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IMA Lectures

Ray Theory for the Elastic Wave Equation

Ray Theory for the Elastic Wave Equation

March 4-6, 2002

Mathematics in Geosciences, September 2001 - June 2002

* Robert Burridge*Earth Resources
Laboratory

Massachusetts Institute of Technology (MIT)

burridge@erl.mit.edu

Slides: pdf

Syllabus

Introduction.

Ray theory for the scalar wave equation and scalar Helmholtz equation

- The ray theory ansatz, amplitude and phase, slowness and travel time
- The eikonal equation
- The transport equation(s)
- The ray-tube-area method to obtain the amplitudes
- Caustics

Dynamic ray theory

- The second derivatives of the phase function
- The first variation of the ray equations
- The transverse components of the Hessian of the phase
- The dynamic ray equation and the simultaneous solution of the ray and dynamic ray equations
- The solution of the transport equation using the results of dynamic ray theory

The elastic wave equation

- Isotropy
- Isotropy: P and S waves
- The transport of the S polarization
- Anisotropic ray theory

Ray theory for a symmetric hyperbolic system

- Group velocity: various properties

The aim for the course is to be self contained at the expense
of being formal. For instance, methods of calculating coefficients
in an asymptotic series will be treated without proving that
the series is indeed asymptotic.

Prerequisites: The material
will be strictly classical. The audience is expected to be familiar
with the solution of first order PDE's by the method of characteristics,
cartesian tensors, vector and dyadic notation, and the theory
of the eigenvalues and eigenvectors of symmetric matrices.

Name | Department | Affiliation |
---|---|---|

Santiago Betelu | Mathematics | University of North Texas |

Robert Burridge | Earth Resources Laboratory | Massachusetts Institute of Technology |

Jamylle Carter | Institute for Mathematics & its Applications | |

Christine Cheng | Institute for Mathematics & its Applications | University of Minnesota |

Dacian Daescu | University of Minnesota | Institute for Mathematics and its Applications |

Gregory S. Duane | University of Minnesota | Institute for Mathematics and its Applications |

Michael Efroimsky | University of Minnesota | Institute for Mathematics and its Applications |

Selim Esedoglu | Institute for Mathematics & its Applications | |

Daniel Kern | ||

Anna Mazzucato | Mathematics | Yale University |

Aurelia Minut | University of Minnesota | Institute for Mathematics and its Applications |

M. Yvonne Ou | University of Minnesota | Institute for Mathematics and its Applications |

Jianliang Qian | Institute for Mathematics & its Applications | |

Toshio Yoshikawa | University of Minnesota | Institute for Mathematics and its Applications |

Slides: pdf

Mathematics in Geosciences, September 2001 - June 2002