The velocity $(v)$ of water waves may depend upon 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:

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$v^{2} \propto g λ$

$v^{2} \propto λ g^{- 1} ρ^{- 1}$

$v^{2} \propto g λ ρ$

$v^{2} \propto g^{- 1} λ^{- 3}$

answer is C.

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Detailed Solution

Using Dimensional method:

Let $v \propto λ^{a} g^{b} ρ^{c}$
$[{LT}^{- 1}] = {[L]}^{a} {[{LT}^{- 2}]}^{b} {[{ML}^{- 3}]}^{c} = [M^{c} L^{a + b - 3 c} T^{- 2 b}]$

$a + b - 3 c = 1$

$\begin{aligned} c = 0 \\ - 2 b = - 1 \\ b = 0.5 \end{aligned}$

$\Rightarrow v \propto \sqrt{g λ}$

$∴ v^{2} \propto g λ$

Alternative method:

$g λ = [L T^{- 2}] \times [L] = [L^{2} T^{- 2}] = υ^{2} (B y int r o s p e c t i o n)$

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