What are Waves?
Waves of different forms are all around us, ranging from sound waves, electromagnetic waves, including visible light and radio waves, to large scale high and low pressure systems in the atmosphere, called Rossby waves, that affect our weather. Surface waves are another familiar type of wave which can bring a lot of joy to children and surfers, or in large storms can batter boats and shorelines. Waves affect our lives in many ways.
Waves also have nonlinear behaviours. This can manifest itself in different ways:
One example is surface water waves for which the wave crests can propagate faster than the troughs. This can been easily seen at the beach where shoaling waves break and topple over as the wave crest moves ahead of the base of the wave.
Nonlinearity can also result in waves of different amplitude propagating at different speeds and interacting in other strange ways.
This shows a large wave catching up to a slower moving smaller wave. Just before it reaches it, the large wave transfers mass and energy to the small one. The large wave shrinks and slows down while the small wave grows and accelerates.
These are special types of waves called solitons.
Surface waves of this type were first discovered in the 1834 by John Scott Russell on the Edinburgh-Glasgow canal (Russell was a boat designer among other things, interested in designing fast boats for horses to pull along the canals). The fascinating interaction properties of these waves weren't discovered until the 1960s. Since then many other types of waves with soliton solutions have been found, including electromagnetic waves in fiber optic cables. This is part of the field on nonlinear optics.
Waves and Applied Mathematics
Some waves have very strange properties.
Did you know, for example, that the energy associated with some types of waves can move with a different velocity than wave crests? Sometimes it can even go in a different direction!
For some types of waves, if you see the crests coming towards you the wave packet (a group of waves) can actually be going away from you. How can this be you ask? As a student in Applied Mathematics you will have the opportunity to learn about this and other aspects of the weird and fascinating world of waves.
An introduction to waves begins in courses in Partial Differential Equations and continues in courses in Continuum and Fluid Mechanics.