Description 
There is a long history of mathematicians working on canonical diffraction (or scattering) problems. The mathematical theory of diffraction probably started with the work of Sommerfeld at the end of the 19th century and his famous solution to the diffraction of acoustic waves by a solid halfplane. Since, some very ingenious mathematical methods have been developed to tackle such problems. One of the most famous being the WienerHopf technique.
However, despite tremendous efforts in this field, some canonical problems remain open, in the sense that no clear analytical solution is available for them.
One of this problem is the quarterplane problem, the problem of diffraction of acoustic waves by a solid quarterplane. Thus far, it has not been possible to apply classical methods such as the WienerHopf method successfully in that case, and hence some new mathematical methods need to be developed in order to tackle this problem. This makes it very interesting as it implies that many different types of mathematics can be used and it makes the subject intrinsically multidisciplinary.
Many industrial problems can be linked to the theory of diffraction, for example the noise generated by a jet engine (acoustic waves) or radar detection (electromagnetic waves) and defect detection in materials (elastic waves).
PhD projects are available in this field.
References:  R. C. Assier and N. Peake. On the diffraction of acoustic waves by a quarterplane. Wave Motion, 49(1):6482, 2012  R. C. Assier and N. Peake. Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarterplane. IMA J. Appl. Math., 77(5):605625, 2012.
