equation of motion in elastic media, wave equations, P and S waves
  14. vibration of a homogeneous elastic sphere, surface gradient, vector spherical harmonics
  15. spheroidal and toroidal fields for displacement and traction, separated equations
  16. toroidal modes of a homogeneous sphere, l>0, rigid rotation for l=1 and n=0. 0T2 gravest mode
  17. toroidal modes: mode shape for different l and n, Love wave, deep penetrating S wave
  18. spheroidal modes of a homogeneous sphere, radial mode (l=0), 0S0; rigid translation of 0S1; 0S2 gravest mode
  19. sphere with a liquid core, eigenfrequency equation for toroidal mode, dispersion diagram, comparison with uniform sphere, S vs. ScS, mode shapes
  20. sphere with a liquid core, spheroidal modes, dispersion diagram, PKP modes, Stoneley mode, mode shapes.
  21. SNREI with self-gravity: continuum mechanics, Lagrangian and Eulerian descriptions, material time derivatives; conservation of mass.
  22. conservation of linear momentum; gravity and gravitational potential; linearized equations of motions.
  23. free oscillation of a uniform sphere with self-gravity, effect of gravity; numerical integration for realistic models, PREM, slichter mode 1S1.
  24. atmospherical modes. tsunami modes.
  25. source representation, stress glut, earthquake fault, moment tensor
  26. moment tensor for an ideal fault, double couple, decomposition of a general moment tensor, CLVD
  27. synthetic seismogram by normal mode summation.
  28. effect of anelasticity, attenuation
  29. effect of rotation, Corriollis force, ellipticity
  30. effect of rotation, perturbation theory, splitting
  31. effect of lateral heterogeneities, variational method
  32. direct solution method.
  33. inversion for earth structure using normal mode data