Wavenumber Calculator
Enter a wavelength or wavenumber and this calculator instantly returns the spectroscopic wavenumber (cm-1), angular wavenumber (rad/m), frequency (Hz), photon energy in both joules and electron-volts, and the electromagnetic-spectrum region. Switch the solve direction to go from wavenumber back to wavelength, or use frequency as the starting point. All constants are CODATA 2018 values.
What is wavenumber and why do spectroscopists use it?
Wavenumber (symbol nu-tilde or k-bar, unit cm-1) is the number of complete wave cycles that fit into one centimetre of space. It is the reciprocal of wavelength expressed in centimetres, so a 500 nm visible photon has a wavelength of 5 x 10-5 cm and a wavenumber of 1 / (5 x 10-5) = 20 000 cm-1. Scientists who work with infrared spectroscopy, Raman spectroscopy, and molecular vibrations prefer wavenumber over wavelength because it is directly proportional to energy: doubling the wavenumber exactly doubles the photon energy. That proportionality makes it easy to compare transition energies across a spectrum without mental division. Most laboratory IR spectra are plotted on a wavenumber axis running from about 400 to 4000 cm-1, covering characteristic bond stretches and bends of organic molecules.
Spectroscopic wavenumber vs. angular wavenumber
Two quantities share the name "wavenumber." The spectroscopic wavenumber (nu-tilde) equals 1 / lambda expressed in centimetres and has the unit cm-1. The angular wavenumber k equals 2*pi / lambda in SI units (metres) and has the unit radians per metre. Angular wavenumber appears in physics and engineering when describing the phase of a wave: the wavefunction of a photon or particle is written as exp(i k x), where x is position. In chemistry and spectroscopy the spectroscopic cm-1 form dominates; in optics and quantum mechanics the angular form is standard. This calculator reports both. To convert: k (rad/m) = 2*pi * nu-tilde (cm-1) * 100.
Key IR absorption bands and common wavenumber ranges
Mid-infrared spectroscopy (400-4000 cm-1) is the workhorse of organic chemistry because most molecular vibrations fall in this range. Carbonyl groups (C=O) absorb strongly near 1650-1800 cm-1 depending on the functional group: esters near 1740, ketones near 1715, carboxylic acids near 1710, and amides near 1650. O-H stretches produce a broad band around 3200-3600 cm-1, while N-H stretches appear near 3300-3500 cm-1. Aliphatic C-H stretches cluster just below 3000 cm-1, and aromatic C-H stretches sit just above. The fingerprint region below 1500 cm-1 contains complex skeletal vibrations unique to each molecule, making it useful for compound identification by spectral matching.
How this calculator converts between quantities
All calculations use CODATA 2018 exact values: speed of light c = 299 792 458 m/s and Planck constant h = 6.626 070 15 x 10-34 J s. Starting from a wavelength lambda the sequence is: (1) wavenumber nu-tilde = 1 / lambda[cm]; (2) angular wavenumber k = 2*pi / lambda[m]; (3) frequency f = c / lambda = c * nu-tilde * 100; (4) photon energy E = h * f = h * c * nu-tilde * 100. Starting from a frequency, wavelength = c / f, and the rest follows. The convenient shortcut E[eV] * lambda[nm] = hc[eV nm] = 1239.84 eV nm (often rounded to 1240) comes from the same equation. When working in cm-1, E[eV] = nu-tilde / 8065.54 is another frequently used shorthand.
Electromagnetic spectrum quick reference
| Region | Wavelength range | Wavenumber (cm-1) | Energy per photon (eV) |
|---|---|---|---|
| Radio | > 1 mm | < 10 | < 0.000012 |
| Microwave | 1 mm - 0.1 m | 10 - 100 | 0.000012 - 0.0012 |
| Far IR | 25 - 1000 µm | 10 - 400 | 0.0012 - 0.05 |
| Mid IR | 2.5 - 25 µm | 400 - 4000 | 0.05 - 0.5 |
| Near IR | 0.7 - 2.5 µm | 4000 - 14300 | 0.5 - 1.8 |
| Visible | 400 - 700 nm | 14300 - 25000 | 1.8 - 3.1 |
| UV | 10 - 400 nm | 25000 - 1000000 | 3.1 - 124 |
| X-ray | 0.01 - 10 nm | 10^6 - 10^9 | 124 - 124000 |
| Gamma ray | < 0.01 nm | > 10^9 | > 124000 |
Typical wavenumber, wavelength, and photon energy ranges for each EM region.
Frequently asked questions
What is wavenumber in cm-1?
Wavenumber in cm-1 (read "reciprocal centimetre" or "per centimetre") tells you how many complete wavelengths of a wave fit into one centimetre. It is calculated as 1 divided by the wavelength expressed in centimetres. For example, green light at 500 nm has a wavelength of 0.00005 cm, so its wavenumber is 1 / 0.00005 = 20 000 cm-1. Chemists prefer cm-1 over nanometres because it is directly proportional to energy.
How do I convert wavelength (nm) to wavenumber (cm-1)?
Use the formula: wavenumber (cm-1) = 10 000 000 / wavelength (nm). This comes from converting nanometres to centimetres (1 nm = 10-7 cm) and taking the reciprocal. Example: 600 nm gives 10 000 000 / 600 = 16 667 cm-1. To go the other way: wavelength (nm) = 10 000 000 / wavenumber (cm-1).
What is the difference between spectroscopic wavenumber and angular wavenumber?
Spectroscopic wavenumber (nu-tilde) = 1 / lambda in centimetres, with units cm-1. Angular wavenumber (k) = 2*pi / lambda in metres, with units rad/m. They describe the same physical wave but use different normalisation conventions. Chemists and spectroscopists use cm-1; physicists use rad/m in wave equations and quantum mechanics.
What wavenumber range covers the infrared region?
The mid-infrared region - the most important for chemical identification - spans roughly 400 to 4000 cm-1 (wavelengths of 2.5 to 25 micrometres). The near-infrared extends from about 4000 to 14 300 cm-1 (0.7 to 2.5 µm), and the far-infrared covers 10 to 400 cm-1 (25 µm to 1 mm). Visible light falls between about 14 300 and 25 000 cm-1.
How do I convert wavenumber to photon energy in eV?
Divide the wavenumber (cm-1) by 8065.54 to get photon energy in electron-volts (eV). This constant comes from h * c expressed in eV / cm-1. Example: 20 000 cm-1 / 8065.54 = 2.48 eV. You can also use the shortcut E (eV) = 1239.84 / wavelength (nm). This calculator displays both eV and joule values directly.
Can wavenumber be used for sound waves?
Yes, but the relationship is the same: wavenumber = 1 / wavelength (in the chosen unit), or angularly, k = 2*pi / lambda. For acoustic waves the phase velocity is the speed of sound rather than light, but the wavenumber formulas are identical. This calculator is designed for electromagnetic waves, where frequency and photon energy are linked via Planck constant; for sound you would need to supply the speed of sound instead of c.