Speed of Sound: Newton’s Formula and Laplace’s Correction: MDCAT Physics notes

General formula

The speed of a mechanical wave in a medium depends on its elasticity (modulus $E$) and its density $\rho$:

$$v = \sqrt{\frac{E}{\rho}}$$

For solids $E$ is Young's modulus; for liquids and gases it is the bulk modulus. Sound travels fastest in solids and slowest in gases. A solid is denser than air, but its modulus is enormously larger, so the ratio $E/\rho$ is much greater for solids than for air.

Newton's formula

Newton assumed that the compressions and rarefactions in air take place at constant temperature (isothermal), so Boyle's law applies. For an isothermal change the bulk modulus equals the pressure, $E = P$. Hence

$$v = \sqrt{\frac{P}{\rho}}$$

At STP, $P = 1.01 \times 10^5\ \mathrm{Pa}$ and $\rho = 1.29\ \mathrm{kg\,m^{-3}}$, which gives $v \approx 280\ \mathrm{m\,s^{-1}}$. The measured value is about $332\ \mathrm{m\,s^{-1}}$, so Newton's formula underestimates the speed by roughly 16%.

Laplace's correction

Laplace pointed out that compressions and rarefactions happen so rapidly, and air is such a poor conductor of heat, that no heat can flow in or out. The process is therefore adiabatic, for which $PV^\gamma$ is constant and the bulk modulus is $E = \gamma P$:

$$v = \sqrt{\frac{\gamma P}{\rho}}$$

For air $\gamma = C_p/C_v = 1.4$, giving $v = \sqrt{1.4} \times 280 \approx 331\ \mathrm{m\,s^{-1}}$, in agreement with experiment.

	Newton	Laplace
Medium considered	Gas (air)	Gas (air)
Process assumed	Isothermal	Adiabatic
Modulus	$E = P$	$E = \gamma P$
Speed at STP	about 280 m/s (too low)	about 331 m/s (correct)

Factors affecting speed in a gas

• Temperature: $v \propto \sqrt{T}$ (T in kelvin). Near room temperature, $v_t \approx v_0 + 0.61\,t$ m/s, where $t$ is in °C.
• Pressure: at constant temperature, $P/\rho$ stays constant, so pressure has no effect.
• Density / molar mass: $v \propto 1/\sqrt{\rho}$; sound is faster in hydrogen than in oxygen.
• Humidity: moist air is less dense than dry air, so sound travels slightly faster.

Key formulas

• $v = \sqrt{E/\rho}$
• Newton: $v = \sqrt{P/\rho}$
• Laplace: $v = \sqrt{\gamma P/\rho}$
• $v_1/v_2 = \sqrt{T_1/T_2}$

Common MDCAT traps

• Newton's error was assuming isothermal changes; Laplace used adiabatic.
• Laplace's correction concerns a gas, not solids or liquids, and not an isobaric process.
• Sound is faster in solids because $E/\rho$ is larger, even though $\rho$ is larger.
• Changing pressure alone does not change the speed of sound in a gas.

Quick revision

• Newton: isothermal, 280 m/s.
• Laplace: adiabatic, factor $\sqrt{\gamma}$, about 332 m/s.
• $\gamma = 1.4$ for air.
• $v \propto \sqrt{T}$.
• Speed order: solids > liquids > gases.