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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?

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The actual question seems unrelated to water:

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?

From your formulation, I assume you don't accept an answer which circumvents wavepackets.

So: Choose two different k values you want to compare and then build a wavepacket for each k value with a certain width, but choose this k-width small enough such that they dont overlap in k.

then let those two wavepackets race against each other :)

the closer your chosen k values were (and hence the smaller your k-width and hence the longer the wavepacket), the more space and time you will need to have meaningfull separation and see a clear winner.

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