The Mathematics of Waves

Aug 26, 2022 | Cherry Hill

This summer has been the perfect opportunity to get out to the beach and enjoy the sun, sand, and waves. It's amazing to see how rich the structure of ocean waves is. The structure of a wave is the web of math and physical processes that go into making a wave. Everything, from the speed of the wave to the pressure of the water, is connected. Ocean waves come in very different sizes, as can all waves. To fully understand waves, we need to understand measurements associated with these waves, such as how often they repeat (their frequency), how long they are (their wavelength), and their vertical size (amplitude).

Waves and circles

Due to the fact that waves repeat themselves over a consistent period of time and distance, it is possible to analyze their shape mathematically. This behavior mirrors the repetition of the circle. Imagine drawing a circle on a piece of paper. If you had drawn that same shape while your friend slowly pulled the paper out from under your pencil, the line you would have drawn would have traced out a wave. Cycles of rising and falling in the wave are completed by one rotation around the circle. Mathematicians use the sine function (Sin) to express the shape of a wave. The mathematical equation representing the simplest wave looks like this:

y = Sin(x)

What Causes Water Waves

Waves are formed by energy passing through water, causing it to move in a circular motion. Water particles travel only in a small circle as a wave passes. The Wind provides energy to the waves. The Wind causes waves to travel in the ocean and the energy is released on coastlines. Three factors influence how big these waves are:

– The speed of the wind.

– The distance the wind travels over the water is called the “fetch”.

– The length of time the waves travel for.

Out in the open ocean, storms create the biggest surf waves. Storms are caused by gradients between low pressure and high pressure, just like the gradient between cold air in Alaska and warm air in the Pacific Ocean. A storm wave starts as a huge, choppy wave that gradually merges into a smooth line of peaks called a "swell". Large waves with the most energy are formed by a strong wind blowing over a long period of time over a long distance (or “fetch”). Eg. Wave Energy = Wind Speed x Wind Duration x Fetch Distance

There will be a long "period" of time between successive waves so that breaking waves will not interfere with each other as they evolve into lines of long waves (called "swell"). As the longer period waves travel faster, when they eventually reach a shoreline and break, they produce more kinetic energy.

Myth Buster: You might have heard that waves are caused by the moon’s gravitational pull. That’s not entirely true. While the moon does influence low and high tides through something called a tidal force, and that can affect the movement of the earth's oceans, it is not the main cause for waves. Wind and Temperature fluctuation are.

Understanding the mathematics of waves is big business and it is used for designing artificial surfing reefs, computer wave modeling for testing ship designs, bridges, and coastal construction, and for converting wave energy into electricity.