Using dimensional analysis, obtain an expression for the speed of sound waves in a medium.

Speed of sound in a medium depends on bulk modulus (B) and density of medium ($\mathrm{\rho }$).

Dimensional formula for Bulk modulus, density and speed are respectively,

$\begin{array}{l}\left[\mathrm{B}\right]=\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-2}\right];\\ \left[\mathrm{\rho }\right]=\left[{\mathrm{ML}}^{-3}\right]\end{array}$

$\left[\mathrm{V}\right]=\left[{\mathrm{L}}^1{\mathrm{T}}^{-1}\right]$

$\mathrm{Let}, \mathrm{V}={\mathrm{KB}}^{\mathrm{\alpha }}{\mathrm{\rho }}^{\mathrm{\beta }}$ , Where K is Proportionality constant

$\begin{array}{l}\Rightarrow \left[\mathrm{V}\right]=\mathrm{K}{\left[\mathrm{B}\right]}^{\mathrm{\alpha }}{\left[\mathrm{\rho }\right]}^{\mathrm{\beta }}\\ \Rightarrow \left[{\mathrm{LT}}^{-1}\right]={\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-2}\right]}^{\mathrm{\alpha }}{\left[{\mathrm{ML}}^{-3}\right]}^{\mathrm{\beta }}\\ \Rightarrow \left[{\mathrm{LT}}^{-1}\right]=\left[{\mathrm{M}}^{\mathrm{\alpha }+\mathrm{\beta }}{\mathrm{L}}^{-\mathrm{\alpha }-3\mathrm{\beta }}{\mathrm{T}}^{-2\mathrm{\alpha }}\right]\end{array}$

Comparing above equation, 

$-2\alpha =-1 \Rightarrow \alpha =\frac{1}{2} ;$

$\mathrm{\alpha }+\mathrm{\beta }=0 \Rightarrow \mathrm{\beta }=-\mathrm{\alpha }=-\frac{1}{2} ;$

$\therefore \mathrm{V}=\mathrm{K}\sqrt{\frac{\mathrm{B}}{\mathrm{\rho }}}$