Wave Number of Periodic Wave

Theorem
Let $\phi$ be a periodic wave expressed as:
 * $\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$

The wave number $k$ of $\phi$ can be expressed as:
 * $k = \dfrac 1 \lambda$

where $\lambda$ is the wavelength of $\phi$.

Proof
By definition, $k$ is the number of complete wavelengths of $\phi$ per unit distance along the $x$-axis.

By definition, $\lambda$ is the period of the wave profile of $\phi$.

So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.

The result follows by definition of wave number.