We have modelled, designed and printed a metamaterial with a structure that reduces the effective speed of sound in air by half, opening up the potential for space-saving noise cancellation devices.

Noise pollution can reduce our quality of life, and even our life expectancy.  A 2014 report from the European Environment Agency estimates that 10,000 premature deaths and 43,000 hospital admissions for coronary heart disease and stroke can be attributed to noise exposure in Europe each year. (1)  The urgent need for efficient noise cancellation devices is clear.

Noise cancellation devices often work on the principle of destructive interference, where sound waves are combined so that they cancel each other out.
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Here’s how it works:

One tool that may be employed to reduce noise from engines, fans and other devices is a Quarter Wavelength Resonator (QWR).  This is a side branch in a duct that redirects sound waves so that they interfere destructively.  The length of the QWR is ¼ of the wavelength of the noise frequency to be attenuated. 

​The animation below shows how a QWR functions:

Part of the sound wave is diverted into the QWR, reflects off the end wall and then recombines with the undiverted wave.  The diverted wave has travelled an additional ½ wavelength, so the waves now re-combine destructively, cancelling the noise. 
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There are limitations to this approach: the size of the resonator depends on the wavelength of the noise we want to reduce.  Low frequency noises have long wavelengths, so the size of the resonator required soon becomes impractically large.

We used a mathematical mapping process called Transformation Acoustics to design and 3D print a metamaterial that has the potential to overcome the size limitations of QWRs for reducing low frequency noise.

Metamaterials are special materials with properties that are not found in conventional materials.  Unlike conventional materials, the properties of metamaterials are defined by their structures rather than by their chemical make-up.  Acoustic metamaterials have geometries that allow the manipulation of sound waves in ways not previously achievable.

Our metamaterial has arrays of elliptical cylinders which stretch the apparent space inside the resonator, with the result that the effective speed of sound is reduced by half.  Crucially, it does this while maintaining a close match to the impedance of sound in air across a range of frequencies. 

​One of the most rewarding experiences of my PhD is being able to attend to conferences and the opportunity to introduce myself and present my research to a wide audience of professionals. After two and a half years of PhD studies, I have had the chance to visit some great places within the UK and Europe, and now Australia! KOZWaves is a biennial conference focused on the study wave science and is always held somewhere within Australia and New Zealand. I first heard of it in my PhD first year and thought how amazing it would be to get to know a new community and give a talk at this event. Two years later I was very happy to receive the news that my talk: "Wave Propagation in Thermo-Visco-Elastic Continua" had been accepted to KOZWaves 2020 which was held at the University of Melbourne from the 17-19th of February.
 
The conference exceeded my expectations with great speakers talking from gravitational waves to water waves as well as light, sound and vibrations (to name just a few!) They showcased - the well known fact of - how waves describe so many physical phenomena of the world that surrounds us.

​The next step of the journey was a visit to the University of Adelaide in South Australia which is an 80 minute flight from Melbourne's Tullamarine Airport. As a keen surfer myself, before taking the flight I paid a visit to URBNSURF, a new facility for surfing artificial waves (again waves!) in a big pool right by the airport which was super fun!

I spent three fantastic days at the University of Adelaide visiting Dr. Luke Bennetts. I gave a talk to the Mechanics group on "Modelling Thermo-Viscous Damping in Continua". 

Finally, I visited Dr. Stuart Hawkins at Macquarie University in Sydney and some of his colleagues including Dr. Elena Vynogradova and her PhD student Martin Sagradian. I gave an hour long talk similar to the one I presented in Adelaide and had some interesting conversations. Stuart showed me some very impressive numerical computations via the use of the T-matrix approach for multiple scattering problems.

On International Women's Day in the International Year of Sound we want to acknowledge the extraordinary life and work of mathematician and philosopher Marie-Sophie (Sophie) Germain (1776-1831).

