Speed of Sound Calculator

**Speed of sound in dry air (linear approximation):** v ≈ 331.3 + 0.606 × T Where T is temperature in °C, v in m/s. **More accurate formula (ideal gas):** v = 331.3 × √(1 + T/273.15) Or: v = √(γ × R × T / M) Where: - γ = ratio of specific heats (1.4 for air) - R = 8.314 J/(mol·K) - T = absolute temperature (K) - M = molar mass (0.02897 kg/mol for air) **Worked example: sound speed at 20°C** Linear: v = 331.3 + 0.606 × 20 = 331.3 + 12.12 = 343.4 m/s. Exact: v = 331.3 × √(1.0732) = 331.3 × 1.0360 = 343.2 m/s. Both agree well at moderate temperatures. **Temperature effect (air):** | Temperature (°C) | v (m/s) | v (mph) | |---|---|---| | -40 | 306.5 | 686 | | -20 | 319.1 | 714 | | 0 | 331.3 | 741 | | 10 | 337.4 | 755 | | 20 | 343.4 | 768 | | 25 | 346.4 | 775 | | 30 | 349.5 | 782 | | 40 | 355.5 | 795 | | 100 | 387.0 | 866 | About +0.6 m/s per °C. **Speed in different media:** | Medium | v (m/s) | Notes | |---|---|---| | CO₂ | 259 | heavier than air | | Air | 343 | at 20°C | | Helium | 1,007 | lighter than air (why "voice high"?) | | Hydrogen | 1,310 | lightest gas | | Water (fresh, 20°C) | 1,481 | | | Water (sea, 25°C) | 1,533 | more dense | | Rubber | 60-1,500 | depends on density | | Wood (typical) | 3,800-4,800 | along grain | | Concrete | 3,600-5,000 | | | Brass | 3,500 | | | Aluminum | 6,420 | high | | Iron | 5,950 | | | Steel | 5,960 | | | Granite | ~6,000 | | | Glass | 5,500-6,000 | | | Diamond | 12,000 | very stiff | | Beryllium | 12,890 | metal with highest v | **Why sound is faster in stiffer media:** v = √(K/ρ) for fluids, where K = bulk modulus, ρ = density. v = √(E/ρ) for solids, where E = Young's modulus. Higher stiffness → faster sound. Higher density → slower. Steel: high stiffness, moderate density → very fast. Lead: moderate stiffness, very high density → moderate (1,200 m/s). **Mach number:** M = v_object / v_sound | M | Regime | Examples | |---|---|---| | < 0.8 | Subsonic | most commercial aircraft, cars | | 0.8-1.0 | Transonic | airliners just below sound speed | | 1.0 | Sonic | sound speed exactly | | 1.0-5.0 | Supersonic | Concorde, fighter jets | | ≥ 5.0 | Hypersonic | reentry vehicles, missiles | Mach 1 at sea level (20°C): 343 m/s ≈ 1,235 km/h ≈ 767 mph. **Sonic boom physics:** When v_object > v_sound, pressure waves pile up into shock wave. Conical shock front trails the object. Mach cone angle: sin θ = 1/M. M = 2: cone angle ~30°. M = 5: cone angle ~11.5°. Listener hears sudden boom as shock passes, even if aircraft has already left. **Worked example: thunder distance** Lightning to thunder delay: 5 seconds. Distance? Speed of sound: 343 m/s. Distance = 343 × 5 = 1,715 m ≈ 1 mile. The "5-second rule" for thunder: count seconds, divide by 5 for miles (or 3 for km). **Sound in atmosphere:** | Altitude | Temperature | Speed of sound | |---|---|---| | Sea level | 15°C | 340 m/s | | 1 km | 8.5°C | 336 m/s | | 5 km | -17.5°C | 320 m/s | | 11 km (tropopause) | -56.5°C | 295 m/s | | 20 km | -56.5°C | 295 m/s | | 50 km | -2.5°C | 329 m/s | Speed depends only on temperature (in dry air), so it can increase or decrease with altitude depending on atmospheric layer. **Humidity effect:** Water vapor is lighter than dry air (M = 18 vs 29). Humid air has lower effective density → slightly faster sound. | Humidity | Speed change at 25°C | |---|---| | 0% | reference | | 50% | +0.15% | | 100% | +0.30% | Tiny effect compared to temperature. **Pressure effect:** For an ideal gas at constant T, v is independent of pressure. Real gases have small departures (~0.1% at extreme pressures), but for normal atmospheric variations, ignore. **Wavelength-frequency:** c = f × λ for sound: Audible (20 Hz - 20 kHz) wavelengths in air: - 20 Hz: 17.2 m. - 100 Hz: 3.43 m. - 1 kHz: 34.3 cm. - 10 kHz: 3.43 cm. - 20 kHz: 1.72 cm. Wavelength matters for: - Speaker design (subwoofers must be large; tweeters small). - Microphone placement (interference at half-wavelength). - Room acoustics (standing waves). - Ultrasonic imaging (high f → small λ → fine detail). **Underwater sound:** Sound travels 4× faster in water than air. Used for: - **Sonar**: detection by echo. - **Whale communication**: across ocean basins. - **Submarines**: stealth and detection. - **Marine biology**: tracking species. Underwater speed varies with temperature, salinity, depth (pressure). **Sound in solids (seismology):** Earth's interior characterized by sound wave speeds: - P-waves (compression): 5-13 km/s (fastest). - S-waves (shear): 3-7 km/s. Used to map Earth's structure (mantle, core boundaries). **Helium voice effect:** Helium has lower density than air → sound travels faster → resonance frequency of vocal tract is higher → high-pitched voice. Doesn't actually change vocal cord frequency, just the resonant filtering. Sulfur hexafluoride (SF₆): opposite effect — very dense gas → low pitch.