 It was during Fourier's time in Grenoble that his mathematical and scientific work reached its peak. He had developed an intense interest in heat energy and used to keep his home unpleasantly warm and wore thick, heavy coats. This eccentricity enabled him to achieve significant developments in his theory of heat flow. He applied mathematical techniques to the theory and wrote a memoir entitled On the Propagation of Heat in Solid Bodies. It was while he was working on this theory that he proposed the idea that Any function of a variable, whether continuous or discontinuous, can be expanded in a series of signs of multiples of the variable. We'll look at what that actually means in a moment. But this is the idea which became known as the Fourier series. He presented his memoir to the Paris Institute in 1807 and also to a committee that comprised of Le Grange, Mange, Le Coing and Laplace. Although this memoir is highly regarded now, at the time of its proposal it was controversial. Le Grange and Laplace both challenged the idea and it wasn't until 1829 that the German mathematician Peter Gustav Nejeune de Richelais actually managed to demonstrate Fourier's ideas and lay down the conditions under which the theory held. It was de Richelais' work that provided the foundation of what would become known as the Fourier transform. So what does Fourier's theory actually mean? Any function of a variable, whether continuous or discontinuous, can be expanded in a series of signs of multiples of the variable. I personally find abstract concepts like this difficult to grasp. I like to be able to apply the things I learn to something practical. I admit that this is not always possible. However, where Fourier is concerned, his theory is so useful that there are many everyday applications to which we can apply it. So we need an everyday variable that we can find a function of. A function is another way of saying a signal. And one of the variables we can use when looking at signals is time. Throughout this course we're going to apply Fourier's theory using one particular type of time-dependent signal. A signal that is very dear to my heart as a musician. Sound. So with my apologies to Fourier, I'm going to rewrite his theory in a slightly simpler way, as it applies to sound signals. Any sound can be broken down into a series of sine waves at many different frequencies. Sounds are made when objects such as this violin string vibrate. The two-and-fro motion of the string pushes the air molecules around it together and then pulls them apart again, causing a wave of pressure changes in the air. These pressure changes are picked up by our ears and interpreted by our brains as sound. Many objects can generate sound, such as the vibrating cone of a speaker. But what does sound actually look like? This might seem like a bit of a strange question. Sound is something we hear, not see. In the next lecture, we're actually going to see a sound wave, as we set out on our journey into the maths behind the magic of the Fourier transform. So let's begin, like all good scientists, with a question. What is sound?