Goal

Course listing

Short syllabus for W4945x (Chapters 1 to 11)

Short syllabus for W4946y (Chapters 12 to 19)

Text

• Chapter 1: Rotational Mechanics

  (used to introduce notation, and to give an example of a second-order Cartesian tensor)
  
  
  1. Angular velocity is a vector, although finite rotations are not.
  2. Suffix notation, Einstein summation convention. Kronecker delta function. Rotational energy, and its geometrical interpretation as a property of the momental ellipsoid, independent of any coordinate system.
  3. Angular momentum (an example of two vectors related by a tensor Š the tensor is in a sense defined by this relation), and its geometrical interpretation.
  4. Effect of applied torque (rate of change of momentum).
  5. Deduction that the momental ellipsoid is independent of the coordinate system used. So, a tensor has definition independent of any coordinate system.
  
  
  

• Chapter 2: Definition of vectors and tensors

  1. Discussion of vector and tensor components, and how these change when axes are rotated. The transformation rules become the definition.
  2. Discussion of the "quotient rule", i.e., the way in which a tensor relates two vectors.
  3. Notes on uses of the "substitution tensor," the "alternating tensor", and their various products.
  
  
  

• Chapter 3: Nutation of the Earth

  1. Distinguish from precession (which requires forcing).
  2. Rate of change of vector components in a rotating coordinate system.
  3. Euler equations.
  4. Solution of these, in absence of forcing torque, for an axially symmetric body.
  5. Facts (observed) on Chandler wobble.
  6. Larmor's explanation for the way that yielding (non-rigid behavior) changes the period.
  7. Summarize problems of the stimulation of the Chandler wobble.
  8. Review of Milankovich's ideas, the associated facts about Earth-orbital changes, and the underlying mechanics. Look toward Chapter 5.
  
  
  

• Chapter 4: Potential theory and integral equations of Gauss and Stokes

  1. Notes on Gauss's "divergence theorem" (true, component-by-component), and on Stokes' theorem.
  2. Physical meaning of grad, div, curl. (Gibbs' notation --- nabla.)
  3. Helmholtz representation of a vector.
  4. Notes on solid angle.
  
  
  

• Chapter 5: Poisson's equation, MacCullagh's formula; Precession

  1. Potential for the gravitational attraction between two point masses.
  2. Derive MacCullagh's formula for external potential, giving the approximate effect of departures from spherical symmetry, in terms of the moment of inertia.
  3. Precession and "nodding" each have two cycles per year, due to the Sun acting on the Earth's equatorial bulge; but precession has a non-zero mean.
  4. GaussÕs Theorem: If the closed surface S contains a total mass M (which may be a combination of point masses and continuous density distributions) (and there may be mass outside S) then the flux of the gravitational force out of S equals -4 x pi x (the universal gravitational constant) x M.
  5. Introduce a scalar potential for the gravittional force, and see that it satisfies Poisson's equation, i.e. the differential form of Gauss's theorem.
  
  
  

• Chapter 6: Generalized orthogonal curvilinear coordinates

  1. Notes on the general case. Spherical polars as an example.
  2. Scaling functions, to relate small increment (in just one coordinate) to small distance.
  3. Elementary derivation of operations grad, div, nabla².
  
  
  

• Chapter 7: Separation of Laplace's equation in spherical polars (r, theta, phi)

  1. Notes on the method of separation of variables.
  2. Reasons why the theta -dependence is (for axial symmetry) in the form of a polynomial in cos theta. Legendre polynomials.
  3. Associated Legendre functions. Reason why azimuthal order is never greater than angular order.
  4. Interior potential, exterior potential: expansion of (reciprocal distance) as a sum over Legendre polynomials.
  5. Rodrigues' formula.
  6. Demonstration that Legendre polynomials are the best polynomials for representing axially symmetric functions on the surface of a sphere, since they give equal weight to each element of area.
  7. Orthogonality. Concept of eigenvalue/eigenfunctions, in differential equations.
  8. Full normalized surface harmonics.
  9. Wavelength of Y_n^m, for large n.
  
  
  

• Chapter 8: Figure of the Earth, and Clairaut's Theory

  1. History of the subject: Newton, Cassini, French expeditions to Lapland and Peru.
  2. How flattening is measured. Geocentric and geographic latitudes.
  3. Best estimates of the Earth's flattening.
  4. First-order Clairaut theory, to obtain the relation between flattening, differences in principal moments of inertia, and the effect of rotation; gravity on the surface of a rotating spheroid.
  
  
  

• Chapter 9: Step-function and Delta-functions

  1. Revise concepts of finite and infinite Fourier series for functions defined on a range from -L/2 to L/2. Mention Gibbs' phenomenon (the series representation of a discontinuity, and how it doesn't nicely converge on the discontinuity as the series number goes up).
  2. Extend discrete sampling to a continuum, as L tends to infinity, obtaining a Fourier integral transform pair.
  3. Delta-functions as step-function derivatives.
  4. Property of a delta function as a weighting function.
  5. The defining properties of a delta - function. Extension to 2 and 3 dimensions.
  6. Delta functions defined over the surface of a sphere.
  7. Green's theorem, used to solve Poisson's equation.
  8. Interpretation of the (reciprocal distance) function as a solution of Laplace's equation --- except where distance equals zero.
  9. Three-dimensional delta function as illustrated by the density distribution of a point mass.
  10. Extension of Poisson's equation to this case. Green's functions.
  
