Independent Study: Fall 2003 Wave Equations
The course will cover wave equations in physics and geophysics and their associated boundary-value problems. The course will have a lecture component with tests, and a computer-programming component with graded projects.
Rough Outline:
1. One dimensional wave equation:
A. Waves on a string
B. Derivation of the wave equation for 1-D string
C. Boundary conditions
D. Fourier series solutions
E. Properties of Fourier series
F. Numerical Fourier series (computer project)
2. Fourier analysis of Waves and spectral density methods:
A. Real transforms and BVP’s
B. Complex numbers and Euler’s identity
C. Complex Fourier transform and inverse transforms
D. Power spectra
E. Correlation Functions
F. Fast Fourier Transforms
3. Numerical Fourier transforms and Spectra:
A. Slow Fourier Transforms
B. Fast Fourier Transforms
C. Numerical Transforms and Correlations (computer project)
4. Two and Three dimensional wave equations:
A. Waves on a membrane
B. Separation of variables
C. Boundary Conditions
D. Cylindrical and spherical coordinates
E. Separation of variables and orthogonal functions
F. Numerical Solutions of the wave equation for a circular membrane (computer project)
5. Wave equations in physics, engineering, and oceanography
A. Elasticity
B. Longitudinal and transverse oscillations of a beam
C. Acoustic waves
C. Tidal Waves in the Ocean
D. Numerical project on seismic or ocean wave data (computer project)
Texts: Physics of Waves, Elmore and Heard
Fourier Series and Boundary Value Problems, either J. Hanna or R. Churchill
Numerical Analysis Burden, Faires, and Reynolds
(I may want to use the math methods book the school has as well, but 90% of my material is covered in these texts)