Wave Number of Harmonic Wave/Proof 2

Theorem

Let $\phi$ be a harmonic wave expressed as:

$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$

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 displacement 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.

$\blacksquare$