Date: Tuesday, November 26, 2013
Location: 3096 East Hall (3:10 PM to 4:00 PM)
Abstract: Many PDEs that describe physical systems admit wave-like phenomena. A system is said to be dispersive if different wave modes travel at different speeds. That is, what might initially be a coherent wave packet will “disperse”, or separate, to form an oscillatory wave-train. Solutions of this type are ubiquitous in nature and are readily observed in optics, structural beams, and water waves, just to name a few examples.
This presentation is intended to be an introduction to the theory of linear dispersive waves. Most of the talk will be spent working with the simple, yet powerful, ideas of phase and group velocity. I will show how they arise from simple kinematic considerations and more generally from simple asymptotics. I will also talk about some of the conserved quantities that arise in dispersive systems. I will finish by showing how the results obtained in our discussion give us the correct qualitative framework to study the propagation of dispersive waves in nature. In particular, I will use simple results from the theories of capillary, deep water, and shallow water waves to provide physical examples (with movies). This leads to some surprising results for those of us who are used to non-dispersive waves. The talk is intended to provide mathematical underpinnings for physical intuition, so it should be accessible to all graduate students.
Speaker: Alfredo Wetzel
Institution: University of Michigan