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Department of Mathematics

Waves and metamaterials

Waves are an important aspect of physics and applied mathematics because they are the fundamental mechanism by which all information is transmitted. Metamaterials are complex media that manipulate waves in ways that are deemed unphysical.

Research into wave phenomena at Manchester has a very strong history. Horace Lamb (Beyer Professor 1888-1920) conducted research into waves in elastic media (among other aspects) including surface waves, the Rayleigh-Lamb waves in elastic plates that now carry his name and the theory of sound. Sir James Lighthill was Beyer Professor from 1950-1959 and is known, among other work, for his theory of aeroacoustics and nonlinear acoustics. Together with Gerald Whitham (who obtained his PhD at Manchester in 1953 under the direction of Lighthill) Lighthill developed a comprehensive theory of kinematic waves. The notable contributors to wave theory, DR Bland (viscoelasticity) and D.S. Jones (electromagnetic waves and acoustics) also spent time at Manchester. After Lighthill left Manchester, Fritz Ursell took up the Beyer Chair. Ursell carried out important research into water waves while at Manchester, first working with Goldstein and then as Beyer Professor (1961-1990). Ursell had numerous PhD students and postdoctoral assistants at Manchester who went on to do important research in wave mechanics. These include Frank Leppington (1964), David Evans (1966) and Douglas Gregory (1967). Paul Martin worked at Manchester from 1976-1999.

The current focus of wave research at Manchester lies within the Mathematics of Waves and Materials (MWM) Research Group, established in 2010 and led by Dr Raphael Assier and Prof William Parnell. The group is funded by a variety of sources including the EPSRC, the Royal Society, the Leverhulme Trust and industry, including Thales UK and Dyson. It has a thriving group of PhD research students and postdocs working on broad range of research problems in wave mechanics. The group also welcomes research visitors from around the globe.

Of specific importance to the group is the study of waves in heterogeneous media, of which good examples are composite materials and bone. The complex microstructure of such materials gives rise to complex wave fields and an important area of research is how to predict the effective wave as it propagates through the medium in question. In order to study problems in this domain, a variety of mathematical techniques are required including the development of fundamental analytical techniques, computational methods, probability theory, complex variable methods and the use of many aspects of the theories of acoustics, elasticity and water waves.

Over recent years the group has focussed on various aspects of metamaterials. These media are composite materials with bulk properties determined by their microstructure rather than their composition. They can be designed to have quite remarkable properties. This enables manipulation of waves in ways that have not been achieved with standard, or natural materials. Metamaterials are normally composed of a host material with periodic arrays of inclusions with alternative material properties. The inclusions can be solid scatterers or can be locally resonant structures themselves.

Some of the current exciting research projects at Manchester are:

  • Canonical scattering problems
  • Multiple complex variable approaches to solving wave propagation problems
  • Effective propagation of waves in random heterogeneous media, including composite media
  • Waves in periodic heterogeneous media
  • Scattering of waves by multiple obstacles (multiple scattering theory)
  • Hydroacoustics
  • Wave propagation in pre-stressed media
  • Thermo-visco-acoustics and thermo-visco-elastodynamics
  • Cloaking theory and slow-sound metamaterials
  • Active cloaking, anti-sound and anti-vibration

More details of these projects can be found on the MWM website.

More information about our research, and some papers, can be found by browsing the webpages of academic staff members. Potential PhD students may email staff directly to discuss possible projects.