Applied Hyperbolic Partial Differential Equations

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract: 0405899, J. Rauch, University of Michigan Applied Hyperbolic Partial Differential Equations The principal investigator will investigate mathematical problems involving hyperbolic partial differential equations. These equations appear in the description of waves which propagate with finite speed, for example acoustics, electromagnetism, fluid and aero dynamics, plasmas, lasers, telegraphy, radio, television, cell phones, shock waves, music, power lines, .... etc. Typical mathematical problems involve showing that specific problems, motivated by applications, have solutions and studying their qualitative properties, for example stability and methods for effective computation. A first set of problems in the proposal concerns the damping of acoustic waves by enclosing the acoustic cavity in a diffusive medium. Two situations easy to understand are the protection of pilots and passengers in airplanes from the noise of the engines and the absorption of sound in a submarine so that it will not escape to the sea. There are also applications to the automatic control of mechanical systems immersed in fluids. A second problem concerns the scattering of solutions of a strongly nonlinearly damped wave equation. A third problem is to show that the phenomenon of continuum generation in focused laser pulses is a consequence of nonlinear electromagnetic models without ionization. Self focused laser beams exhibit continuum generation. The band of frequencies present in the focused beam is much broader and spans a continuum of frequencies (in contrast to a discrete set) than before the focusing. Current explanations are frankly not very convincing. This is a fundamental problem long recognized in the laser physics community. It is a place where mathematics may add important insights. Focused beams are used in many situations including a surprising application to reduce resistance in hypersonic aircraft. The proposer will study the behavior of solutions of the differential equations that govern the propagation of waves. One problem investigated is how well sound can be excluded from a region by enclosing the region is an absorbing layer. Here the absorption in the layer is supposed to be of a diffusive nature, like one would have for a gel. Applications include sonic insulation of airplane fuselages and submarines and the control of immersed mechanical systems. A second problem to be studied is that of white light generation in focused lasers. When high power lasers focus, their color changes from the color of the carrier frequency to nearly white light. There are a variety of explanations of this, none very convincing. The PI will try to find a better one. There is convincing experimental evidence that the phenomenon does not require ionization of the propagation medium and this will help to keep the equations under study manageable and well founded.