Wave Propagation
List of researchers in CAMS working on problems related to Wave Propagation:
Booty,
Elmer,
Goodman,
Jiang,
Kriegsmann,
Michalopoulou,
Miura,
Moore,
Petropoulos,
Wang.
The analysis of wave propagation has a long and storied tradition in the
history of applied mathematics, and the exploration of wave behavior
has been a source of countless problems that have changed our
understanding of acoustics, hydrodynamics, electromagnetics, optics,
and even matter itself. These studies have also led to the
development of powerful new mathematical and computational techniques,
which have on occasion revolutionized entire fields of study. Several
members of the CAMS faculty have research interests in the area of
wave propagation; the following is a brief overview of the field and
of their particular interests.
For obvious reasons, water waves have been studied the longest, and
are still regarded as the point of reference for wave phenomena in
other fields. George Stokes' notoriously intractable equations
describing the motion of water waves were rendered far more accessible
by the various small-amplitude limits considered by Joseph Boussinesq,
D. J. Korteweg, and Gustav de Vries. Their explorations laid the
groundwork for a discovery that would prove to have far-reaching
consequences in several fields: the soliton,
a solitary wave with special self-preserving properties.
This exotic ``nonlinear mode'' propagates without spreading due to
the competing influences beween nonlinearity and dispersion. Even
more important than the solitons themselves is the structure that
makes their existence possible. Their study and the study of
equations that support them now fall generally under the heading of
``integrable systems'', and have given rise to such mathematical tools
as the inverse scattering transform.
One field that has been affected very profoundly by the relatively new
science of nonlinear waves is optical communications. Pulse-like
waveforms that maintain their shape for long times and over great
distances are of obvious interest to an industry seeking to ensure
error-free transmission of digital information. Every environment is
subject to some form of noise, whether it be thermal noise, electronic
noise, or quantum noise, so these pulses must also be tested for their
resistance to external influences. Richard Moore is currently
using perturbation theory and statistical techniques to develop
efficient ways to characterize the effect of perturbations on solitons
used for optical communications. The same nonlinear and dispersive
properties that give rise to solitons can be manipulated to condition
light for use in novel devices that will ultimately replace
the electronics upon which telecommunications and computing still
depend. Roy Goodman uses Hamiltonian mechanics and asymptotic
methods to explore how light can slowed, delayed
or ``trapped'' by engineering defects into nonlinear periodic
structures.
The simple cylindrical geometry of an optical fiber lends itself to
analytical treatment of the electromagnetic wave propagating inside of
it; however, the vast majority of electromagnetic scattering problems
have far more complexity due to complicated geometries and inhomogeneous
material properties with disparate spatial scales. The treatment of
transient electromagnetic signals such as those arising in signal
analysis, spectroscopic applications, and the nondestructive testing
of structures requires sophisticated numerical techniques that are
stable, fast, and accurate, and that have reasonable memory
requirements. Peter Petropoulos is conducting research on a
variety of approaches that address these restrictions, including
high-order finite difference schemes, boundary integral methods, and
perfectly matched layers. Shidong Jiang investigates nonreflecting boundary
conditions and scattering problems for acoustic and electromagnetic waves by
open surfaces. He employs fast algorithms, including the fast multipole method,
iterative solvers, and integral equation formulation of boundary value problems
for such problems and for related large-scale problems in physics and
engineering. Sheldon Wang has been working on problems involving wave
propagation in moving materials.
Even in cases where deterministic wave propagation is relatively well
understood, the related inverse problem is far more
challenging. The identification of
certain characteristics of a source of
acoustic waves, such as its location and intensity, through the
analysis of information gathered by receivers placed strategically or
at random within the same medium, is of obvious use in national
defense, in environmental studies, in seismology, etc. Zoi-Heleni
Michalopoulou has developed a localization-deconvolution approach
based on Gibbs sampling that explores the space of allowable
configurations with improved speed and accuracy over conventional
approaches.
Finally, the propagation of waves through materials is often
influenced by parameters that depend on the waves in a way
that requires fundamentally different physics. The microwave heating
of ceramics or the passage of optical fields through
photorefractive crystals, for instance, couples hyperbolic
equations to parabolic equations governing the evolution of thermal
profiles and chemical species. In optics, this can lead to the generation
of self-guided optical beams and, given the difference in time scales
dominating the hyperbolic and parabolic behaviors, bistability. In
the case of microwave heating of ceramics, it can lead to the
formation of weak spots that compromise the quality of the material.
Gregory Kriegsmann and Richard Moore are investigation asymptotic
and numerical methods to treat such coupled hyperbolic-parabolic systems.
The rest of this page contains links to examples of the research projects
that have been recently considered by the CAMS members. The links to individual
faculty web pages that contain more information can be found at the top of
this page.