# wavelength The wavelength is graphically illustrated the distance between two neighboring wave crests or, more generally, between two neighboring points of the same phase (these are points with the same deflection and the same slope ).

The wavelength ( Greek : lambda ) of a periodic wave is the smallest distance between two points in the same phase . Two points have the same phase if they have the same deflection (elongation) and the same direction of movement over time. The wavelength is the spatial analogue of the time period . ${\ displaystyle \ lambda}$ In general

${\ displaystyle \ lambda = {\ frac {c} {f}} \,}$ where is the phase velocity and the frequency of the wave. ${\ displaystyle c}$ ${\ displaystyle f}$ However, at a given frequency, the phase velocity and wavelength depend on the propagation medium and the geometry of the wave. If necessary, the distinction between vacuum wavelength and free space wavelength is used if one does not mean the wave in the medium or the wave in a waveguide .

## Wavelength of sound waves

The human ear is sensitive to frequencies of a maximum of around 16  Hertz to 20,000 Hertz (this corresponds to a wavelength range of around 21 m to 17 mm with a sound propagation speed in the medium of air of  = 343 m / s), whereby the ability to perceive higher frequencies in the Usually wears off with age. Since the wavelength is proportional to the speed of sound propagation in the propagation medium, a sound with a frequency of 16 Hertz in water (  = 1484 m / s) has a wavelength of around 90 m. The auditory impression, the pitch , is given by the frequency, not by the wavelength in a medium outside the ear, since the sound propagation speeds of the media in the inner ear - and thus the wavelengths of a certain tone occurring there - are independent of the media through which the sound is produced reaches the eardrum . Certain animal species can also perceive sound waves with lower or higher frequencies, hence also sound of other wavelength ranges. ${\ displaystyle c}$ ${\ displaystyle c}$ ### Visible light wavelengths: colors

The human eye is sensitive in a wavelength range from around 380  nm ( violet ) to 780 nm ( red ). Bees also see shorter-wave radiation ( ultraviolet ), but cannot perceive red light.

### Wavelength of electromagnetic waves in the medium

The following applies to the wavelength in a medium:

${\ displaystyle \ lambda ^ {\ prime} = {\ frac {\ lambda _ {0}} {\ sqrt {\ mu _ {\ rm {r}} \ varepsilon _ {\ rm {r}}}}} = {\ frac {c} {f}} {\ frac {1} {\ sqrt {\ mu _ {\ rm {r}} \ varepsilon _ {\ rm {r}}}}}}$ It is the speed of light in a vacuum, the magnetic permeability and the relative permittivity of the medium. If electromagnetic waves cross a medium with a refractive index greater than , this reduces the wavelength and the speed of propagation. The frequency of the wave remains the same. The wavelength in the medium is ${\ displaystyle c}$ ${\ displaystyle \ mu _ {\ rm {r}}}$ ${\ displaystyle \ varepsilon _ {\ rm {r}}}$ ${\ displaystyle n}$ ${\ displaystyle 1}$ ${\ displaystyle \ lambda ^ {\ prime} = {\ frac {\ lambda _ {0}} {n}},}$ where is the wavelength of the electromagnetic wave in vacuum. ${\ displaystyle \ lambda _ {0}}$ ## De Broglie wavelength

Louis de Broglie discovered that all particles can be described by matter waves . The wavelength of such a matter wave is called the De Broglie wavelength and depends on the momentum p of the particle. For a relativistic particle, the wavelength can be determined using the following equation:

${\ displaystyle \ lambda = {\ frac {h} {p}} = {\ frac {h} {mv}} {\ sqrt {1 - {\ frac {v ^ {2}} {c ^ {2}} }}}}$ Here h is Planck's quantum of action , c the speed of light , m the mass and v the speed of the particle.