# New Methods and Analysis for Wave Propagation Problems

Lead Research Organisation:
University of Bath

Department Name: Mathematical Sciences

### Abstract

Our understanding of wave phenomena underpins many technologies upon which our society depends, for example, radar, sonar, mobile phones, ultrasound, optical fibres, and crack-detection in structures.Many wave phenomena can be described mathematically by Partial Differential Equations'' (PDEs); and information about the physical processes can be obtained by studying these mathematical models. Mathematicians have a toolkit of techniques to study PDEs and extract useful information. This project seeks both to sharpen'' two tools, which the investigator has played a key role in developing, and to combine them, not only with each other, but also with other cutting-edge techniques from different areas of mathematics. This combination will increase the power of these techniques and allow mathematicians to apply them in new situations, further increasing our knowledge of waves.Some examples of problems this project will investigate are:- The scattering of sound and electromagnetic waves from obstacles with sharp corners and edges. These problems are of fundamental importance to many engineering applications, and hence have been extensively studied for many years. However, the current mathematical tools are still not powerful enough to solve many important practical problems.- The propagation of waves through so-called meta-materials'', artificial materials engineered to produce properties not found in nature, and periodic media'', which have applications in photonic crystals used in optical communication.- The detection of cracks in the surface of materials; this is obviously important for testing the integrity of many engineering structures, but in particular nuclear and chemical reactors.

### Planned Impact

Our understanding of acoustic, electromagnetic and elastic wave propagation is used in a plethora of technologies upon which our society depends, for example: radar, sonar, mobile phone technology, ultrasound, noise barriers on motorways, optical fibres, seismic imaging, and crack-detection in structures. The results of this Fellowship will enhance our understanding of wave phenomena, and thus ultimately contribute to improved technologies which will benefit the general public. In the short term, several results this Fellowship seeks to obtain will almost immediately be able to be applied in an industrial context. In the longer term, results from this Fellowship will form the basis of future investigations into wave propagation problems. Examples of short term potential industrial impacts of this Fellowship include: -Improved modelling of acoustic and electromagnetic scattering by obstacles with sharp edges and corners. This will lead to improved radar and sonar technologies. An example of one such company that will benefit is BAE Systems through their improved computation of radar cross-sections. -Improved detection of cracks in materials. When a material is put under stress, cracks form in regions of high stress intensity, often near the surface. When these cracks break the surface, the integrity of the material can be compromised, thus detecting this cracks is crucial. One important example where detection is vital is in safety inspections of nuclear and chemical power plants. -New understanding of elastic waves propagating through materials. This is important both in cases where one seeks to eliminate unwanted oscillation (e.g. flutter'' in airplane wings), or excite it (e.g. in ultrasonic non-destructive testing). This Fellowship will also have an impact on the general public in the short term through increasing their understanding of science. The applicant has already been involved in several outreach activities, including the high profile Royal Society Summer Exhibition 2010, and is committed to continuing this public engagement.

### Publications

Baskin D
(2016)

*Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations*in SIAM Journal on Mathematical Analysis
Betcke T
(2013)

*Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering*in IMA Journal of Numerical Analysis
Chandler-Wilde S
(2012)

*Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering*in Acta Numerica
Fokas A
(2012)

*Synthesis, as Opposed to Separation, of Variables*in SIAM Review
Graham I
(2014)

*When is the error in the $$h$$ h -BEM for solving the Helmholtz equation bounded independently of $$k$$ k ?*in BIT Numerical Mathematics
Moiola A
(2014)

*Is the Helmholtz Equation Really Sign-Indefinite?*in SIAM Review
Spence E
(2014)

*Bounding acoustic layer potentials via oscillatory integral techniques*in BIT Numerical Mathematics
Spence E
(2014)

*Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering*in SIAM Journal on Mathematical Analysis
Spence E
(2015)

*Coercivity of Combined Boundary Integral Equations in High-Frequency Scattering*in Communications on Pure and Applied MathematicsDescription | Wave propagation is currently studied by the research community from a wide variety of perspectives; on the one hand there are still unanswered theoretical questions about how waves behave in certain circumstances, and on the other hand there are many practical questions about how to best simulate and compute the behaviour of the waves used in, e.g., radar, sonar, and telecommunications. The findings of this project fall, broadly speaking, into the following three categories: 1) results on theoretical issues in wave propagation (e.g. how do waves behave when they hit obstacles with corners and edges? what happens as the frequency increases?) 2) results increasing our understanding of existing methods for the practical solution of wave propagation problems (in particular, in the difficult case when the frequency of the waves is very high), and 3) results that provide new methods for solving wave propagation problems. |

Exploitation Route | The findings are already being used by the research community (since many of the findings relate to commonly-used methods for solving wave propagation problems). It is possible that some of the new methods introduced in the project may one day form the basis for industrial codes used to solve wave propagation problems, but this will require further development of these ideas, both by the PI and other members of the research community. |

Sectors | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software) |

Description | EPSRC Fellowship |

Amount | £1,027,193 (GBP) |

Funding ID | EP/R005591/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Academic/University |

Country | United Kingdom |

Start | 10/2017 |

End | 09/2022 |

Description | Research collaboration with Baskin and Wunsch |

Organisation | Northwestern University |

Department | Department of Mathematics |

Country | United States |

Sector | Academic/University |

PI Contribution | I visited Northwestern in April 2014 and began to collaborate with two researchers there. |

Collaborator Contribution | Jared Wunsch and Dean Baskin were both at Northwestern at the time (Baskin is now at Texas A&M University) and collaborated with me. |

Impact | The paper: D. Baskin, E.A. Spence, J. Wunsch, Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations SIAM J. Math. Anal. vol. 48, no. 1, 229-267 (2016) |

Start Year | 2014 |

Description | Research collaboration with Gander |

Organisation | University of Geneva |

Department | Section of Mathematics |

Country | Switzerland |

Sector | Academic/University |

PI Contribution | Along with Ivan Graham (also at Bath), I collaborated with Martin Gander from Geneva (with me visiting him in December 2012). |

Collaborator Contribution | Martin Gander from Geneva collaborated on a project with Ivan Graham and I. |

Impact | The paper: M.J. Gander, I.G. Graham, E.A. Spence, Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed? Numer. Math., vol. 131, issue 3, page 567-614 |

Start Year | 2012 |