Question

The velocity of sound in a gas of pressure p and density roll is written as v=kp^xp^y find x and t and state the relationship

Answer 1

The velocity of a sound in gas is related to the pressure $p$ and density $\rho$ by the formula

$$v=kp^x\rho^y$$

We use the dimensional analysis to find $x$ and $y$.

[v]=LT^{-1}\\ [p]=MT^{-2}L^{-1}\\ [\rho]=ML^{-3}

We get

$$LT^{-1}=(MT^{-2}L^{-1})^x (ML^{-3})^y$$

or

$$-x-3y=1,\\ x+y=0,\\ -2x=-1.$$

Roots:

$$x=1/2,\quad y=-1/2$$

So, the velocity of a sound in gas is given by

$$v=kp^{1/2}\rho^{-1/2}=k\sqrt{p/\rho}$$