Nonlinear Hyperbolic Waves

University Of California-Davis, Davis CA

Investigators

Abstract

Waves are a phenomenon of fundamental physical and technological importance. For example, the observation of waves from distant sources (whether gravitational waves generated by the collision of black holes or sound waves generated by a submarine) allows one to make useful conclusions about the nature of those sources, and the manipulation of waves has many technological applications (such as the use of ultrasonic surface acoustic waves in signal processing devices). <br><br> Nonlinear hyperbolic waves are an especially interesting and important class of waves. They include the shock waves generated by aircraft in transonic or supersonic flight, the seismic waves (and possible Tsunamis) generated by earthquakes, and the gravitational waves predicted by Einstein's general theory of relativity. <br><br> The dominant properties of hyperbolic waves are described mathematically by hyperbolic systems of partial differential equations. In a first approximation, hyperbolic waves are non-dispersive, meaning that the velocity at which they propagate, and their other properties, do not depend on the wavelength of the waves. This contrasts with dispersive waves, such as light waves in an optical fiber or Rossby waves in the atmosphere. The effects of nonlinearity on hyperbolic waves have a unique character; for example, the formation of singularities, such as shock waves, is a ubiquitous phenomenon, and one with significant physical implications. <br><br> The goal of the proposed research is to study nonlinear hyperbolic waves from a unified mathematical perspective in close connection with various physical applications. Specific topics include the reflection of shock waves and related questions in transonic fluid flows; the propagation of nonlinear hyperbolic surface waves, including Rayleigh (or surface acoustic) waves in elasticity; and the effect of nonlinearity on gravitational waves in general relativity.