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Speed of Sound (Ideal Gas)
Local speed of sound in an ideal gas as a function of temperature, used to compute Mach number.
Interpretation
Speed of sound in an ideal gas: a = √(γ·R·T), where γ is specific heat ratio, R is specific gas constant, T is absolute temperature. Example: Air (γ=1.4, R=287 J/kg·K, T=288 K) → a = √(1.4×287×288) ≈ 340 m/s.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| a | Speed of sound | m/s |
| γ | Ratio of specific heats | |
| R | Specific gas constant | J/(kg*K) |
| T | Static temperature | K |
What it means
The speed of sound is the speed at which small pressure disturbances propagate through a medium. In an ideal gas, it depends only on the temperature and gas properties, not on pressure. This formula is derived from the isentropic compressibility. It is fundamental in aerodynamics, as it defines the Mach number and influences compressibility effects. The speed of sound decreases with altitude (lower temperature), which affects aircraft performance. It is also used in meteorology and acoustics. Understanding this relation is essential for any compressible flow analysis and for interpreting flight data.
Worked example
Speed of Sound – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| R | 287 J/kg·K |
| T | 288 K |
| Parameter | Value |
|---|---|
| T | 216.5 K |
Common mistakes
- Speed of sound (ideal gas): a = √(γ·R·T).
- γ: Specific heat ratio (dimensionless).
- R: Specific gas constant (J/(kg·K)).
- T: Absolute temperature (K).
- Units: (m²/s²) → m/s.
Applications
The speed of sound in an ideal gas, a = √(γ·R·T), is a fundamental property that determines the Mach number. It varies with temperature; higher temperatures increase the speed of sound. Engineers use this formula to compute Mach numbers, to design acoustic liners, and to assess the effects of altitude on engine performance. In propulsion, the speed of sound affects the design of turbomachinery (compressor blade speeds). By understanding the speed of sound, aerospace engineers can interpret compressible flow phenomena and ensure that aircraft operate within the desired Mach regime.
- Mach number calculation for flight test and design
- Engine component design (compressor, turbine tip speeds)
- Noise generation and acoustic propagation studies
- High‑speed aerodynamic testing and instrumentation
- Atmospheric property modelling for flight planning
Frequently Asked Questions
It calculates the local speed of sound in an ideal gas, which is needed to compute the Mach number and to analyse compressible flows.
γ = specific heat ratio (dimensionless)
R = specific gas constant (J/kg·K)
T = static temperature (K)
It is proportional to the square root of temperature because sound waves propagate through molecular collisions, and higher temperature means faster molecular motion.
At 288.15 K, with γ=1.4 and R=287.05, a = √(1.4×287.05×288.15) ≈ 340.3 m/s (≈ 1116 ft/s).
- Using sea‑level temperature at altitude instead of the actual (lower) local static temperature.
- Using the wrong gas constant for the gas mixture (e.g., using air’s R for steam).
- Confusing static temperature with stagnation temperature.
At an altitude where T = 220 K, for air γ=1.4, R=287 J/kg·K: a = √(1.4×287×220) = √(88396) ≈ 297.3 m/s.
It decreases with altitude in the troposphere (due to temperature decrease), then becomes roughly constant in the stratosphere (isothermal region).
For helium (γ=1.66, R=2077), at 300 K, a ≈ √(1.66×2077×300) ≈ 1017 m/s, much higher than in air.
Humidity slightly increases the speed of sound because water vapour has a lower molar mass and higher γ? Actually, the effect is small (less than 1%).
Mach number M = V/a, so the speed of sound is the reference speed for compressibility effects.