TAM 518: Wave Motion

Class Description:

Linear waves in one-dimensional homogeneous and inhomogeneous media (both solids and fluids), linear elastic waves in a homogeneous halfspace, scalar waves in a layer and in a layered halfspace, nonlinear diffusive waves, nonlinear dispersive waves, and the inverse scattering transform. Prerequisite: TAM 541 or MATH 556 or equivalent; one of TAM 514, TAM 531, TAM 551 4 graduate hours.

Karl Graff, Wave Motion in Elastic Solids, Dover Publications.(recommended)

G. B. Whitman, Linear and Nonlinear Waves, Wiley. (recommended)


Part I - Linear Waves

The flexible string and related problems(6 hr)
Waves on a string, free oscillations, traveling and standing waves, normal modes, forced motion, use of Fourier and Laplace transforms; the effect of local and extended inhomogeneities, WKBJ methods and/or periodic systems, acoustic waves

Elastic waves in a halfspace(14 hr)
Plane waves; kinematics, reflection and refraction, critical angle effects, Rayleigh surface waves; buried harmonic source, angular spectra, radiation conditions, an analysis of the spatial complex plane; buried transient source, Cagniard-Hoop technique, its relation to the steepest-descent approximation

Linear dispersive waves(12 hr)
Free harmonic waves; direct approach, superposition of partial waves, dispersion, energy propagation and group velocity, leakage; normal-mode solution, role of dispersion, transient solution, stationary phase approximation, Poisson sum formula, ray description; instabilities, stationary waves in fluid dynamics

Part II - Nonlinear Waves

Nonlinear diffusive waves(12 hr)
Kinematic wave equation. Characteristics, wave breaking, shock waves, conservation laws, shock-fitting; the Burgers equation; Cole-Hopf transformation; nonlinear plane waves; Riemann's method, simple waves

Nonlinear dispersive waves and the inverse scattering transform(6 hr)
Solitons and their interactions; KdV equation; the square-well and Dirac-delta potentials, the sech2 potential; the inverse scattering transform; discrete spectra, continuous spectra, Gelfand-Levitan-Marchenko equation; applications of the inverse scattering transform; reflectionless potentials, potentials with reflections

Examinations (2 hr)


All Courses