# Newton’s Formula for Speed of Sound Wave in Gas Medium

Newton assumed that when sound wave propagates through gas medium, compression and rarefraction are alternately formed in the medium. He argued that this is very slow process. Heat is evolved during each compression which is given to surrounding and heat is lost during rarefraction which is  taken from the surrounding. As a result, temperature remains constant.

Since, the temperature of the system remains constant, it is isothermal process. For isothermal process,

PV = Constant

Differentiating above equation:

VdP + PdV = 0

or, PdV = -vdP

$$\text{or, P} = \frac{\text{-VdP}}{\text{dV}}$$

$$\text{or, P} = \frac{\text{-dP}}{\frac{\text{dV}}{\text{V}}}$$

$$\text{Since, bulk modulus (B)} = \frac{\text{-dP}}{ \frac{\text{dV}}{\text{V}}},$$

$$\text{P = B}$$

Velocity of sound in gas is given by:

$$\text{V} = \sqrt{\frac{\text{B}}{\rho}}$$

$$\text{or, V} = \sqrt{\frac{\text{P}}{\rho}}$$

This is the formula for speed of sound wave in gas medium also known as Newton’s formula.

At Normal temperature and pressure:

ρ = 1.295 kgm-3

T = 273 K

P = 1.013 × 105 Pa

Now, $\text{V} = \sqrt{\frac{\text{P}}{\rho}}$

$\text{or,} \text{V} = \sqrt{\frac{\text{1.013 × 10^5}}{1.293}}$

$\therefore \text{V} = 250 ms^{-1}$

But, experimentally, it is found that the speed of sound medium at NTP is 332 ms-1. Since, the velocity of sound obtained from Newton’s formula is very less than the experimentally obtained velocity of sound, this theory was later concluded wrong.

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