The speed of a wave produced in water is given by \(v=\lambda^a g^b \rho^c\). Where \(\lambda\), g and \(\rho\) are wavelength of wave, acceleration due to gravity and density of water respectively. The values of \(a , b\) and \(c\) respectively, are

using dimension formula

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

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

\(\therefore c =0, a + b -3 c =1,-2 b =-1 \Rightarrow b =\frac{1}{2}\)

Now \(a+b-3 c=1\)

\(\Rightarrow a+\frac{1}{2}-0=1\)

\(\Rightarrow a=\frac{1}{2}\)

\(\therefore a=\frac{1}{2}, b=\frac{1}{2}, c=0\)