EPSRC Reference: 
EP/N013719/1 
Title: 
Canonical Scattering Problems 
Principal Investigator: 
Assier, Dr R C 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Manchester, The 
Scheme: 
First Grant  Revised 2009 
Starts: 
30 November 2015 
Ends: 
29 November 2017 
Value (£): 
93,588

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Nonlinear Systems Mathematics 

EPSRC Industrial Sector Classifications: 
Aerospace, Defence and Marine 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
07 Sep 2015

EPSRC Mathematics Prioritisation Panel Sept 2015

Announced


Summary on Grant Application Form 
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 timedependent 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 semiinfinite halfplane, 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 WienerHopf and the SommerfeldMalyuzhinets 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 threedimensional problem of diffraction by a quarterplane and the twodimensional 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 quarterplane 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.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.man.ac.uk 