A to Z

Join us on an alphabetical journey through the research themes and key concepts behind our work.
We will add new entries to this list regularly, so keep checking in to find out more!

A is for Acoustics

Sound is all around us: it accompanies everything we do. Human beings are both sources and the receivers of sounds. Acoustics is the branch of science and engineering concerned with all aspects of sound, including its generation and transmission, its control and use, and its effects on people and the environment.

Vitruvius presenting De Architectura to Augustus. From Vitruvius on Architecture (1684) by Thomas Gordon Smith. Image credit: Sebastian Le Clerc, public domain, via Wikimedia Commons

The scientific study of sound dates at least as far back as the 6th century BCE, with experiments by Greek philosopher and mathematician Pythagoras into harmonics and vibrating strings [1]. Several centuries later, the Roman engineer and architect Vitruvius (d. 15 BCE) is credited with the first formal treatment of architectural acoustics in his multi-volume work De Architectura, in which he discusses sound propagation in theatres [2].

The use of the term “acoustics” to define a branch of science came much later. Although the word had been used in the context of sound previously [3], the definition is usually attributed to physicist and mathematician Joseph Sauveur, who in 1701 proposed a new field of study called acoustique. He described his new discipline as “a science superior to music”, because it encompassed all sound, not simply music and harmonics [4].

When we think about acoustics the first thing that springs to mind might be architectural acoustics, and the optimisation of indoor sound. However, modern acoustics is a multidisciplinary field, encompassing physics, material science, engineering, psychology, physiology, music, architecture, neuroscience, environmental science and mathematics. Some of the many other branches of acoustics include:

Image created by UK Acoustics Network members, reproduced with permission.

Physical acoustics: Physical acoustics overlaps with other fields of acoustics such as engineering and medical imaging. It also encompasses sound beyond the human audible range. Frequencies below the audible are called infrasound. Many natural phenomena such as earthquakes and thunder generate infrasound. Higher frequency sound is ultrasound, whose uses in medical imaging are well known. There are many other interesting physical acoustic phenomena, such as the photoacoustic effect: where sound is generated by the interaction of light with matter.

Musical acoustics: This is a broad field encompassing the design of musical instruments, building acoustics, the human perception of sound and music, and music recording and reproduction.

Psychoacoustics and physiological acoustics: These include the study of people's perception of sound and its physical and pshychological impact on us, and the physiology of hearing and hearing-loss.

Underwater acoustics and oceanography: The study of underwater acoustics and sonar has military and commercial applications, while in marine biology, acoustics can lead to a better understanding of animal communication, and the effect of human-generated noise on marine animals.

Environmental noise control: We know that exposure to loud noise can cause hearing loss, but this is not the only adverse effect. Noise can reduce our quality of life, and even our life expectancy, causing an estimated 12,000 premature deaths in Europe each year [7]. For this reason, noise reduction and control is an extremely active branch of acoustics.

Researchers in the Mathematics of Waves and Materials group apply mathematical modelling to acoustics. Research themes include the design of acoustic metamaterials and phononic crystals: materials whose properties are determined by their structures, rather than their chemical composition. Acoustic metamaterials have structural elements that are smaller than the wavelengths of the sounds they are designed to interact with. This leads to properties that are not found in conventional materials. Other areas of acoustical research within the group include aero- and hydroacoustics: the study of the noise generated by fluid flows both on their own (jet noise) and when interacting with solid boundaries (trailing edge noise) in air and water, and the interaction of soundwaves with materials, including nanofibrous materials.

2020-2021 is the International Year of Sound, a global initiative to highlight the importance of sound and acoustics in society. You can find learning resources and events from acoustics organisations around the world on the Year of Sound website.

