Page "Phase velocity" Paragraph 6

To understand where it comes from, imagine a basic sine wave, A cos ( kx − ωt ).

Given time t, the source produces ωt oscillations.

At the same time, the initial wave front propagates away from the source through the space to the distance x to fit the same amount of oscillations, kx = ωt.

So that the propagation velocity v is v = x / t = ω / k.

The wave propagates faster when higher frequency oscillations are distributed less densely in space.

Formally, Φ = kx − ωt is the phase.

Since ω = − dΦ / dt and k = + dΦ / dx, the wave velocity is v = dx / dt = ω / k.