Air — Speed of Sound vs. Temperature

Speed of Sound in Air — Deep Dive (Temperature, Humidity, and More)

The speed of sound describes how fast small pressure disturbances travel through a medium. For an ideal gas such as air, the fundamental relation is \( c = \sqrt{\gamma\,R\,T} \), where \( \gamma \) is the ratio of heat capacities \( c_p/c_v \), \( R \) is the specific gas constant of the gas mixture, and \( T \) is the absolute temperature in kelvins. This reveals the key dependence: sound gets faster as temperature rises (since \( c \propto \sqrt{T} \)).

Dry-Air Rule of Thumb

Near room temperature, a widely used linear approximation is \( c \approx 331.3 + 0.606\,T_{^\circ\!C} \) m/s. At \(20^\circ\)C this gives about 343 m/s, which matches everyday experience and many classroom measurements. The linear form is simply a convenient fit to the square-root law over a limited range.

Why Humidity Matters (A Little)

Real air contains water vapour. Replacing some dry air with water vapour changes the mixture’s properties: the gas constant increases (water vapour has a larger \( R \)) and the effective heat capacity changes, nudging \( \gamma \) and \( R \) in the formula. The net effect is that humid air carries sound slightly faster than completely dry air at the same temperature and pressure. The change is modest (typically ~0.5–1% from 0% to 50% RH at 20 °C), but it can matter for careful measurements, time-of-flight sensing, and microphone calibration.

Pressure, Altitude, and Composition

In the ideal-gas view, if the gas composition is fixed, sound speed does not explicitly depend on pressure. That’s why at the same temperature, sea-level and mountain-top air have nearly the same \( c \) despite very different densities. However, altitude often goes hand-in-hand with lower temperatures, and it is the temperature drop that reduces \( c \). Long-term composition shifts (e.g., CO\(_2\) fraction) or trace gases also tweak \( R \) and \( \gamma \), though the effect on \( c \) is small in typical atmospheres.

Worked Example

Suppose \( T = 25^\circ\mathrm{C} \) (298.15 K), 50% RH, and 101.325 kPa. A moist-air model blends dry-air and water-vapour properties to estimate \( R_\text{mix} \) and \( \gamma \). Plugging them into \( c = \sqrt{\gamma\,R_\text{mix}\,T} \) yields a value a little above the dry-air linear estimate \( 331.3 + 0.606\times25 \approx 346.45 \) m/s. The difference—fractions of a percent—illustrates humidity’s subtle but measurable role.

From Speed to Wavelength and Mach

Once you know \( c \), you can compute wavelength at frequency \( f \) via \( \lambda = c/f \) (e.g., at 1 kHz and \( c=343 \) m/s, \( \lambda \approx 0.343 \) m). In aerodynamics and audio measurements, Mach number is \( \mathrm{Ma} = v/c \); because \( c \) tracks temperature, the same vehicle speed may correspond to different Mach numbers on a hot runway versus a cold morning.

Common Pitfalls

• Mixing units: Always convert to kelvins before using the square-root formula.
• Over-extending the linear fit: The \(331.3+0.606T\) rule is great near room temperature, less accurate far from it.
• Ignoring humidity in precision work: For time-of-flight sensors or long baselines, include RH (and temperature gradients).
• Confusing density with speed: In ideal gases, higher density does not automatically mean slower sound; temperature and composition are what matter.

This tool offers both the simple linear estimate and a moist-air model using relative humidity and pressure. For metrology-grade results (wide temperature ranges, high humidity, or unusual gas mixes), use full thermodynamic property libraries and consider temperature stratification along the path.