Nonlinear Wave Propagation

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Abstract, Proposal 0309648, Hunter, University of California-DavisTitle: Nonlinear wave propagationThis project addresses fundamental and poorly understood issues in thetheory of hyperbolic conservation laws,such as the ones that modelcompressible fluid mechanics. It also addresses problems inincompressible fluid mechanics and general relativity, and applies theresults to a variety of important physical problems. A common theme ofmuch of the proposed work is how nonlinearity leads to the formationand propagation of various kinds of singularities, such as shock wavesin compressible fluids, or space-time singularities in generalrelativity. Specific topics proposed for study are the following:shock wave reflection; shock wave propagation in random media;nonlinear hyperbolic surface waves in elasticity andmagnetohydrodynamics; the Prandtl boundary layer equations andvorticity wave, mean flow interactions in incompressible fluidmechanics; and the effect of nonlinearity on gravitational waves ingeneral relativity.Waves (such as sound waves, seismic waves, light waves, quantummechanical waves,or gravitational waves) are a central feature of manyphysical systems, from the atomic to the cosmological scales. At lowerintensities, waves superpose and they may be described by lineartheories, but at higher intensities nonlinear effects becomessignificant, leading to qualitatively new phenomena (like shock wavesand solitons). Nonlinear waves are much more difficult to analysemathematically than linear waves, and are often of great practicalimportance. For example, shock waves form around aircraft that travelclose to the speed of sound. One aim of this project is to studybasic, but not fully understood, shock wave phenomena, such as theirreflection from walls. Among other topics, the project also includes astudy of nonlinear surface waves, such as the Rayleigh waves which aregenerated by earthquakes and are used in a variety of ultrasonicdevices, and a study of the formation and propagation of space-timesingularities in gravitational waves.