Frequency of Harmonic Wave/Proof 2

Proof
By definition, $\nu$ is the number of complete wavelengths of $\phi$ to pass an arbitrary point on the $x$-axis in unit time.

Let $x_0$ be that arbitrary point.

By definition, $\tau$ is the time taken for one complete wavelength of $\phi$ to pass $x_0$.

So after unit time, $\dfrac 1 \tau$ wavelengths of $\phi$ pass $x_0$.

Hence the result.