Velocity of Harmonic Wave is Wavelength times Frequency/Proof 1

Theorem

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

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

where $c$ is the velocity of $\phi$.

Then:

$c = \nu \lambda$

where:

$\nu$ is the frequency of $\phi$

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

Proof

By definition, a harmonic wave is an instance of a periodic wave.

Hence Velocity of Periodic Wave is Wavelength times Frequency can be used directly.