Demonstration of group velocity for waves on water.
Waves on deep water satisfy the dispersion relation $\omega = \sqrt{g k}$. For the phase and group velocity we then get
\[ v_{\varphi} = \tfrac{\omega}{k} = \sqrt{\tfrac{g}{k}}, \quad v_g = \frac{d\omega}{dk} = \tfrac{1}{2} \sqrt{\tfrac{g}{k}}. \]
Therefore the relation $v_{\varphi} = 2v_g$ holds – the carrier wave is faster than the amplitude envelope of the wave packet.
Demonstration of group velocity for waves on water. Signal festival 2023.