[1] Zhmud, L. (2012) "8: Harmonics and Acoustics in Pythagoras and the Early Pythagoreans", Oxford Scholarship Online
[2] Maconie, R. (2005) "Musical Acoustics in the age of Vitruvius", Musical Times; London, 146: 75-82.
[3] Dostrovsky, S. (2008) "Sauveur, Joseph", in Complete Dictionary of Scientific Biography vol. 12, New York, NY: Charles Scribner's Sons, 127-129.
[4] Fix, A. (2015) "A Science Superior to Music: Joseph Sauveur and the Estrangement between Music and Acoustics", Phys. Perspect., 17: 173–197.
[5] Rossing, T. D (2014) "A Brief History of Acoustics" in Springer Handbook of Acoustics. New York, NY: Springer, 9-24
[6] Rossing T.D. (2014) "Introduction to Acoustics" in Springer Handbook of Acoustics. New York, NY: Springer, 1-6
[7] European Environment Agency, Environmental noise in Europe — 2020, EEA Report No 22/2019

B is for Boundary Conditions

Just as in high school mathematics or physics, where one imposes conditions on the solution to an ordinary differential equation (such as a simple harmonic oscillator), in continuum mechanics boundary conditions are imposed on the solutions to partial differential equations which represent mathematically the laws governing the generic behaviour of particular materials.

A familiar example is the dripping of a tap. As water slowly accumulates at the outlet of the tap, its surface forms a smooth near-spherical shape as a consequence of surface tension acting to minimize the exposed surface area. Eventually, it becomes energetically expedient for a drop to detach and fall away, and the process repeats. The partial differential equations in this context are the Navier-Stokes equations, and the boundary conditions are the surface tension at the interface between water and air, the pressure gradient or the flow rate within the tap, and also the contact conditions at the meeting point between water, air, and the tap itself. In general, the flow of water, or any Newtonian fluid, is a function only of the Reynolds number and the boundary conditions.

Credit: PumpkinSky, CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0 via Wikimedia Commons

Credit: Spigget, CC BY-SA 3.0: https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

The refraction of a light ray as it passes from one medium into another can also be understood in terms of boundary conditions applied at the interface between the two media. Solving the Euler-Lagrange equation, one can show that \(\frac{sin \theta}{c}\) must be the same on either side of the boundary, where θ is the angle of incidence on one side and the angle of refraction on the other side, and c is the local speed of light in each medium. This result is known as Snell's Law, although it was described by Ibn Sahl almost 600 years before Snell was born [1].

The extent to which the full detail of the boundaries are incorporated within a mathematical model can depend on whether there is a practical need for a quantitative result in a particular case, for example, the stress on a rocket's nose cone, or the circulation of air in a hospital. If the intention of the modelling is mainly to understand the underlying phenomena in a more qualitative sense, then an idealized and representative form of the problem is often developed, typically involving simplifications in the form of assumptions about spatial symmetry, infinite extent, or smoothness of walls etc. Such simplifications are usually intended to allow a certain amount of progress with a purely analytical approach, or for testing of a computational approach at a proof of concept stage. Solving the fundamental equations in full generality may be impossible, but with restrictions on the possible solutions, e.g. an assumption of spherical symmetry, or taking an asymptotic limit in terms of a small parameter, the problem may become tractable. In effect, the simplifications can be interpreted as modifying the boundary conditions. Then the relevance of the idealized solution in the context of the full unsimplified problem will depend on whether the boundary conditions are amenable to the sort of simplifications discussed.

When assessing the robustness and accuracy of a new computational algorithm, certain well-studied and sometimes challenging combinations of boundary conditions are used, known as benchmark problems. These are problems for which the correct solutions are known, and the idea is to highlight any limitations of the algorithm, or any mistakes made in its implementation. For example, the lid-driven cavity is a standard benchmark problem in CFD (computational fluid dynamics) for the validation of numerical methods [2].

N. F. Morrison

[1] Rashed, R. (1990) "A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses", Isis, 81(3): 464-491
[2] Hinch, E. J. (2020) "1: The Driven Cavity" in Think Before You Compute. Cambridge: Cambridge University Press, 3-8

C is for Composite Materials

Composite materials can be defined as materials manufactured by the combination of two or more phases, to perform tasks that neither of the constituent materials can achieve alone. One phase is continuous and is called matrix while the other phase which is discontinuous is known as reinforcement or filler [1] (Figure 1). The matrix material binds the reinforcement and gives composite its net shape. The interfacial region between reinforcement and matrix in composites facilitate the transfer of forces between the relatively weak matrix and stronger reinforcement. A strong interfacial bond is necessary to enhance the mechanical properties of the composites.

