Speed of sound: Newton and Laplace

Sound squeezes and relaxes small volumes of the medium, so the elastic property that matters is the bulk modulus B = −ΔP/(ΔV/V), measured in pascal like pressure.

B/ρ has the dimensions of a speed squared, and the exact result is v = √(B/ρ) for longitudinal waves in any medium.

In a thin solid bar the sideways bulging is negligible, so Young's modulus replaces B: v = √(Y/ρ).

Sound is generally faster in liquids and solids than in gases. They are denser, but their bulk moduli are larger by a much bigger factor, because they are far harder to compress.

Newton assumed the compressions happen at constant temperature. For an ideal gas PV = constant then gives B = P, so v = √(P/ρ): Newton's formula.

Worked example (NCERT): air of molar mass 29.0 × 10⁻³ kg occupies 22.4 × 10⁻³ m³ at STP, so ρ = 1.29 kg m⁻³. Newton's formula then gives about 280 m s⁻¹, roughly 15% below the measured 331 m s⁻¹.

Laplace's correction: the pressure changes in sound are too fast for heat to flow in or out, so they are adiabatic. PV^γ = constant gives B = γP and v = √(γP/ρ), where γ = Cp/Cv.

For air γ = 7/5, and the Laplace formula gives 331.3 m s⁻¹ at STP, in agreement with the measured speed.

The speed of sound in a gas does not change with pressure at constant temperature, since P/ρ stays fixed; it rises with temperature and with humidity.