Formula of Wavelength: λ = v/f, Unit, de Broglie and Examples

The formula of wavelength is $\lambda = \frac{v}{f}$ : divide the speed of the wave by its frequency. The answer is the length of one complete wave, measured from one crest to the next, in metres. A sound wave moving at 340 m/s with a frequency of 425 Hz has a wavelength of 0.80 m.

Light and radio waves cross empty space at c, so $\lambda = \frac{c}{f}$ . For a moving particle such as an electron, quantum physics gives a different wavelength formula physics students meet in Class 12, the de Broglie relation $\lambda = \frac{h}{p}$ . This page covers all three, with the unit conversions that trip people up most: nanometres, megahertz and gigahertz.

Quick facts

Formula
$\lambda = \frac{v}{f}$ , also $\lambda = vT$
Light in vacuum
$\lambda = \frac{c}{f}$
Matter waves
$\lambda = \frac{h}{p} = \frac{h}{mv}$
SI unit
metre (m); light often in nanometres, 1 nm = 10⁻⁹ m
Dimensional formula
[M⁰ L¹ T⁰]
Symbol
λ (Greek lambda)
In a medium
$\lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n}$

Wavelength formula physics symbols

Symbols for finding wavelength
Symbol	Quantity	SI unit
$\lambda$	Wavelength, the length of one cycle	m
$v$	Wave speed in the medium	m/s
$f$	Frequency set by the source	Hz
$T$	Time period of the wave	s
$h$	Planck's constant (value 6.626 × 10⁻³⁴)	J s
$p$	Momentum of a particle, mv	kg m/s

On a displacement against distance graph, the wavelength is the horizontal gap between two crests, two troughs, or any two points moving in step. On a displacement against time graph, the same gap is the period, not the wavelength. Read the axis label before you measure.

Why λ = v / f

In one period T the source completes one oscillation and the wave front moves forward by one wavelength. Distance divided by time is speed, so $v = \frac{\lambda}{T}$ . Since $T = \frac{1}{f}$ , that becomes $v = f\lambda$ . Dividing both sides by f gives the formula of wavelength.

The speed depends on the medium. The frequency depends on the source. The wavelength is whatever fits between them. So when a wave moves into a slower medium its wavelength gets shorter, and in a faster medium it gets longer, while f stays fixed. See the wave velocity formula hub for the speeds in strings, gases and solids, and the formula of frequency for the reverse problem.

Examples with the formula of wavelength

Example 1: a tuning fork

A 425 Hz tuning fork sounds in air where sound travels at 340 m/s. Find the wavelength.

$\lambda = \frac{340}{425} = 0.80\ \text{m}$

Example 2: an FM radio station

A station broadcasts at 92.7 MHz. What is the wavelength of its signal?

Convert: f = 92.7 × 10⁶ Hz.

$\lambda = \frac{3.00 \times 10^{8}}{9.27 \times 10^{7}} = 3.24\ \text{m}$

Example 3: inside a microwave oven

A microwave oven runs at 2.45 GHz. Find the wavelength in centimetres.

$\lambda = \frac{3.00 \times 10^{8}}{2.45 \times 10^{9}} = 0.122\ \text{m} = 12.2\ \text{cm}$

The uneven heating pattern inside an oven has hot spots spaced by about half of this.

Example 4: an electron as a wave

An electron (mass 9.109 × 10⁻³¹ kg) moves at 1.0 × 10⁶ m/s. Find its de Broglie wavelength.

$\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34}}{(9.109 \times 10^{-31})(1.0 \times 10^{6})} = 7.3 \times 10^{-10}\ \text{m}$

About 0.73 nm. That is comparable to the gaps between neighbouring atoms in a solid, so a crystal lattice can diffract a beam of electrons.

Wavelengths across the spectrum

Typical wavelengths
Wave	Wavelength
Long radio waves	100 m to several km
Microwaves	about 1 mm to 1 m
Visible light	about 400 nm (violet) to 700 nm (red)
X rays	about 0.01 nm to 10 nm
Audible sound in air	about 1.7 cm to 17 m

Sound in air spans 20 Hz to 20 kHz, which with v = 340 m/s gives the range in the last row. Why does every electromagnetic wave in the table share one speed? The speed of light formula answers that, while the speed of sound formula covers the sound row.

Where the marks go missing

Leaving nanometres unconverted. Multiply by 10⁻⁹ before using λ in any formula with SI units.
Using MHz as Hz. 92.7 MHz is 9.27 × 10⁷ Hz, not 92.7 Hz.
Measuring a period off a time graph and calling it λ.
Assuming the wavelength stays the same in glass or water. It shrinks by the factor n, as the index of refraction formula explains.
Mixing up the de Broglie p with pressure. Here p is momentum, mass times velocity.

Rough check: a higher frequency in the same medium must give a shorter wavelength.

Frequently asked questions

What is the formula of wavelength?

Wavelength equals wave speed divided by frequency, λ = v/f. For electromagnetic waves in vacuum, λ = c/f with c = 3.00 × 10⁸ m/s.

Which unit is wavelength measured in?

The metre. Light wavelengths are usually quoted in nanometres (10⁻⁹ m) and sound wavelengths in centimetres or metres.

How do you find wavelength from a graph?

On a graph of displacement against distance, measure the distance between two neighbouring crests or troughs. That distance is one wavelength.

Does wavelength change when light enters water?

Yes. The speed drops by the factor n and the frequency stays the same, so the wavelength also drops by n. In water, with n = 1.33, it falls to about three quarters of its value in air.

What is the de Broglie wavelength formula?

For a particle of momentum p, λ = h/p = h/(mv). It applies to electrons, neutrons and other particles and is part of the Class 12 dual nature chapter.