Wavelength

The wavelength is the distance between repeating units of a wave pattern. It is commonly designated by the Greek letter lambda (λ).

In a sine wave, the wavelength is the distance between the midpoints of the wave:

The x axis represents distance, and I would be some varying quantity at a given point in time as a function of x, for instance air pressure for a sound wave or strength of the electric or magnetic field for light.

Relationship with frequency

Wavelength λ has an inverse relationship to frequency f, the number of peaks to pass a point in a given time. The wavelength is equal to the speed of the wave type divided by the frequency of the wave. When dealing with electromagnetic radiation in a vacuum, this speed is the speed of light c. For sound waves in air, this is the speed of sound in air. The relationship is given by:

\lambda = \frac{v_w}{f}

where

λ = wavelength of a sound wave or electromagnetic wave

v_w

is the speed of propogation of the wave, and

f = frequency of the wave in 1/s = Hz.

For radio waves this relationship is approximated with the formula: wavelength λ (in metres) = 300 / frequency (in megahertz).

For sound waves this relationship is approximated with the formula: wavelength λ (in metres) = 343 / frequency (in hertz).

Note that the speed of light $c = {3}\times{10^8} \frac{m}{s}$ in a vacuum, and thus in a vacuum $\lambda = \frac{c}{f}$ or $\lambda = \frac{f}$

In non-vacuum mediums

When light waves (and other electromagnetic waves) enter a medium, their wavelength is reduced by a factor equal to the refractive index n of the medium but the frequency of the wave is unchanged. The wavelength of the wave in the medium, λ' is given by:

\lambda^\prime = \frac{\lambda_0}{n} where:

λ₀ is the vacuum wavelength of the wave

Wavelengths of electromagnetic radiation, no matter what medium they are travelling through, are usually quoted in terms of the vacuum wavelength, although this is not always explicitly stated.

Quantum wavelength of particles

Louis de Broglie discovered that all particles with momentum have a wavelength associated with their quantum mechanical wavefunction, called the de Broglie wavelength:

\lambda = \frac{h}{p}

where

h is Planck's constant, and

p is the momentum of the object.

In general, large particles have much smaller wavelengths than photons. For particles and objects, their wavelength depends on their speed, as well as their mass.

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Relationship with frequency

In non-vacuum mediums

Quantum wavelength of particles

See also