In material science, we refer to a material domain as a matrix and an inclusion as an object inside the matrix with different material properties.

​Neutral inclusions were first introduced by Mansfield in 1953 for holes in plates [1]. The idea is to provide inclusions with an additional coating to remove any undesired perturbations from the physical fields in the matrix. A perturbation is any deviation of a field from its state when the inclusion is not present. 

​In the following example we consider a steady-state, two-dimensional problem. The steady-state assumption removes any time-dependence from the solution. As a result, the only material property we need to consider is the thermal conductivity​, denoted \(k\). The thermal conductivity of a material measures its ability to conduct heat through diffusion.
​
We are interested in the temperature field, \(T(\mathbf{x})\), which must satisfy the heat equation. For this example, we assume the thermal conductivity is isotropic (independent of direction) and homogeneous (independent of position) and is therefore a scalar. As a result, together with the steady-state assumption, the heat equation reduces to Laplace's equation, given by:
$$\begin{equation}
\nabla ^2 T(\mathbf{x}) = 0,
\label{laplace}
\end{equation}$$
​​where \(\nabla ^2\) is the Laplace operator.​

​Consider the linear temperature field in Figure 1, where the black lines represent lines of constant temperature called isotherms. This temperature field is linear with respect to \(x\) and therefore has the form \(T(x)=\alpha x+\beta\) for some constants \(\alpha\) and \(\beta\).

​Let the matrix in Figure 1 have conductivity \(k_m\). Notice in Figure 2 that when a circular inclusion with conductivity \(k_i\), where \(k_i \neq k_m\), is embedded in the matrix, the field becomes perturbed and is no longer linear.

By adding a coating with conductivity \(k_c\), as illustrated in Figure 3, we can remove the perturbations from the matrix and recover the linear temperature field

​We wish to find a solution where \(B_m=0\) as this leads to a linear temperature field throughout the matrix given by \(T_m(r,\theta)=\alpha r \sin \theta = \alpha x\) (since \(x=r \sin \theta\)).
​
Let the radius of the inclusion and coating be given by \(r_i\) and \(r_c\) respectively. Solving the temperature fields in (2) with perfect contact conditions across each interface (continuity of heat flux and temperature) we find that \(B_m=0\) is satisfied when
\begin{equation}
D^2 = \dfrac{(k_m-k_c)(k_i+k_c)}{(k_i-k_c)(k_m+k_c)} \qquad \text{where} \qquad D= \dfrac{r_i}{r_c}
\label{k cond}
\end{equation}​

In terms of limitations, firstly, an isotropic solution may not exist for a given field. For example, to find a neutral inclusion for a pressure field we must relax the isotropic assumption. Secondly, a neutral inclusion is not a perfect cloak as it is tailored for a specific configuration with fixed external forces. Therefore, if we change the nature of the external forces, the properties within each coating will no longer achieve the desired effects.

​A perfect cloak is one which, regardless of the external forces, leads to unperturbed fields in the matrix. Transformation theory is used to design perfect cloaks and other interesting concepts. We will discuss these in a future blog post.

[1] E. H. Mansfield, Neutral holes in plane sheet - reinforced holes which are elastically equivalent to the uncut sheet, The Quarterly Journal of Mechanics and Applied Mathematics, 1953, 6(3), 370–378

​Sound travels as small vibrations of particles, and can be modelled by a wave of oscillating pressure. A single sound pressure wave is governed by an amplitude and frequency. The amplitude dictates the strength of the pressure wave and the frequency gives the number of full oscillations completed in one second. For a given point in space, the oscillating pressure looks like the following when measured against time:

We can see here that the wave peaks at a pressure of 1, which gives us the amplitude, \(A = 1\), and completes a full oscillation in 1 second meaning the wave has a frequency of \(f = 1\text{Hz}\), and the wave is defined by the cosine function of time, \(t\):
$$ \begin{equation} g(t)=A \cos(2\pi f t)  \label{eq:eq1} \end{equation}$$

A wave of this shape would only produce a simple sound of a single note. In reality the majority of audible sound is the result of many individual waves interacting to create a more complex sound. In Figure 2 we can see how three different sound waves with frequencies \(f = 1, 2\) and \(3 \text{Hz}\) interact to create a sound pressure field.


When the peaks of the waves meet, the pressure field is amplified, and where the the peaks meet the troughs the pressure cancels out. We call this process of overlapping waves to create a non wave-like pressure field superposition. Sound recorded by a microphone is much more likely resemble the pressure field on the right hand side of Figure 2, than the wave in Figure 1. However, for us to look at the complex pressure field it it difficult to identify its component frequencies. In fact, not only can certain sound fields be created by the superposition of waves, but any sound field can be produced by a unique combination of waves.

In signal processing and mathematical analysis, it is difficult to analyse a time-pressure field as the individual contributing waves behave independently. For example, different frequencies of sound can be absorbed at different rates by a damping material. Therefore, a tool called the Fourier Transform is used to convert the time-pressure field into a frequency dependent function. The transform was first proposed by a French mathematician, Joseph Fourier, in the 19th century. It converts a time-domain pressure field as discussed above, into a frequency-domain function, which is dependent of the frequency \(f\). The transform is given by the equation
$$\begin{equation} G(f)=\int_\infty^\infty g(t) \exp\{2\pi i f t\}\, dt, \label{eq:eq2}\end{equation}$$
where \(\exp\{-\}\) is the exponential operator, and \( i = \sqrt{-1}\).

Initially, the pressure field is given as a time-domain function, where the pressure changes over time. However, after the Fourier transform is applied, this dependence on time is removed, and the solution becomes a function of frequency, peaking where there are contributing frequencies. An important property of the Fourier transformed function is that the sum of two Fourier transforms is equal the Fourier transform of the sum of two waves. Therefore, we can analyse the behaviour of our Fourier transformed function as sound behaves differently at each frequency, and then transform the final product back into the time domain to get the resulting sound field.

In Figure 3 the first graph shows the resulting field of three waves in superposition, with frequencies \(f = 1/2\pi\), \(f = 1\) and \(f = 2\), given by
$$\begin{equation} g(t)=2\cos(t)+\cos(2\pi t)+\cos(4 \pi t). \label{eq:eq3}\end{equation}$$

The second graph in Figure 3 shows the Fourier transform of \((\ref{eq:eq3})\). The Fourier transform clearly has spikes at the three contributing frequencies in f, and the relative height of the spikes tells us about the relative strength of each wave to each other. In the initial function \((\ref{eq:eq3})\) we see that the \(f=1/2\pi\) wave has twice the amplitude in as the other two contributing waves, which is reflected in the Fourier transform, with the spike at this frequency being twice as large. The inverse Fourier transform can also be performed, to create a sound profile from a given frequency domain function.

In sound analysis the Fourier transform is very useful in helping us understand the composition of sound and how it changes as it passes through certain materials. This is because each sound frequency behaves differently in different materials, so to understand how a given sound changes we must first know which frequencies are contributing to the sound profile. The transform is also very useful in sound editing. If, for example you wanted to remove a high pitched noise from a recording, by taking the Fourier transform the high pitched frequency can be identified. Then by 'squashing' the peak at this high frequency and performing the inverse Fourier transform to return to a time-domain signal, the high pitched note will have been removed.

The Fourier transform is not limited to use in sound processing and analysis: it can be applied to any system where there is known to be an oscillating energy source which can operate at a continuous spectrum of frequencies. For example, the Fourier transform is used in a wide range of spectroscopy techniques such as Nuclear Magnetic Resonance (NMR) spectroscopy which uses the Fourier transform to analyse the magnetic fields of atomic nuclei.