MA 5125 Mathematical Theory of Waves
General solution of wave equation in R, energy methods, waves with a source, Duhamel’s Principle, reflections of waves, wave equation on an interval: separation of variables, wave equation in higher dimensions, Shock wave: weak solution, entropy condition and Riemann’s problem, shock structure.Sound waves: acoustic energy, simple source, acoustic dipole, and source regions in different fields, ripple, scattering, and radiation from various bodies. One-dimensional waves in fluids: theory of longitudinal waves, transmission of waves, and propagation through branching systems. Water waves: surface gravity waves, internal waves, sinusoidal waves on deep water, ripples, group velocity, wave patterns, Fourier analysis of dispersive systems.
1. Evans, Partial differential equations, AMS, 1997,
2. Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, 1973,
3. Waves in fluids, James Lighthill, Cambridge, 2001,
4. Linear and nonlinear waves, Whitham, Wiley-Interscience, 1999,
1. Geophysical Fluid Dynamic, Joseph Pedlosky, Springer, 1987,
2. Internal Gravity waves, B. R. Sutherland, Cambridge University Press, 2010,