What is a wavenumber?

Question

I have been struggling to understand the concept of wavenumber for a number of years. Is a wavenumber we talk about in the context of wave propagation in physics (say ocean waves, or acoustics) same as wavenumber as in Fourier Transform? What is horizontal Fourier transform? What is the meaning of a vertical wavenumber if a wave doesn't travel out radially from an origin as a sphere.

Answer 1

It might be helpful to think of the wavenumber concept in terms of a wavevector, whereby the wavenumbers are the components of that wavevector along the x and y directions in the case of a two-dimensional setting.

The usual formulation is in terms of the dot product of the wavevector $\vec{k}$ with the positional/displacement vector $\vec{r}$ which is how it appears in formulations such as the Fourier Transform:

$$ F\left(k_{x},k_{y}\right)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\left(x,y\right)e^{-i\vec{k}\cdot \vec{r}}\mathrm{d}x \mathrm{d}y $$

where $$ \vec{r}=x\mathbf{\hat{x}}+y\mathbf{\hat{y}}, \vec{k}=k_{x}\mathbf{\hat{x}}+k_{y}\mathbf{\hat{y}} $$.

The above Fourier transform will decompose the function $f\left(x,y\right)$ into waves that point in the directions $\vec{k}$. If there is no component in a given direction, the value along that direction will be zero.

The units of the wavenumber are inverse wavelength, usually something like $\frac{2\pi}{\lambda}$, so longer wavelengths will have smaller wavenumber values.

The usage of wavenumbers/wavevectors in physics and Fourier transforms is identical. The below figure tries to capture all this. I hope this helps.