  
  

• Chapter 10: Some ways of determining density distribution in the Earth

  1. Individually homogeneous mantle and core. Fit to mass and moment of inertia.
  2. The Adams-Williamson method: discuss pros and cons of this method. Derive relationships between a harmonic component of exterior potential, and component of density anomaly on an interior shell (this exercise has many uses of Chapters 7 and 9).
  3. Green's equivalent surface layer, for internal and external potentials. Show following:
     Imagine any mass distribution of finite extent, and enclose it with an equipotential surface. Then, to a point outside the surface, the potential might equally as well be caused by a certain density on the equipotential.
  
  
  

• Chapter 11 : Tides

  1. Descriptive discussion of the tidal-generating force, in terms of "revolution without rotation", following Darwin's approach.
  2. Obtain the formula for the potential of tidal generating force.
  3. Mention resonance phenomena, and the suggestion that deep-ocean tides are exactly out of phase with the tidal generating force.
  4. Response of the solid Earth (dropping the assumption of rigidity).
  5. Definition of two Love numbers: mention the third Love number.
  6. Laplace's separation into three different periods: 14 days, diurnal, semi-diurnal.
  
  
  

Second Semester: expanded syllabus for W4946y

• Chapter 12: Basic elasticity

  1. Definitions of elasticity, viscous behavior, plasticity, yield point.
  2. Analysis of stress and strain.
  3. Geometrical properties of quadric surfaces associated with stress and strain tensors (principal axes, physical significance of direction of normal, and length of radius).
  4. Planes of greatest shear stress, and their relationship to thrust faults/strike-slip faults (an example of the use of Lagrange multipliers).
  5. Stress-strain relations: specialization to linear isotropic elastic media.
  6. Examples of stress and strain fields.
  
  
  

• Chapter 13: Elastic body waves, and body waves in seismology

  1. Waves in general: the one-dimensional wave equation and its general solution.
  2. Obtaining a solution for waves spreading spherically from a point source.
  3. Derivation of the vector equation of elastic motion.
  4. Plane wave solutions propagating along x₁-axis: P (longitudinal); S (transverse).
  5. Terms "dilatational, distortional, equi-voluminal, solenoidal, irrotational".
  6. Nomenclature of body wave phases in the Earth.
  7. Describe seismograms and major features of velocity in Earth. Show how the travel-time slope is found in practice from arrival-time picks.
  8. Ray parameter.
  9. SV and SH component of the S-wave.
  10. Potentials for P and S waves in homogeneous material.
  11. Discuss the three-dimensional scalar wave equation, and plane wave solutions in any direction.
  12. Reflection of SH from a free horizontal surface. P-SV coupling.
  13. Ray theory. Rays not straight, in general. Ray parameter, travel-time triplications, use of arrays. Eikonal equation for travel-time. Geometrical spreading.
  14. Travel time and distance as functions of ray parameter.
  15. Weichert-Herglotz inversion.
  16. Summarize known features of velocity in the Earth.
  
  
  

• Chapter 14: Seismic surface waves

  1. The defining property of surface waves. Inhomogeneous waves.
  2. Coupled P-SV in a half-space with a free surface -- the Rayleigh wave.
  3. A half-space with a surface layer -- existence of Love waves, always dispersed.
  4. Particle motion in surface waves.
  5. Group velocity, stationary phase approximation. Obtaining dispersion from seismograms.
  6. Features of dispersion curves: differences between oceanic/continental paths.
  
  
  

• Chapter 15: General methods of studying a scalar wave equation

  1. Separation of variables (4, in general, for space and time), leading to the Helmholtz equation, after separation of time.
  2. In cartesians, separation gives plane waves.
  3. In cylindrical polars, find Bessel functions and a simple dependence on azimuth & depth. Brief summary of Bessel/Hankel function properties.
  4. In spherical polars, find Legendre functions again, and spherical Bessel functions.
  5. Free oscillations of a fluid sphere. Extend qualitatively to free oscillations of the Earth.
  
  
  

• Chapter 16: Seismometry

  1. Uses of pendulums, and different ways to increase the period. "Zero-length spring."
  2. Vertical seismometer/gravimeter. Horizontal seismometer (hanging gate).
  3. Modern "feedback" systems as a basis for broadband response.
  4. Bits and bytes and decibels.
  
  
  

• Chapter 17: Earthquake source theory

  1. Reid's theory of elastic rebound.
  2. P-wave radiation pattern for a shear dislocation.
  3. "Fault-plane solutions", and how obtained from observations.
  4. Double couple sources, to model slip on a fault. S-wave radiation pattern.
  5. Differences in the seismograms generated by bombs and earthquakes.
  6. Seismic monitoring of test ban treaties. Yield estimation, discrimination, and a little geopolitics.
  7. Dilatancy?? The possible role of pressurized fluids in triggering earthquakes.
  
  
  

• Chapter 18: Heat Flow

  1. Description of surface measurements of heat flow, and the reason for knowing that heat sources must lie near the Earth's surface.
  2. Heat flux vector, thermal conductivity, derivation of diffusion equation for temperature.
  3. Discussion of observation (within one heat flow province) that surface radioactivity is linear with heat flow. Meaning of slope and intercept.
  4. Solution of the diffusion equation, for dyke injection, by integral transforms.
  5. Fundamental differences between diffusing properties and wave-propagating properties.
  6. Discussion of continental and oceanic observations, and implications for mantle.
  
  
  

• Chapter 19: Water Waves

  1. Kinematics of flow fields: concept of the "material derivative".
  2. Continuity. Simplification for incompressible flow.
  3. Equation of motion.
  4. Kelvin's circulation theorem, for incompressible inviscid flow.
  5. Velocity potential, and its Laplace equation for irrotational flow.
  6. Surface and bottom boundary conditions for water waves.
  7. Small amplitude water waves, and approximations for shallow water/deep water waves (linear theory).
  8. In what ways do non-linearities develop?
  [back to the first-semester syllabus]
  [back to the beginning of the second-semester syllabus]
  [back to the top]