Composite materials offer superior properties than their base materials. Humans have been making composites since ancient times. Adobe brick is an early example of a composite, made of straw and mud. Wood is also an example of composite material, made from long cellulose fibres held together by lignin. In recent times, the use of composite materials has increased in the aerospace, automotive, marine, defence, wind/energy and construction industries due to their light weight nature, high specific stiffness and strength, corrosion resistance, buoyancy, damage tolerance, impact resistance, and damping and sound absorbing properties.

Fig. 1 Schematic of a composite material, showing the matrix and dispersed, or filler phases.

Fig. 2 Young’s modulus and density of metals, polymers and composites, from [2]. Reproduced under a creative commons CC BY-NC-ND 3.0 license: https://creativecommons.org/licenses/by-nc-nd/3.0/

The Young’s modulus and density relation of metals, polymers and composites is presented in Figure 2, which shows that composite materials can offer better specific stiffness (E / ρ) than metals.

The properties of composite materials can be tailored by selecting appropriate combinations of matrix and filler/reinforcement. The volume fraction of reinforcements also plays an important role in determining the performance of composites. Based on the type of matrix material, composites are generally classified as polymer matrix composites (PMCs), metal matrix composites (MMCs), ceramic matrix composites (CMCs) and carbon matrix composites (CAMCs)

PMCs are most widely used type of composites. They are further divided into different categories, namely: thermosets, thermoplastics and elastomeric composites. Thermoset matrices are used in high performance composite materials. They become rigid after curing and can be used at elevated temperatures without losing structural rigidity. Thermoplastics, unlike thermosets can be melted on heating. Examples of thermosetting polymer matrices include polyester, vinyl ester, epoxy, phenolic, cyanate ester, polyurethane, polyamide and bismaleimide. Polyethylene, polypropylene, polyamide, PEEK, thermoplastic polyamide, thermoplastic polyurethane, polycarbonate, PLA, polysulfone, polyphenylene sulphide are all examples of thermoplastic polymer matrices. Elastomers achieve their cross linking as a result of the vulcanization process. Rubber is a well-known elastomeric material. Metal, ceramic and carbon matrices are used for highly specialised objectives. For example, ceramics are used for high temperature applications, and carbon is used for applications which are expected to undergo friction and wear.

A variety of filler/reinforcement materials are available. Fibre reinforced composites are popular for high strength/weight and high modulus/weight ratios. Different fibres like glass, carbon, boron, ceramic, metal and natural fibres like flax, sisal and hemp are used as reinforcements in many applications. Composites made from carbon fibres are popular in aerospace, owing to their strength and weight reduction characteristics. Components made from carbon fibre reinforced composites can be five time stronger than 1020 grade steel, while utilising only 20% of the weight of steel. In aerospace applications, high strength and weight reduction are highly desirable properties, for structural integrity and fuel economy. The use of composite materials in Boeing 787 aircraft is depicted in Figure 3.

Fig. 3 Usage of composite materials in Boeing 787, from reference [3], reproduced under a creative commons cc by 4.0 license: https://creativecommons.org/licenses/by/4.0/deed.en

Composites manufactured by the combination of hollow microballoons (of glass, ceramic and plastic) and polymer matrix, generally known as syntactic foams, are popular for their acoustic, buoyancy, low moisture absorption and weight saving characteristics, and find their applications in marine and aerospace structures.

Numerous methods are available to manufacture composites, for example resin transfer moulding (RTM), vacuum infusion (VI), vacuum assisted resin transfer moulding, compression moulding and 3D printing. The selection of manufacturing method will mainly depend on the materials, part design and application.

Z. Yousaf

[1] Issac, M., Daniel O. I. (2006) Engineering mechanics of composite materials, New York: Oxford University Press
[2] Kickelbick, G. (2014) Hybrid Materials – Past, Present and Future, Hybrid Mater, 1: 39-51. DOI: 10.2478/hyma-2014-0001
Image reproduced under CC BY-NC-ND 3.0, https://creativecommons.org/licenses/by-nc-nd/3.0/
[3] Alemour, B., Badran, O. & Hassan, M. R. (2019), A Review of Using Conductive Composite Materials in Solving Lightning Strike and Ice Accumulation Problems in Aviation, Journal of Aerospace Technology and Management, 11: e1919.
Image reproduced under CC By 4.0, https://creativecommons.org/licenses/by/4.0/deed.en

D is for Diffraction

The term diffraction was coined in 1666 by Francesco Maria Grimaldi, after experimenting with light. He shone a light through a small hole in an opaque plate, and found that the light cast into the room covered an area larger than the hole it travelled through. This suggests that the light is not travelling in straight lines through the hole but is bent around the corners of the hole [1]. This can be seen in Figure 1, where the wave becomes curved as it passes through the slit, covering an area larger than the slit on the other side. The theory of diffraction can be extended beyond light passing by opaque surfaces, to any system of waves travelling past or around an object. Whether those waves be acoustic, elastic, electromagnetic or even oceanic waves - all can be diffracted.

