A selection of potential PhD, MSc, and undergraduate projects are summarised below. However, I am always happy to discuss the possibility of developing novel projects with potential students.

PhD & MSc Projects

I am always interested in recruiting excellent PhD & MSc students. Brief outlines of a couple of potential research projects can be found here. However, this list is not exhaustive and I am happy discuss the possibility of developing bespoke projects with potential students. If you are interested in reading for a PhD or MSc under my supervision, please get in touch.

Surface wave control

This project involves the mathematical modelling and design of mechanical structures capable of controlling the propagation of surface and bulk waves in elastic solids. The project will employ both numerical and asymptotic analysis to study a combination of continuous and discrete structures in one-, two-, and three-dimensions. The research programme includes scattering, homogenisation, and spectral problems for finite and infinite systems and has a broad range of applications including, filtering of waves, lensing, and cloaking.

Further reading

Skelton EA, Craster RV, Colombi A, Colquitt DJ. 2018. New J Phys, 20, 053017. (doi: 10.1088/1367-2630/aabecf)

Carta G, Colquitt DJ, Movchan NV, Movchan AB, Jones IS. 2020. J Mech Phys Solids, 137, 103866 (doi: 10.1016/j.jmps.2020.103866)

Colombi A, Colquitt, DJ, Roux P, Guenneau S, Craster RV. 2016. Sci Rep, 6, 27717. (doi: 10.1038/srep27717)

Colquitt DJ, Colombi A, Craster RV, Roux P, Guenneau SRL. 2017. J Mech Phys Solids, 99, 379-393. (doi: 10.1016/j.jmps.2016.12.004)

Skelton EA, Craster RV, Colombi A, Colquitt DJ. 2018. New J Phys, 20, 053017. (doi: 10.1088/1367-2630/aabecf)

Carta G, Colquitt DJ, Movchan NV, Movchan AB, Jones IS. 2020. J Mech Phys Solids, 137, 103866 (doi: 10.1016/j.jmps.2020.103866)

Colombi A, Colquitt, DJ, Roux P, Guenneau S, Craster RV. 2016. Sci Rep, 6, 27717. (doi: 10.1038/srep27717)

Colquitt DJ, Colombi A, Craster RV, Roux P, Guenneau SRL. 2017. J Mech Phys Solids, 99, 379-393. (doi: 10.1016/j.jmps.2016.12.004)

discrete defects

This project involves the analysis of finite and semi-infinite defects in elastic lattice systems. These defects may be dislocations or variations in inertial properties and, for finite defects, will have an associated spectrum of eigenstates. The focus of the research programme is on the analysis of these eigenstates and the fields in the vicinity of the defect sites; algorithms will be developed to study the solutions in various asymptotic regimes. The research programme will involve both analytical and numerical models. The project may also include the study of edge and interfacial waves in mechanical lattices.

Further reading

Colquitt DJ, Nieves MJ, Jones IS, Movchan AB, Movchan NV. 2013. Proc. R Soc. A 469: 20120579. (doi: 10.1098/rspa.2012.0579)

Colquitt DJ, Nieves MJ, Jones IS, Movchan NV, Movchan AB. 2012. Int J Eng Sci 61, 129–141. (doi: 10.1016/j.ijengsci.2012.06.016)

Madine KH, Colquitt DJ. 2021. Q J Mech Appl Math, hbab006. (doi: 10.1093/qjmam/hbab006)

Colquitt DJ, Nieves MJ, Jones IS, Movchan AB, Movchan NV. 2013. Proc. R Soc. A 469: 20120579. (doi: 10.1098/rspa.2012.0579)

Colquitt DJ, Nieves MJ, Jones IS, Movchan NV, Movchan AB. 2012. Int J Eng Sci 61, 129–141. (doi: 10.1016/j.ijengsci.2012.06.016)

Madine KH, Colquitt DJ. 2021. Q J Mech Appl Math, hbab006. (doi: 10.1093/qjmam/hbab006)

undergraduate projects

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.

elastodynamics

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.

Further reading

Graff, K. F. (1975). Wave motion in elastic solids. Clarendon Press.

Graff, K. F. (1975). Wave motion in elastic solids. Clarendon Press.

phoXonics

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.

Further reading:

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.

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.

current and previous students

Katie Madine

PhD (2018- )

Invisible Visibility Cloaks: Analysing the scattering and transmission properties of regularised cloaks

PhD (2018- )

Invisible Visibility Cloaks: Analysing the scattering and transmission properties of regularised cloaks

Rawabi Shujaa A Aljoufi

MSc (2020-2021)

Analysis of dynamic Green’s functions of discrete flexural systems

MSc (2020-2021)

Analysis of dynamic Green’s functions of discrete flexural systems

Georgia Hayes

MMATH (2020-21)

A Direct Method to Extract the Dispersion Curve for Scattering Waves from a Grating

MMATH (2020-21)

A Direct Method to Extract the Dispersion Curve for Scattering Waves from a Grating

Samuel Wilson

BSc (2020-2021)

Elastodynamics: The effects of fluid-loading on thin elastic plates

BSc (2020-2021)

Elastodynamics: The effects of fluid-loading on thin elastic plates

Valentina-Marina Bucur

MMATH (2019-20)

Acoustic Rayleigh-Bloch Waves

MMATH (2019-20)

Acoustic Rayleigh-Bloch Waves

Jared Jorgenson

MSc (2018—2020)

Mechanical metasurfaces

MSc (2018—2020)

Mechanical metasurfaces

Michael Ransom

MSc (2017-2018)

Multiple scattering of acoustic waves in two-dimensional isotropic cylindrical media

MSc (2017-2018)

Multiple scattering of acoustic waves in two-dimensional isotropic cylindrical media

Rachel Staples

MSc (2017-2018)

Cracks in lattice structures

MSc (2017-2018)

Cracks in lattice structures

Yifan Hu

MSc (2016-2017)

Multiple scattering by two cylinders

MSc (2016-2017)

Multiple scattering by two cylinders