Worked example 1.3: Dimensional analysis

Question: The speed of sound $v$ in a gas might plausibly depend on the pressure $p$, the density $\rho$, and the volume $V$ of the gas. Use dimensional analysis to determine the exponents $x$, $y$, and $z$ in the formula

$$v = C p^x \rho^y V^z,$$

where $C$ is a dimensionless constant. Incidentally, the mks units of pressure are kilograms per meter per second squared.
 
Answer: Equating the dimensions of both sides of the above equation, we obtain

$$\frac{[L]}{[T]} = \left(\frac{[M]}{[T^2][L]}\right)^x\left( \frac{[M]}{[L^3]}\right)^y [L^3]^z.$$

A comparison of the exponents of $[L]$, $[M]$, and $[T]$ on either side of the above expression yields

$$1 = -x -3 y+ 3z,$$

$$0 = x + y,$$

$$-1 = -2 x.$$

The third equation immediately gives $x=1/2$; the second equation then yields $y=-1/2$; finally, the first equation gives $z=0$. Hence,

$$v = C \sqrt{\frac{p}{\rho}}.$$