Question

Assume that the speed (v) of sound in air depends upon the pressure (P) and density of air, then use dimensional analysis to obtain an expression for the speed of sound.

Answer 1

Answer:

The expression of the speed of sound in terms of pressure and the density of the air is given by

$v=K\sqrt{\dfrac P\rho}$

Explanation:

Given that,

The speed $(v)$ of sound in air depends upon the pressure $(P)$ and density of air $(\rho)$.

Therefore,

$v=v(P,\rho)$.

Let, $v\propto P^{\alpha}\rho^{\beta}\ \ \ \ ........\ (1).$

• Dimensions of velocity, $[v]=[M^0L^1T^{-1}]$.

• Dimensions of pressure, $[P]=[M^1L^{-1}T^{-2}]$.

• Dimensions of density, $[\rho]=[M^1L^{-3}T^0]$.

Comparing the dimensions of both sides of relation (1), we get,

$[M^0L^1T^{-1}]=[M^1L^{-1}T^{-2}]^{\alpha}[M^1L^{-3}T^0]^{\beta}\\=[M^{\alpha}L^{-\alpha}T^{-2\alpha}][M^{\beta}L^{-3\beta}T^0]\\=[M^{\alpha+\beta}L^{-\alpha-3\beta}T^{-2\alpha}]$

which gives,

$-2\alpha = -1\\\Rightarrow \alpha =\dfrac 12.\\ \alpha+\beta =0\\\dfrac 12 + \beta =0\\\Rightarrow \beta = -\dfrac 12 .$

Using these values,

$v\propto P^{\frac 12} \rho^{-\frac 12}\\v=K\sqrt{\dfrac P\rho}$

where, $K$ is the constant of proportionality.

Answer 2

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