Using dimensional analysis, angular frequency $(ω)$ of surface waves in a lake as a function of surface tension $σ$ , water density Q and the wave number $k = \frac{2 π}{λ}$
( $λ$ is the wavelength) is derived. Assuming the dimensionless coefficient is equal to two (in SI units) in this case, find the value of group velocity $(v_{g})$ where $v_{g} = \frac{\partial ω}{\partial k} .$ If your answer is $v_{g}$ (in m/s) then fill value of $10 v_{g}$
$(U s e : σ = 64 m N . m^{- 1}, Q = 320 π k g m^{- 3}, λ = 1 c m)$

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Detailed Solution

Lets assume $ω = 2 σ^{α} Q^{β} k^{γ} .$
Using dimensional analysis
$T^{- 1} = {(M T^{- 2})}^{α} {(M L^{- 3})}^{β} {(L^{- 1})}^{γ}$
On comparing
$0 = α + β, 0 = - 3 β - γ, - 1 = - 2 α . α = 1 / 2, β = - 1 / 2 a n d γ = 3 / 2 ω = 2 \sqrt{\frac{σ k^{3}}{Q}} S o, v_{g} = \frac{\partial ω}{\partial k} = 3 \sqrt{\frac{σ k}{Q}}$