UNDERGRADUATE RESEARCH IN APPLIED MATHEMATICS
SUMMER REU AT MISSISSIPPI STATE UNIVERSITY
June 9-August 7, 2004
A mathematical model for wave propagation in nonhomogeneous, elastic media is described by the plasma-wave equation
where u(x,t) denotes the wave amplitude at location x at time t. Here, V(x) denotes the nonhomogeneity, i.e. the density of the elastic force the medium exerts on the propagating wave, and c denotes the (constant) speed of propagation. A plane wavefront sent onto the nonhomogeneity will be partly reflected back and partly transmitted in the forward direction. In other words, as the propagating wave interacts with the nonhomogeneity, it starts gaining a tail trailing the wavefront.
The direct scattering problem consists of determining the reflection and transmission when V(x) and c are known. On the other hand, the inverse problem consists of the determination of V(x) and c when the reflection is known. A more challenging problem is to consider the above model by replacing c with c(x), where now the propagation speed varies from location to location. The problem becomes even more challenging when c(x) changes abruptly at certain locations corresponding to discontinuities in the nonhomogeneous medium.
Now consider all possible incident waves other than a plane wave and ask the following question: Can we choose or prepare an incident wave (consisting of a wavefront plus a certain tail) such that its tail, at a desired instant in time, completely vanishes as a result of the interaction with the nonhomogeneity? When this happens the wave is said to focus at the wavefront. Wave focusing can be used to determine the nonhomogeneity at the focusing location in a remote fashion, i.e., by performing a measurement on the incident wave at any chosen moment in time before the wave enters the nonhomogeneous medium. This provides an alternative approach to solve the inverse scattering problem. In the conventional setting, a wavefront is sent onto the nonhomogeneous medium and the nonhomogeneity is recovered via a measurement performed after the wave interacts with the medium, whereas, by using wave focusing, one can try to fine-tune the tail in the initial wave before the wave enters the medium so that the wave focuses at a desired location in the medium; then, it is possible to recover the nonhomogeneity of the medium at the focusing location via a measurement prior to the entrance of the wave into the nonhomogeneity. There are some other interesting aspects of wave focusing. For example, by choosing the focusing location behind a compactly supported nonhomogeneity, it is possible to separate the tail from the wavefront spatially; i.e., the nonhomogeneity causes the creation of a spatial gap between the wavefront and the tail.
There are various projects for the undergraduate students with the following aims:
understanding the scattering of a plane wave from the nonhomogeneity V(x) in the plasma-wave equation
mathematical description of reflection and transmission
obtaining focusing solutions to the plasma-wave equation analytically and numerically
using Mathematica and Matlab to simulate wave focusing for various nonhomogeneities
understanding solutions of the plasma-wave equation and representing them as Fourier integrals of certain solutions of the time-independent Schrödinger equation d2y/dx2+k2y=V(x)y of quantum mechanics
recovering the nonhomogeneity V(x) at desired locations via measurements on the incident wave prior to its interaction with the nonhomogeneity
understanding and analysis of the spatial gap between the wavefront and the tail
analyzing similar problems when the propagation speed c in the plasma-wave equation depends on x and perhaps contains discontinuities