Wavelength

In physics and mathematics, wavelength is a fundamental way to describe a wave. It is defined as the spatial period of a wave or a periodic function. This means it is the distance over which the wave's shape repeats itself.

You can measure wavelength by finding the distance between any two consecutive corresponding points of the same phase. For example, you might measure from one crest to the next crest, or from one trough to the next. You can also measure between two zero crossings. Scientists often use the Greek letter lambda (λ) to represent wavelength.

How a wavelength behaves depends on the wave's speed and its frequency. If a wave moves at a constant speed, wavelength is inversely proportional to frequency. This means that waves with a higher frequency will have shorter wavelengths. Conversely, waves with a lower frequency will have longer wavelengths.

This relationship is expressed mathematically by the formula where wavelength equals the phase speed divided by the frequency. In a dispersive medium, the phase speed itself changes based on the frequency, which makes the relationship between wavelength and frequency nonlinear.

Waves also change when they move from one medium to another. A medium is the substance, such as air, water, or a vacuum, that a wave travels through. When a wave enters a new medium, its speed often changes. This change in speed causes refraction, which is a change in the wave's direction.

You can imagine this like a column of marching soldiers moving from solid ground into mud. For electromagnetic waves, this change in angle is governed by Snell's law. If the speed of light in a medium is lower than in a vacuum, the wavelength will also decrease.

This change in speed can also depend on the specific wavelength, a process called dispersion. Dispersion is responsible for the way a prism separates light into different colors. This happens because the refractive index of the prism varies with the wavelength. As a result, different wavelengths travel at different speeds and refract at different angles.

This phenomenon is described by a dispersion relation, which is the mathematical rule for how speed varies with wavelength.

We can see many different types of wavelengths in the natural world. For example, visible light has very tiny wavelengths. It ranges from roughly 700 nm for deep red to about 400 nm for violet.

Sound waves are much larger than light waves. In air at room temperature, the speed of sound is 343 m/s. The frequencies humans can hear, between 20 Hz and 20 kHz, result in wavelengths between 17 m and 17 mm. Bats use even higher frequencies to help them resolve targets smaller than 17 mm.

Sometimes waves do not travel, but instead stay in one place. This is known as a standing wave. A standing wave is an undulatory motion that stays in one position. It can be viewed as the sum of two traveling waves moving in opposite directions.

These waves include stationary points called nodes, where there is no motion at all. In a sinusoidal standing wave, the wavelength is exactly twice the distance between these nodes.

Finally, wavelength can behave differently in complex environments. In the ocean, a wave approaching the shore may have a varying local wavelength. This depends on the height of the wave compared to the depth of the sea floor.

In crystalline solids, waves move through a regular lattice of atoms. Because these atoms are arranged in discrete positions, the wavelength can be viewed in multiple ways, a concept known as wavelength indeterminacy.

Even in shallow water, waves can form non-sinusoidal shapes, such as cnoidal waves, which have sharper crests and flatter troughs.