Shallow Water Waves: Definition & Speed Calculation

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, we'll go over a simple formula that lets you easily calculate the wave speed for waves that occur in shallow water. What's shallow water? Read on to find out.

Shallow Water Waves

Did you know that waves don't just happen at the shoreline? They actually travel. Surfers know this and they wait for just the right wave to come from the ocean towards the shore.

Waves in different parts of the oceans are referred to by different names. Those differences are dependent on the wavelength.

A deep water wave is one that occurs at depths greater than half a wave's wavelength.

A shallow water wave is one that occurs at depths shallower than the wavelength of the wave divided by 20. A wave with a wavelength of 100 meters is a shallow water wave at depths less than 100 / 20, or 5 meters, while a wave with a wavelength of 200 meters is called a shallow water wave at depths less than 200 / 20. or 10 meters.

The Formula

The formula to calculate the speed of a shallow water wave is:

shallow water waves


  • g is the acceleration of gravity, which is considered a constant in most physical and mathematical calculations on earth and is about 9.8 m/s2.
  • d is the depth where the shallow water wave is occurring.

Water Depth and Wave Speed

Looking at the formula, you can see that the wave speed really depends on the depth of the ocean water where the wave is happening. This formula works in any body of water, but it is idealized in that it doesn't take into consideration other particles and objects such as big fish or giant algae that can impede the flow of a wave.

However, it still gives a good approximation of the actual wave speed, and is therefore useful in finding a rough estimate of actual speed.


Let's look at an example of this formula in use. A shallow water wave has a wavelength of 120 meters. What is its speed at a depth of 5 meters?

Using the formula, go ahead and multiply 9.8 m/s2 by 5 meters.

Then take the square root to get roughly 7 m/s.

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