Question

Natural frequency (f) of a closed pipe depends on the above three factors

find the relation for frequency (f) of a closed pipe using dimensional analysis

Answer 1

$f\propto {\left(l\right)}^a{\left(\rho \right)}^b{\left(\rho \right)}^{c}f\propto {\left(L\right)}^a{\left(M{L}^{-3}\right)}^b{\left(M{L}^{-1}{T}^{-2}\right)}^{c}f\propto L^{a-3b-c}{M}^{b+c}{T}^{-2c}butf\propto T^{-1}\therefore bfc=0b=-\frac{1}{2};a=2\Rightarrow f=k{\rho }^2{\rho }^{-\frac{1}{2}}{P}^{\frac{1}{2}}f=K{\rho }^2\sqrt{\frac{P}{\rho }}$