The speed of ripples ( v ) on water surface depends on surface tension ( σ ) , density ( ρ ) , and…

Speed of ripples, $v$ depends on $\sigma ,\rho ,\lambda$

$\Rightarrow v\propto {\sigma }^a{\rho }^b{\lambda }^c$

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

$\Rightarrow \left[M^{0}L{T}^{-1}\right]\propto {\left[M\right]}^{a+b}{\left[L\right]}^{-3b+c}{\left[T\right]}^{-2a}$

By equating the powers, $\Rightarrow a+b=0, -3b+c=1, -2a=-1$

$\Rightarrow a=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right., b=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right., c=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

$\Rightarrow v\propto {\sigma }^{\frac{1}{2}}{\rho }^{-\frac{1}{2}}{\lambda }^{-\frac{1}{2}}$

$$\begin{gathered} \Rightarrow v\propto \sqrt{\frac{\sigma }{\rho \lambda }} \\ \Rightarrow v^2\propto \frac{\sigma }{\rho \lambda } \end{gathered}$$