Wave Tank

  Turn the handle to create five waves. Measure how much overspill this makes.
  Try the different coastal defences: vertical wall, slopes, steps, rocks, curved wall. Which results in the least amount of flooding?
  What are the pros and cons of each defence? Are they cheap or expensive? Do they create a lot of spray? Would they look attractive on the beach?

Ocean waves are created by wind and storms. Studying the motion of waves is important in the creation of effective coastal defences.
Important features of a wave include its height, known as amplitude and the distance between crests (or peaks), known as the wavelength.

The speed of the wave can then be calculated by dividing the wavelength, by the time taken for the wave to travel this distance, known as the period.

Although the crest of the wave is travelling, underneath the surface, small volumes of water are relative stationary, and are simply moving in circles. These circular orbits become more squashed and elliptical with depth. The shape of this wave is called a trochoid.

Modelling wave motion can be complicated, especially when you include factors such as turbulence. However, a sine wave can be used as a good approximation to calculate the height, shape, speed and energy of the wave.

Coastal engineers use experimental data from around the world, coupled with the maths of wave motion to predict flooding. Simulations can be designed to find the most effective coastal defences for each beach.

In general, the most effective coastal defences are rocks or steps as these create turbulence and dissipate wave energy. Alternatively, a slope combined with a curved wall will deflect any overspill back towards to the sea.


Coastal defences have existed since ancient times. Two-thousand years ago, Romans built a seawall on the Mediterranean sea, creating an artificial harbour which still exist today.

In the 19th century, mathematicians began describing the motion of waves mathematically. This included the shape and speed of a wave, as seen on oceans and used by coastal engineers today. Solutions to these problems can be difficult to approximate, so these equations are often simplified to give us approximations that are good enough for practical applications.

George Airy 1801 – 1892
George Airy was an English mathematician and astronomer famous for his work on planetary motion and calculating the mean density of the Earth.
In 1841, Airy was the first to describe wave motion mathematically. From this description we can derive properties such as shape, speed and wave energy.
George Stokes 1819 – 1903
George Stokes was an Irish mathematician famous for his work in fluid dynamics, which describes the motions and properties of waves.
The Navier-Stokes equations describe the direction and speed of fluid at each point of the fluid and at each point in time. These equations can help describe everything from the ripples in a pond, to why clouds hang in the air.

As sea levels rise, coastal defences will become even more important to prevent flooding. Coastal engineers use experimental data, and the maths of wave motion, to design coastal defences.

Thousands of experiments have been performed that measure the effectiveness of various types of coastal defences. This data is then used to predict the expected flooding that might occur at a given beach, allowing coastal engineers to design the most effective defences.

In computer animation, the maths of wave shape and wave motion essential in creating realistic-looking oceans.

Maths at Home

The shape of a wave is called a trochoid, and it is the shape traced by a point on a circle as it rolls along the ground.

The point tracing the wave doesn’t have to be on the edge of the circle, it can be a point inside the circle as well. This will create smaller, flatter waves.

Try to make a wave shape yourself using this online Spirograph (called Inspirograph) https://nathanfriend.io/inspirograph/

What do you need to make a wave shape? How does the shape change when you use a point closer to the edge or closer to the centre?

From the circle, what is the distance between peaks? What is the height of the wave?