Canonical Scattering Problems

Scattering problems (or diffraction problems) consist in studying the field resulting from a wave incident upon an obstacle. This can for example be an acoustic or an electromagnetic wave. In general, these are complicated time-dependent problems, but often a justified hypothesis can be made, which allows time considerations to be dismissed. As a consequence, the wave fields encountered in such problems all satisfy the same equation called the Helmholtz equation. The adjective "canonical" in the title of the project derives from studying simple obstacles, generally of infinite size, with particular characteristics such as sharp edges or corners. Although "simple", these canonical geometries can be used to evaluate the scattered field of more complicated finite obstacles subject to high frequency incident waves.

The first such canonical problem to be considered was the problem of diffraction by a semi-infinite half-plane, and it was solved very elegantly by Arnold Sommerfeld in 1896. This was the start of the mathematical theory of diffraction. Since then, some very ingenious mathematical methods have been developed to tackle such problems, the most famous being the Wiener-Hopf and the Sommerfeld-Malyuzhinets techniques. However, despite tremendous efforts in this field, some canonical problems remain open mathematically, in the sense that no clear analytical solution is available for them. In particular, this is the case for two such problems, the three-dimensional problem of diffraction by a quarter-plane and the two-dimensional problem of diffraction by a penetrable wedge. The word penetrable means that waves can propagate inside the wedge region as well as outside, but with dissimilar wave speeds in the two regions.

The aim of this project is to find a mathematical solution to these two problems, and to use these in concrete applications. It is motivated by a need to address environmental and economical issues linked to both climate change and the near future extinction of fossil fuels. In particular, results on the quarter-plane will be used to understand noise generation within a new type of aeroengine (predicted to drastically reduce the fuel consumption of civil aircraft) and underwater propulsors. This will have a significant impact in these fields of engineering, and will help to cement the UK's position as one of the leading countries for aero and underwater propulsor design. Results on the penetrable wedge will be used in collaboration with climate scientists at the University of Manchester to improve current models for quantifying the effect of light diffraction by ice crystals in clouds. This is a particularly important application since, due to the complex shapes of ice crystals, this problem currently represents one of the biggest uncertainties in predicting climate change. Furthermore, both aspects of the project will enhance the UK's reputation for high quality interdisciplinary applied mathematics research.

This project is motivated by a need to address environmental and economical issues linked to both climate change and the near future extinction of fossil fuels. The first part of the project focuses on the problem of scattering by an ideal quarter-plane with a strong emphasis on the understanding of noise generation by blade-gust interaction. The second part of the project focuses on the problem of scattering by a penetrable wedge, with a strong emphasis on the understanding of light diffraction by ice crystals in clouds.

By shedding some light on noise generation within a new type of aeroengine, the first part of the project will certainly inspire methods to reduce the overall noise level of such engine. This is currently one of the main issues to overcome before its commercial use and will lead to an obvious impact on the environment. Indeed, this engine is predicted to lead to a reduction of fuel consumption by 25-30% compared to current civil aircraft, which would result in lower emissions of polluting gases. Moreover, this noise in itself is a source of a different kind of pollution, a reduction of which will have a very beneficial impact on society (especially people living or working close to airports or air passengers).

The economic impact of this part of the project is also worth noting. Indeed, despite living in a time of economic and environmental crisis, travelling has become an essential part of our life. It is also generally admitted that fossil fuels will continue to be used for the foreseeable future. However, it is known that such resources are running out, leading to inevitable price increases. Hence, reducing the amount of fuel required by civil aircraft would be a major step towards keeping flight-related costs to a low level. A more direct economic impact that will result from this project arises from the collaborative work with the project partner Thales and the understanding of noise generation within water-propulsors. This will lead to a significant advancement in both the understanding and relevance of the theoretical models used in the early design phase of major military systems. As such, it will assist in the development of low Technology Readiness Level (TRL) to high TRL enhancements, with the effect of maintaining science and technology at the forefront of research in support of related UK national defence activities.

The environmental impact of the second part of the project is substantial since its long term aim is to improve prediction tools for global warming. Indeed, due to their complex shapes, the modelling of light diffraction by ice crystals in clouds is currently one of the biggest uncertainties in predicting climate change. Such an improvement will also allow for a better prediction of the beneficial effects resulting from different solutions designed to reduce global warming.
Naturally, this part of the project also has an economic impact. Indeed, not only will climate change have an effect on the environment and on people, but it will also massively affect businesses and the global economy. The current cost of climate change to the world economy is estimated to be almost $1.3 trillion, while predictions for 2030 suggest that the US would spend 2% of its GDP, and China $1.2 trillion, due to natural catastrophes alone.