The velocity v of water waves may depend on their wavelength (λ), the density of water (ρ) and the acceleration due to gravity (g). The method of dimensions gives the relation between these quantities as

Velocity $v = k λ^{a} ρ^{b} g^{c}$

$\Rightarrow [M^{0} L T^{- 1}] = [L^{a}] [M^{b} L^{- 3 b}] [L^{c} T^{- 2 c}]$

$o r [M^{0} L T^{- 1}] = [M^{b} L^{a - 3 b + c} T^{- 2 c}]$

-2c = -1

Again, $a - 3 b + c = 1, b = 0, c = \frac{1}{2}$

$∴ v = k λ^{1 / 2} ρ^{0} g^{1 / 2}$

$o r v^{2} \propto g λ$

The velocity $v$ of water waves may depend on their wavelength $(λ),$ the density of water $(ρ)$ and the acceleration due to gravity (g). The method of dimensions gives the relation between these quantities as