Wavelength to Frequency Calculator

Every travelling wave obeys one relation between its speed, frequency and wavelength:

$v = f\lambda \quad\Longrightarrow\quad f = \frac{v}{\lambda}, \quad \lambda = \frac{v}{f}$

Symbols and SI units:

• $v$ — wave speed, metres per second (m/s). For light in vacuum $v = c = 2.998 \times 10^8$ m/s; for sound in air at 20 °C, $v \approx 343$ m/s.
• $f$ — frequency, hertz (Hz), meaning cycles per second
• $\lambda$ — wavelength, metres (m)

When it applies: to any periodic wave in a uniform medium.

The assumption people forget: $c$ is the speed of light in vacuum. Inside a medium of refractive index $n$ the speed is $v = c/n$, so the wavelength shortens to $\lambda/n$ while the frequency stays the same. Frequency is fixed by the source; wavelength is not.

Two further ways to state the same timing are the period $T = 1/f$, in seconds, and the angular frequency $\omega = 2\pi f$, in radians per second. A $50$ Hz mains wave has a period of $20$ ms.

For electromagnetic radiation, frequency also sets the energy of a single photon:

$E = hf = \frac{hc}{\lambda}$

• $E$ — photon energy, joules (J); divide by $1.602 \times 10^{-19}$ to convert to electronvolts (eV)
• $h$ — Planck's constant, $6.626 \times 10^{-34}$ J·s

Spectroscopists often quote the wavenumber instead, the number of cycles per centimetre:

$\tilde{\nu} = \frac{1}{\lambda} \quad [\text{cm}^{-1}]$

with $\lambda$ expressed in centimetres.

Unit trap: wavelengths are usually quoted in nanometres ($1\ \text{nm} = 10^{-9}$ m) and frequencies in MHz or GHz ($1\ \text{MHz} = 10^6$ Hz). Convert everything to metres and hertz before dividing, or the exponent will be wrong by several orders of magnitude.

The assumption people forget: $E = hf$ is the energy of one photon, not of a beam.

• Using $c$ for sound — sound in air travels at about $343$ m/s, roughly a million times slower than light. Using $3 \times 10^8$ m/s for a sound problem is off by six orders of magnitude.
• Forgetting the nano prefix — $500$ nm is $5.00 \times 10^{-7}$ m, not $500$ m.
• Assuming wavelength is unchanged in glass or water — the frequency is what stays constant when light enters a medium; the wavelength divides by $n$.
• Mixing metres and centimetres in the wavenumber — $\tilde{\nu}$ in cm⁻¹ needs $\lambda$ in cm.
• Rounding $c$ too early — using $3 \times 10^8$ m/s instead of $2.998 \times 10^8$ m/s introduces a $0.07\%$ error, which matters in spectroscopy.
• Confusing period with frequency — they are reciprocals, so a period of $0.02$ s corresponds to a frequency of $50$ Hz.
• Using the wrong speed for a wave on a string — that speed is set by tension and linear density, $v = \sqrt{T/\mu}$, and has nothing to do with the speed of light.