Define and explain wave dispersion.

Wave dispersion is the phenomenon where waves of different frequencies travel at different speeds in a medium.

In more detail, wave dispersion is a characteristic of a system where the phase velocity (the speed at which a wave phase propagates) is dependent on frequency. This means that waves with different frequencies (or wavelengths) will travel at different speeds. This is often observed in waves travelling through a medium, such as light through glass or sound through air.

The most common example of wave dispersion is probably the rainbow. When white light (which is a combination of light of different colours, and therefore different frequencies) passes through a prism or raindrops, it is dispersed into its constituent colours. This happens because the speed of light in the medium (glass or water) is dependent on its frequency - a phenomenon known as refractive dispersion.

Dispersion is also a crucial concept in telecommunications, where signals of different frequencies must travel long distances without losing their information. In an optical fibre, for instance, dispersion can cause a broadening of signal pulses, which can lead to information loss. To counteract this, engineers must carefully design the fibre and the signal to minimise dispersion.

In physics, the study of wave dispersion can provide important insights into the properties of the medium the waves are travelling through. For example, seismologists use the dispersion of seismic waves to learn about the Earth's interior. The speed of these waves changes with their frequency, and by analysing these changes, scientists can infer details about the Earth's layers.

In summary, wave dispersion is a fundamental concept in wave physics, with wide-ranging applications from the creation of rainbows to the design of telecommunications systems. It describes the dependence of wave speed on frequency, leading to the spreading out of a wave form in space and time.