I currently offer two undergraduate projects in Applied Mathematics, both of which are suitable for Math399, Math499, or Math490. A summary of the two projects is given below. I will not normally supervise more than two projects each academic year, so it's best to get in touch early if you'd like to do a project with me. For more information on undergraduate projects, please see this page.
In this project we will study the propagation of mechanical waves in elastic solids from first principles. Elastic waves in solids posses many interesting features that are not present in electromagnetism; such waves play a pivotal role in a range of fields from non-desructive testing, to seismology, and metamaterials.
We will begin by studying waves in simple one-dimensional structures such as strings, rods, and beams. We will derive the equations of motion from first principles and introduce the necessary elementary concepts of wave propagation. We will then move on to two-dimensional problems such as membranes and shells covering infinite, semi-infinite, and finite bodies.
Finally, we will consider wave propagation in three-dimensional elastic bodies. Time permitting, problems of scattering, diffraction, reflection and transmission will be studied, as well as external loading (e.g. line and point loads on elastic half-spaces).
Students may find the content of the following modules useful: Math224, Math225, Math243, Math323, Math324, Math427.
Graff, K. F. (1975). Wave motion in elastic solids. Clarendon Press.
The study of wave propagation in periodic media underpins cutting edge research in a wide variety of fields including photonics, phononics, plasmonics, metamaterials, geophysics, solid-state physics, and non-desructive testing.
In this project we will examine various aspects of of wave propagation in periodic media, beginning with elementary one-dimensional problems. We will introduce Bloch-Floquet theory along with the concepts of Bloch waves, reciprocal space, Brillouin zones, group and phase velocity, dispersion, band structure, and resonances.
Initially, we will consider wave propagation in infinite domains. Time permitting, we will examine more advanced problems such as transmission problems for semi-infinite domains, higher dimensional systems, and multi-phase systems.
Students may find the content of the following modules useful, but not essential: Math224, Math323, Math324, and Math427. Some experience with MATLAB would also be useful.
Joannopoulos, J. D., Johnson, S. G., Winn, J. N., & Meade, R. D. (2011). Photonic crystals: molding the flow of light. Princeton University Press.
Brillouin, L. (1953). Wave propagation in periodic structures: electric filters and crystal lattices, 2nd Edn. Dover Publications.