# Wavelength vs Wavenumber etiquette

When am I supposed to use the terminology of EM "wavenumber", instead of "wavelength" (or frequency)?

The concepts of wavelength and frequency are no problem for me, but wavenumber (number of wavelengths per unit length) seems redundant to me as a student engineer and proto-physicist. And then there's use of energy levels at higher frequencies.

• Read this: en.wikipedia.org/wiki/Wavenumber – docscience Jul 31 '15 at 17:46
• Wavenumber is just a convenience -- it's a lot easier to write $e^{ikx}$ than it is to write $e^{2\pi i x\frac\lambda}$. And if you're writing the first thing, you need a name with which to refer to $k$. They are 100% the same thing, though. – Jerry Schirmer Jul 31 '15 at 17:46

## 3 Answers

Traditionally wavenumber is used in molecule spectrums such as infrared spectrums in organic chemistry where it is given in the incoherent SI-unit $\textrm{cm}^{-1}$. Mostly because one obtains convenient numbers on the axis. Also in most of the wave equations it is used, because again you can make the convenient substitution $k \equiv \frac{2\pi}{\lambda} = \frac{p}{\hbar}= \frac{\sqrt{2 m E }}{\hbar}$ which is commonly done in solving the Schrödinger equation for simple boundary constraints. Thus it depends on the exact context if you would use the first or latter.

I don't think there's much to say beyond the obvious: You should use whatever terminology is most helpful in communicating the information that you want to communicate.

• That has to do with the audience you're talking to. Just like how you use °F when talking to Americans and °C when talking to non-Americans ... similarly it's often wise to use cm^-1 when talking to IR spectroscopists, and nm when talking to visible-light photographers, and eV when talking to atomic spectroscopists and so on.

• It also has to do with the concept you're talking about. Just like how you use frequency when talking about aliasing and the Nyquist limit but time-durations when talking about leap-seconds, and meters when talking about height vs diopters when talking about lens design ... similarly it's often wise to use wavenumbers when talking about the Laue equation for diffraction, and wavelengths when talking about the size of a resonant cavity or etalon, and so on.

Wavelength and wavenumber are redundant terms, as it sounds like you know. Their use is a matter of convention, which in my experience changes from field to field which you won't know until you've been around.

So...if you know which one people use, go with the flow. Otherwise, use which one you know, and be confident; for questions of order-of-magnitude accuracy, it's good to keep in mind they're different by ~5x, ~10x for on the fly conversion.