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 $\phi$.

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:

$\lambda = c \tau$

and so:

$\tau = \dfrac \lambda c$

Hence the result.

$\blacksquare$

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 period $\tau$ of $\phi$ can be expressed as:

$\tau = \dfrac \lambda c$

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

1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$