The Significance of a Linear Phase

We know that a phase shift is directly proportional to a time shift in a sinusoid.

$$\begin{align*} \cos \left(\omega t +\phi\right)&=\cos \left[\omega \left(t +\frac{\phi}{\omega}\right)\right]\\ &=\cos \left(\omega \left(t -\tau\right)\right) \end{align*}$$ ...