Period of Harmonic Wave

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

$\tau = \dfrac \lambda c$

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

Proof 1

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

Hence Period of Periodic Wave can be used directly.

$\blacksquare$

Proof 2

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

From Equation of Harmonic Wave, we have:

$(1): \quad \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$

From Wavelength of Harmonic Wave:

$\lambda = \dfrac {2 \pi} \omega$

Hence:

$\omega = \dfrac {2 \pi} \lambda$

and we can express $(1)$ in the form:

$(2): \quad \map \phi {x, t} = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }$

From $(2)$ it follows that $\dfrac {2 \pi} \lambda \paren {x - c t}$ must pass through a complete cycle of values as $t$ is increased by $\tau$.

Thus:

$\dfrac {2 \pi c \tau} \lambda = 2 \pi$

and so:

$\tau = \dfrac \lambda c$

Hence the result.

$\blacksquare$