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Dispersion (water waves)

physical science Maturity 11-13 Next article →

In the study of fluid dynamics, dispersion describes how waves behave as they move through a medium. Specifically, frequency dispersion occurs when waves with different wavelengths travel at different phase speeds. Water is considered a dispersive medium because waves on its surface move at speeds determined by their physical properties. These waves are often driven by restoring forces like gravity or surface tension. When a force acts to return the water surface to a flat state, it creates the movement we observe.

To understand this, we must look at the mechanics of wave propagation. A simple wave that maintains a constant shape is called a sine wave. This wave is defined by its amplitude, which is its height, and its phase function. Scientists use several variables to describe these waves, including wavelength, the period of the wave, and angular frequency. For a wave to exist in a medium, the angular frequency and the wavenumber must satisfy a specific mathematical relationship known as the dispersion relation.

Sine wave amplitude
Sine wave amplitude

There are different types of waves depending on which force is acting on the water. Surface gravity waves occur at the air-water interface where gravity is the primary restoring force. For these waves, the phase speed depends on both the wavelength and the depth of the water. In contrast, capillary waves are driven by surface tension rather than gravity. These smaller waves behave differently, as they actually propagate faster when they have shorter wavelengths.

Interfacial wave rigid boundaries
Interfacial wave rigid boundaries

Water depth plays a critical role in how gravity waves move. In shallow water, where the wavelength is much larger than the water depth, waves do not exhibit frequency dispersion. This means their speed remains constant regardless of their wavelength. However, as the water gets deeper, the rules change. Once the water reaches a depth greater than half the wavelength, it is considered deep water. In these deep-water conditions, waves with longer wavelengths travel faster than those with shorter wavelengths.

Dispersion gravity 1
Dispersion gravity 1

When multiple waves interact, they can form a wave group. This happens when two sinusoidal waves with slightly different wavelengths interfere with each other, creating a beat pattern. A wave group is a collection of waves that moves together as a single unit. This unit travels at a specific rate called the group velocity. It is important to distinguish this from the phase velocity, which is the speed of an individual wave crest. In deep water, the group velocity is exactly half of the phase velocity.

Wave group
Wave group

This difference in speed leads to surprising visual effects in the ocean. Because the group moves slower than the individual crests, new waves appear to emerge at the back of the group. These waves grow in height as they reach the center of the group and then vanish at the front. The group velocity is also significant because it represents the energy transport velocity. This is the speed at which the actual energy of the wave field moves horizontally through the water.

Wave group space time
Wave group space time

Measuring waves in a group can be tricky due to dispersion. If you count the waves in a group at a single moment in space, you will get one number. If you count the waves passing a single fixed point over a period of time, you will get a different number. In deep water, a wave group actually contains twice as many waves when measured in time as it does when measured in space. This mathematical reality is a direct consequence of the dispersion relation.

Wave group space time
Wave group space time

In the real world, the ocean is a complex system known as a sea state. A sea state is not just one wave, but a superposition of many different waves. These waves have different wavelengths, amplitudes, and directions all moving at once. Scientists use a power spectrum to describe the statistics of these complex surface patterns. By understanding how each individual component follows the dispersion relation, researchers can better understand the movement of the entire ocean.

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🖼️ Images & Media (8)
File:Sine wave amplitude.svg
Sine wave amplitude.svg
File:Dispersion gravity 1.svg
Dispersion gravity 1.svg
File:Dispersion gravity 2.svg
Dispersion gravity 2.svg
File:Wave group.gif
Wave group.gif
File:Wave group space time.svg
Wave group space time.svg
File:Wea00816.jpg
Wea00816.jpg
File:Wave disp.gif
Wave disp.gif
File:Interfacial wave rigid boundaries.svg
Interfacial wave rigid boundaries.svg
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