# Wavelength

In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenomena are light, water waves, and sound waves.

File:Wavelength.svg
Wavelength of a sine wave.

In a wave, a property varies with the position. For example, this property can be the air pressure for a sound wave, or the magnitude of the electric or the magnetic field for light. The wavelengths of frequencies audible to the human ear (20 Hz–20 kHz) are between approximately 17 m and 17 mm, respectively. Visible light ranges from deep red, roughly 700 nm to violet, roughly 400 nm (430–750 THz). For other examples, see electromagnetic spectrum.

## Relationship with frequency

Wavelength λ is inverse proportional with the frequency, the number of periods per time unit passing a given point, as in

$\lambda = \frac{v_w}{f},$

where $v_w$ is the phase velocity of the wave. In the case of electromagnetic radiation, such as light, in a vacuum, this speed is the speed of light, 299,792,458 m/s or about 3x108 m/s. For sound waves in air, this is the speed of sound, 345 m/s (1238 km/h) in air at room temperature. Usually, SI units are used, where the wavelength is expressed in meters, the frequency in Hz, and the propagation velocity in meters per second.

It should be noted that for many wave phenomena, wavelength is not the distance that particles travel during a period. For instance, in acoustics and water waves, the particle displacements during a period are only a small fraction of the wavelength, apart from extreme conditions like breaking waves and shock waves.

### In non-vacuum media

The speed of light in most media is lower than in vacuum, which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum. The wavelength in the medium is

$\lambda'=\frac{\lambda_0}{n},$

where n is the refractive index of the medium. Wavelengths of electromagnetic radiation are usually quoted in terms of the vacuum wavelength, unless specifically indicated as the "wavelength in the medium". In acoustics, unless otherwise specified, the term wavelength is used to mean the wavelength in the medium.

## De Broglie wavelength of particles

Louis de Broglie postulated that all particles with momentum have a wavelength

$\lambda = \frac{h}{p}$

where h is Planck's constant, and p is the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a De Broglie wavelength of about 10-13 m.