Period of Harmonic Wave/Proof 2

Proof
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.