If you've ever marvelled at a rainbow, you have witnessed *dispersion* in action. Dispersion is where the speed at which a wave moves depends on its frequency (and so wavelength).

Dispersion appears in many settings: including water waves, ocean currents, atmospheric circulation, digital communication and quantum physics. Recent experimental observations are stimulating new results in the mathematical understanding of dispersion, and new mathematical techniques are helping to explain physical phenomena. These are all being explored in a six month programme running at the Isaac Newton Institute for Mathematical Sciences in Cambridge (INI). The programme *Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves* aims to encourage interactions amongst mathematicians, physicists, and engineers to bring together the latest theoretical advances and applications in this area.

Find out more about the programme below, including some accessible introductions to the concepts involved to help you explore this exciting area.

Why sine (and cosine) make waves — Find out why the perfect wave comes from a triangle, and a circle.

Give us a wave! — Ripples on a pond, the swell of ocean waves, your favourite song – these can all be described using sine waves. But how do we describe a sine wave?

Maths in a minute: Dispersion — Find out what dispersion is and get a glimpse of the maths it involves in this brief introduction.

From rainbows to rogue waves — Discover the fascinating maths behind rainbows, rogue waves and many more applications that is being explored by researchers at the INI.

Maths in a minute: Cartesian coordinates — Remind yourself of this important mathematical field of play.

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