Question

Assume that the speed of sound v in air depends upon the pressure p and density rho of air then use dimensional analysis to obtain an expression for the speed of sound

Answer 1

Explanation:

Equating the dimensions of both sides of the above equation, we obtain

\begin{displaymath}

\frac{[L]}{[T]} = \left(\frac{[M]}{[T^2][L]}\right)^x\left(

\frac{[M]}{[L^3]}\right)^y [L^3]^z.

\end{displaymath}

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

$\displaystyle 1$ $\textstyle =$ $\displaystyle -x -3 y+ 3z,$

$\displaystyle 0$ $\textstyle =$ $\displaystyle x + y,$

$\displaystyle -1$ $\textstyle =$ $\displaystyle -2 x.$

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

\begin{displaymath}

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

\end{displaymath}

Answer 2

Answer:

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