Newton’s formula for the Velocity of Sound Waves in Air

Newton assumed that sound waves travel through air under isothermal conditions (i.e) temperature of the medium remains constant.

The change in pressure and volume obeys Boyle’s law.

PV = constant

Differentiating, P.dV + V.dP = 0

dV = -V dP

So, P = -dp / (dV/V)

P = k (volume elasticity), therefore under isothermal condition, P = k

then, V = √(k/ρ) = √(P/ρ)

where P is the pressure of air and ρ is the density of air. The above equation is known as Newton’s formula for the velocity of sound waves in a gas.

At NTP, P = 76 cm of mercury

= (0.76 x 13.6 x 108 x 9.8) N m-2

ρ = 1.293 kg m-3.

Velocity of sound in air at NTP is;

V = √[(10.76 x13.6 x108 x 9.8) / 1.293]

= 280 ms-1

The experimental value for the velocity of sound in air is 332 m s-1. But the theoretical value of 280 m s-1 is 15% less than the experimental value. This discrepancy could not be explained by Newton’s formula.

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