Period 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 period $\tau$ of $\phi$ can be expressed as:
 * $\tau = \dfrac \lambda c$

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

Proof
By definition, $\tau$ is the time taken for one complete wavelength of $\phi$ to pass an arbitrary point on the $x$-axis.

We have:


 * $\map \phi {x, t} = \map f {x - c t} = \map f {x - c t + \lambda}$

It follows that $x - c t$ must pass through a complete cycle of values as $t$ is increased by $\tau$.

Thus:
 * $\dfrac {c \tau} \lambda = 1$

and so:
 * $\tau = \dfrac \lambda c$

Hence the result.