Think of a wave – what comes to mind? The ocean swell rocking a boat? Ripples on a pond? The loop moving down a whip as it is cracked? Or perhaps your favourite song pumping out of your stereo speaker? Waves happen when a disturbance (of, say, the flat surface of the water by the ripple, the stillness of the air by a sound wave) moves over time.

We had a wonderful time playing with waves when we writing about a research programme that is currently taking place at the Isaac Newton Institute for Mathematical Sciences. Mathematically the simplest and most perfect wave is described by a sine (or cosine) function. You can read a simple introduction to them in Why sine (and cosine) make waves. For example, the function $$y(x,t)=A\sin (\frac{2\pi }{\lambda }x-vt)$$ describes a sine wave travelling in the $x$ direction with speed $v$ over time $t$. Here $A$ is the amplitude – the maximum displacement of the wave from equilibrium. And $\lambda$ is the wavelength – the distance between two peaks of the wave.

Exert your power over the waves by finding out all the different ways you can describe them in the longer version of this article!

Return to the Plus advent calendar 2022.

This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from our collaboration here. The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.