Factors Affecting the Speed of Sound in Air: MDCAT Physics notes
Factors Affecting the Speed of Sound in Air notes for MDCAT: Laplace formula, effects of temperature, pressure, density and humidity, 0.61 m/s per °C.
Speed of sound in a gas
Sound is a longitudinal mechanical wave, so its speed depends on the elasticity and density of the medium. For a gas, compressions and rarefactions happen too quickly for heat exchange, so the process is adiabatic. The Laplace-corrected Newton formula is
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
where $\gamma = C_p/C_v$ is the adiabatic index (1.4 for air), $P$ the pressure and $\rho$ the density. (Newton's original $v = \sqrt{P/\rho}$ assumed isothermal changes and gave about $280\ \text{m s}^{-1}$, too low.)
Effect of each factor
| Factor | Effect on speed | Reason |
|---|---|---|
| Pressure (temperature constant) | No effect | By Boyle's law $P/\rho$ stays constant |
| Temperature | Increases; $v \propto \sqrt T$ | $P/\rho = RT/M$ grows with $T$ |
| Density (same pressure) | $v \propto 1/\sqrt\rho$ | Denser gas is slower |
| Humidity | Increases | Moist air is less dense than dry air |
| Frequency, wavelength, amplitude | No effect | Speed is set by the medium |
Since $P/\rho = RT/M$ for an ideal gas, $v = \sqrt{\gamma RT/M}$: the speed of sound in an ideal gas depends essentially on its temperature (and on the nature of the gas through $\gamma$ and $M$). So at the same temperature, sound travels at the same speed at sea level and on a mountain top, even though the pressure differs.
Temperature correction
$$\frac{v_t}{v_0} = \sqrt{\frac{T}{T_0}}$$
For small temperature changes near $0^\circ\text{C}$, the speed of sound in air rises by about $0.61\ \text{m s}^{-1}$ per $^\circ\text{C}$:
$$v_t \approx v_0 + 0.61\,t$$
with $v_0 \approx 332\ \text{m s}^{-1}$ at $0^\circ\text{C}$.
Example: at $22^\circ\text{C}$, $v = 332 + 0.61 \times 22 \approx 345\ \text{m s}^{-1}$.
Example: raising the absolute temperature four times doubles the speed.
Different media
Sound is generally fastest in solids, slower in liquids and slowest in gases (roughly $5000$, $1500$ and $340\ \text{m s}^{-1}$ in steel, water and air). It cannot travel through vacuum. No wave, including sound, can travel faster than light.
Key formulas
- $v = \sqrt{\gamma P/\rho} = \sqrt{\gamma RT/M}$
- $v \propto \sqrt T$ (T in kelvin)
- $v_t \approx v_0 + 0.61t$
Common MDCAT traps
- Changing pressure at constant temperature does not change the speed of sound.
- Speed rises with higher temperature and higher humidity.
- Increase per degree is 0.61 m/s, not 0.51 or 0.81 m/s.
- Use kelvin in $v \propto \sqrt T$.
- Frequency, wavelength and amplitude do not set the speed; the medium does.
Quick revision
- Laplace: sound in gases is adiabatic, $\gamma = 1.4$ for air.
- $v_0 \approx 332\ \text{m s}^{-1}$ at $0^\circ\text{C}$.
- Moist air carries sound faster than dry air.
- Same temperature: same speed at any altitude.
- Sound is fastest in solids.