# Dispersion of water waves

The particular dispersion relation for water waves or gravity waves dictates that waves with a longer wavelength travel faster than those with a shorter wavelength https://en.wikipedia.org/wiki/Dispersion_(water_waves). Which is very much clear from from $$\omega-k$$ plots. When we define group velocity we add sinusoids with close wavelengths and frequencies which results in a wave packet. My question is, as the wave packet is superposition of many such waves of various wavelengths and what we actually see is the packet itself moving 'as a whole', modulating the component waves then how can we actually say some waves (smaller k) are hitting the coast earlier than the rest?