Question

The speed of sound v in a medium depends on its wavelength λ, the young modulus E, and the density ρ, of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium

Answer 1

$v=\lambda ^\alpha E^\beta \rho ^\gamma$

$[\lambda]=[m]=L$

$[E]=[Pa]=ML^{-1}T^{-2}$

$[\rho]=[kg/m^3]=ML^{-3}$

$[v]=[m/s]=LT^{-1}$

$LT^{-1}=L^\alpha(ML^{-1}T^{-2})^\beta(ML^{-3})^\gamma$

We have

$\alpha-\beta-3\gamma=1$

$\beta+\gamma=0$

$-2\beta=-1$

The solution of the system gives

$\alpha=0$

$\beta=1/2$

$\gamma =-1/2$

$v=\lambda ^0 E^{1/2} \rho ^{-1/2}=\sqrt{\frac{E}{\rho}}$ Answer.