Fig. 1. Diffraction through a slit. Credit: Inductiveload, Public domain, via Wikimedia Commons

Fig 2: Diffraction (1) and Refraction (2). Credit: Cmglee, CC BY-SA 3.0 , via Wikimedia Commons

A practical use of diffraction can be seen in the use of diffraction gratings. Diffraction gratings are periodic structures, which split light into its component wavelengths. This phenomenon occurs as the diffraction of a wave is dependent its wavelength, hence the multiple wavelengths of visible light are diffracted at different rates as they pass through the grating. Therefore, after the light passes through the grating we can see each colour component. Diffraction gratings can be used as a tool in spectrometers - used to analyse the light emitted by a material. They can also be seen in items such as CDs, where the data engraved into the CD causes light to be diffracted when shined upon it.

The theory of diffraction is not to be confused with refraction, which is the warping of waves due to passing through a material, rather than around a material. In Figure 2, we can see light being diffracted by a grating in the top image, and light being refracted by a glass prism below.

Diffraction can not only be seen in light waves, but in any travelling wave front. For example, in the animation on the right, we see the diffracted wave field of a plane wave travelling from the right hand side, by what we call a half-plane. The half-plane is an infinitely thin line, starting at \(x = 0\), and extending to negative infinity, \(-\infty\), to the left. On the right hand side we see a field which is reflected by the edge of the half-plane and looks like a cylindrical plane wave. However, on the left hand side, where the wave is travelling past the half-plane, we see that the wave front is no longer constant due to the bending around the plane.

Scientists have gained a lot of insight about the world around us by the theory of diffraction. Most interesting is the occurrence of diffraction in the quantum world. Quantum physics is the study of matter and energy at the most fundamental level. In general, quantum theory considers the smallest of particles, and since these particles are the building blocks of the solid world around us, you would maybe expect that they behave like solid objects. However, US physicists Clinton Davisson and Lester Germer performed an experiment where electrons were fired in a stream through a small slit and their position recorded on the other side [2]. If each particle were to behave like a discrete solid object, we would expect to see the shape of the slit replicated in the image on the other side. However, it was found that the particles are actually diffracted and show wave-like behaviour, creating an image greater than the size of the slit when recorded.

T. White

[1] Cajori, F. (1899) A history of physics in its elementary branches: including the evolution of physical laboratories, New York: Macmillan
[2] Ball, P. (2018) Two slits and one hell of a quantum conundrum, Nature, 560(7717):165-166

Animation credit: M. Nethercote

E is for elasticity

Elasticity is the property of a material that enables it to recover its original form when it is stretched, compressed, or otherwise deformed. All materials are elastic; stiff materials like steel and even diamond exhibit elasticity to a certain extent but the property is more easily visualised in materials that are highly deformable, e.g. rubber (elastic) bands or soft foams, see Fig. 1.

We usually describe the elasticity of a material by its relationship between force and extension, or equivalently by its stress-strain relationship, where stress is the force applied per unit area, and strain is the degree of deformation. With reference to Fig. 2, many materials exhibit a linear force- extension, or stress-strain relationship, and this type of

Fig. 1. Credit: Sander van der Wel, CC BY-SA 2.0, https://creativecommons.org/licenses/by-sa/2.0, via Wikimedia Commons

elasticity is therefore known as linear elasticity. The origin of this was the work of Robert Hooke, who in 1660 discovered that the extension of a spring is proportional to the weight that is applied to the spring, see Fig. 3. Obviously, if we keep adding weight to the spring, at some point it is no longer able to sustain this loading and it will break. Just prior to this in fact, the material from which the spring is made will have undergone so-called plastic deformation. This is where the atoms comprising the material are forced apart permanently. This is illustrated in Fig. 2, where we indicate that all of this behaviour is beyond the elastic limit. If weights (or more generally, forces) remain within the regime of elasticity then when the weight is removed, the spring will return back to its original rest state.