The daughter of a bourgeois Parisian silk merchant, Sophie’s early interest in mathematics grew during the reign of terror, which kept her confined to the family home.  Sophie found intellectual stimulation in her father’s library, and spent long hours studying mathematics, Greek and Latin; teaching herself the latter in order to understand texts by Newton and Euler. This interest in intellectual pursuits was initially strongly disapproved of by her parents, who removed her candles and her fire in a vain attempt to discourage her studies.  They eventually relented, however, realising that their daughter was serious in her work.
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Image source: http://mathshistory.st-andrews.ac.uk/PictDisplay/Germain.html

As a young woman, Sophie was barred by her sex from attending the newly established Ecole Polytechnique, but was able to obtain the lecture notes and submit work under the borrowed name of Monsieur Leblanc, a former student who had left the city.  She was to use this same pseudonym in her initial correspondence with mathematicians including Legendre, Lagrange and Gauss, fearing that as a woman, she would not be taken seriously. On discovering Germain’s true identity, however, Gauss responded with admiration for her tenacity.

Germain was particularly interested in number theory, but a visit to Paris by Ernst Chladni in 1808 turned her attention in another direction.  Chladni reproduced his classic experiment, producing nodal figures in sand on vibrating plates. In response, in 1809 the Institut de France offered a kilo of gold to any person who could formulate a mathematical theory of elastic surfaces.

Germain submitted her first anonymous entry to the competition in 1811 (the only submission received), but her work, while radical in approach, was flawed, and the competition extended a further two years.  Her second entry was given an honourable mention, but contained errors and was deemed insufficient for the prize. Finally, in 1816, Germain submitted an entry in her own name which, although still imperfect, was awarded the prize.  The Institut did not publish her paper, however, and Germain later published it herself, pointing out the errors in her work (Recherche sur la théorie des surfaces élastiques,  1821). 

It is likely that, being self taught, and lacking in formal education and guidance, there were gaps in Germain’s knowledge that affected her work in mathematical physics more than her work in number theory.  However, this makes her achievements in the field all the more extraordinary.  Germain took a different approach to that of her rival in the early days of the challenge, Siméon-Denis Poisson, whose initial work on the problem was based on molecular theory.  Poisson was admitted to the institute early in the competition (something not permitted to Germain) and became a judge rather than a competitor.  As such, he had access to Germain’s work, and she consulted with him on the subject. However Poisson did not regard Germain as a serious scholar, and did not acknowledge her in his own 1814 work on elasticity.
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In later life, Germain continued her work in both number theory and elasticity.  She contributed to the body of work on Fermat’s Last Theorem with what became known as “Sophie Germain’s Theorem”, published in a supplement to the second edition of Legendre’s Théorie des Nombres.  Before her death from breast cancer in 1831, Germain published her work on surface curvature, Mémoire sur la courbure des surfaces.  After the intervention of Gauss, Sophie Germain was eventually awarded a posthumous honorary degree from the University of Göttingen.

References and further reading

• Del Centina, A., Fiocca, A., (2012), ‘The correspondence between Sophie Germain and Carl Friedrich Gauss’, Arch. Hist. Exact Sci. 66:585–700
• Laubenbacher, R., Pengelley, D.,  (2010), ‘Voici ce que j’ai trouvé: Sophie Germain’s grand plan to prove Fermat’s Last Theorem’, Historia Mathematica 37 (2010) 641–692
• Frize, M. (2010). The Bold and the Brave: A History of Women in Science and Engineering. Ottawa: University of Ottawa Press.
• GRAY, M. (2005). Sophie Germain. In CASE B. & LEGGETT A. (Eds.), Complexities: Women in Mathematics (pp. 68-74). Princeton; Oxford: Princeton University Press.
• Bucciarelli L.L., Dworsky N. (1980) An Award with Reservations. In: Sophie Germain. Studies in the History of Modern Science, vol 6. Springer, Dordrecht.
• https://scientificwomen.net/women/germain-sophie-39
• http://mathshistory.st-andrews.ac.uk/Biographies/Germain.html
• https://simonsingh.net/books/fermats-last-theorem/sophie-germain/
• https://www.thoughtco.com/sophie-germain-biography-3530360