# Deep 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, you will see how the wavelength of a particular wave impacts the wave's speed in deep waters. You will also learn why this calculation only works for deep water waves.

## Deep Water Waves

Surfers know waves. They wait for the perfect wave to come to shore and then they're off. But just how fast do these waves travel? Just think, a wave might have traveled across the whole ocean before coming to shore. In an idealized ocean, how fast this wave travels across the ocean's waters can be easily calculated with a simple formula. This simple calculation does give you a good estimate of its actual speed. In the real world, though, this actual speed is different from the idealized speed because there are other factors that play a role, such as weather and animals getting in the way.

Mathematically, deep water waves are defined as those occurring in ocean depths greater than twice the wavelength of the wave. For example, if the wavelength of a particular wave is 10 meters, then the wave is considered a deep water wave if the depth of the ocean is greater than 5 meters where the wave is occurring.

## Speed Calculation

In an idealized ocean with no other effects on the wave, the wave's speed in deep water can be calculated with this formula.

This is a limiting case of the formula for traveling waves at all depths.

At depths greater than half the wavelength, the hyperbolic tangent function approaches 1 and you get the aforementioned formula for the wave speed in deep waters. To use this formula, all you need to do is plug in the acceleration of gravity, g, along with the wavelength of the wave and then you'll have your wave speed, v.

## Wavelength and Wave Speed Relationship

As you can see, the wave's speed is related to its wavelength. Longer wavelengths have higher wave speeds. Waves with shorter wavelengths travel at a slower speed. Since the acceleration of gravity is treated as a constant, you can find the relationship between the wavelength of a wave and the speed by plugging it into the formula and evaluating that along with the 2 pi. You get this.

So the wave speed is approximately 1.249 times the square root of the wavelength. Reversing it, you can say that the wavelength is approximately the square of the wave speed divided by 1.249. This is for units of meters and seconds.

On the other hand, if you know or also know the period, T, of the wavelength, you can use this formula to calculate the wave speed.

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