Fig. 2. Stress-strain curve illustrating the elastic and plastic deformation and eventual fracture of a material.

Fig. 3. Hooke's Law. Credit: Svjo, CC BY-SA 3.0, https://creativecommons.org/licenses/by-sa/3.0 via Wikimedia Commons

In reality, perfect elasticity is not possible, and according to the second law of thermodynamics there will always be some form of loss in the system, e.g. via friction or heat. This amount of loss is measured by the area between the load and unload curves in the stress-strain relationship, see Fig. 4. However we can get very close to perfect elasticity in many systems (the spring is a very good example) and especially so if we deform the material very slowly.

The elasticity of a material can be defined by its so-called elastic modulus, which is frequently termed Young’s modulus, named after Thomas Young and his work in the 19th century. When put under tension, as we have already discussed, many materials can be defined by their stress-strain relationship, i.e. \(\sigma = E e\) here \(\sigma\) is the stress applied in tension, \(e\) is the extensional strain and \(E\) here is Young’s modulus. The larger \(E\) is, the more stiff the material, given that it takes more stress (or force) to yield a fixed strain. In this context then, stretching your ear is much more straightforward (try it, but don’t pull too hard!) than stretching a piece of concrete or diamond.

Fig. 4. Stress strain curves for perfectly elastic and real materials. The red shaded area represents energy loss in the system.

To put some numbers on this, the Young’s modulus of a soft material such as rubber is around 6 times smaller than that of diamond. Therefore if a given stress of say 1MPa is applied to a piece of rubber it would yield a strain of around 10%, whereas the strain in diamond would be only 0.0001%. This is precisely why then you can visualise elasticity in soft tissue and rubber but not in very stiff materials such as diamond. This is also why there is a general perception that stiff materials such as wood, steel and diamond are not elastic. In reality they are, it is just that one cannot see this with the naked eye.

Now, not all materials are linear elastic for their entire regime of elastic deformation, although they usually are in some limit, close to the origin of the stress-strain curve. Indeed, we just described the example of soft tissue and this is very nonlinear, by which we mean that the material is not Hookean and does not exhibit proportionality between stress and strain. Its behaviour is defined by a more complex relationship between stress and strain. The study of such materials is known as nonlinear elasticity. Understanding this nonlinear relationship is extremely important and assuming that an elastic response is linear when it is not can be very dangerous. Consider for example the case of a bungee cord, responsible for the safe but exciting launch of someone over the side of a bridge into a ravine below. Typically bungee cords are nonlinear

Fig. 5. Graph of force vs extension for a linear elastic (blue curve), and non-linear elastic (red curve) material.

elastic, exhibiting the kind of stress strain relationship depicted in the red curve of Fig. 5. As we can see, for a given force (weight) the bungee cord would extend much further than it would if it were linear elastic (the blue curve in Fig. 5), with its Young’s modulus assumed to be that which is measured in the small strain regime. If we are relying on the cord to stop us at the correct distance (depending on what is below us!) then we should ensure that we have a good understanding of the elasticity of the cord! More generally, nonlinear elasticity is critical for the design of many soft materials, e.g. polymers, isolation mounts, maxillofacial prosthetics, artificial tendons amongst others.

Understanding a material’s elastic response is fundamental to humankind. It plays a critical role in almost everything that we do from taking a walk (made possible by the elasticity of our tendons and ligaments), to driving a car (the elasticity of tyres is crucial, as well as the suspension of the car), to understanding how earthquakes (elastic waves that propagate through the earth) could potentially cause damage to entire cities. In the context of designing materials therefore, understanding and subsequently modifying the design of the elasticity of the material is something that is absolutely critical. Mathematical models are important for this purpose, particularly in order that we can reduce the amount of experimental testing and therefore reduce the impact on the environment. Modelling allows us to consider the impact of a huge parameter space of properties and how they impact on the overall elasticity of complex materials. Our research group has carried out a variety of work over the last decade in the areas of composite materials and elastic metamaterials.

W. J. Parnell