How to compute the Mach number: introduced by Ernst Mach (Austria-Czechoslovakia 1838-1916) physicist philosopher University Vienna pioneer high-speed photography supersonic shock waves 1887, it is a fundamental dimensionless parameter measuring the ratio between an object/flow velocity and the speed of sound propagation in the medium. Fundamental formula: Ma = v / c_sound, where v object/flow velocity (m/s), c_sound speed of sound in medium (m/s). For ideal gas: c_sound = sqrt(gamma * R * T), gamma specific heat ratio (air 1.4), R specific gas constant (dry air 287.05 J/(kg*K)), T absolute temperature (K). For air at 15°C sea level: c=340 m/s=1224 km/h=761 mph=661 knots. For water: c=1481 m/s, mercury 1450, steel 5960, aluminum 6420. Essential tool for: aerospace aerodynamics (Boeing 737 Airbus A320 cruise Ma 0.78-0.85, Concorde Tu-144 historical Ma 2, Lockheed F-22 F-35 fighters, SR-71 Blackbird Ma 3.3 record, NASA X-15 Ma 6.7), missilery (ICBM reentry Ma 25, Space Shuttle), turbomachinery (axial transonic compressors turbines), acoustics (subsonic incompressible objects Ma1), oceanography (underwater acoustic sonar c=1481 m/s). Regime classification: Ma<0.3 incompressible, 0.3<Ma<0.8 subsonic compressible, 0.8<Ma<1.2 transonic, 1.2<Ma<5 supersonic, Ma>5 hypersonic.

Mach cone: angle mu=asin(1/Ma) for Ma>1, basis sonic boom supersonic objects.

Mach number formulas

Basic definition: Ma = v/c_sound. Dimensionless, system-unit independent. Perturbations propagate at c_sound speed in medium. For Ma1: object faster than its own perturbations, shock wave Mach cone formation. Direct analogy Froude number Fr=v/c_grav (gravitational waves) for free-surface fluids.

Speed of sound ideal gas: c = sqrt(gamma*R*T) = sqrt(gamma*p/rho). Air 15°C 1 atm: gamma=1.4 (cp/cv=1.005/0.718), R=287.05 J/(kg*K), T=288.15 K -> c=sqrt(1.4*287.05*288.15)=340.3 m/s. Simplified approximation: c_air(m/s) = 331.3 + 0.6*T(°C) linear for T 0-30°C. Helium gamma=1.67 R=2077: c=970 m/s at 0°C (3x air because lighter molecule). Hydrogen c=1270 m/s. CO2 c=259 m/s at 0°C.

ISA International Standard Atmosphere: T_ISA=288.15-0.0065*h K for h<11000 m (troposphere, lapse rate 6.5°C/km). 11-20 km isothermal lower stratosphere T=216.65 K. 20-32 km T=216.65+0.001*(h-20000). 32-47 km T=228.65+0.0028*(h-32000). 47-51 km isothermal T=270.65. Commercial aircraft cruise 35000 ft=10670 m T=-55°C, c=295 m/s, Ma 0.85 = 250 m/s = 900 km/h.

Mach cone and shock waves: For object Ma>1, perturbations concentrate in cone half-angle mu=asin(1/Ma). Examples: Ma=2 -> mu=30°, Ma=3 -> mu=19.47°, Ma=5 -> mu=11.5°. Sonic boom cone-ground intersection: front-rear double bang Whitham theory. Supersonic fighters regulations: Ma>1 land overflight prohibited US EU, USA altitude limit 50000 ft. Normal shock Ma=1: Rankine-Hugoniot relations density pressure velocity post-shock.

Stagnation properties: T_t = T*(1+(gamma-1)/2*Ma^2). For Ma=2 air: T_t/T=1.8 (stagnation 80% higher). For Ma=5 X-15: T_t/T=6, T_amb=216 K -> T_stag=1296 K=1023°C thermal ablation. Stagnation pressure p_t/p=(T_t/T)^(gamma/(gamma-1)). Density rho_t/rho=(T_t/T)^(1/(gamma-1)). Important for atmospheric reentry design Apollo Soyuz Space Shuttle Starship: heating shock layer ablative thermal shield phenolic carbon graphite.

History and context

Ernst Mach (Austria-Czechoslovakia 1838-1916) Brno-Vienna physicist philosopher University Vienna developed 1887 first high-speed photography supersonic shock wave projectile Schlieren technique introducing dimensionless number Ma=v/c_sound bearing his name. Positivism philosophy influenced young Einstein. William Froude (Great Britain 1810-1879) had already introduced 1868 Fr=v/sqrt(g*L) gravitational analog inspiring Mach. Christian Doppler (Austria 1803-1853) Vienna 1842 Doppler effect description acoustic waves changing tone with velocity. Hermann von Helmholtz (Germany 1821-1894) Berlin acoustic wave theory, Helmholtz vortices. Lord Rayleigh Strutt (Great Britain 1842-1919) Cambridge "Theory of Sound" 1877 first systematic acoustics. Ludwig Prandtl (Germany 1875-1953) Gottingen University 1929 boundary layer theory and Prandtl-Glauert subsonic correction. Theodore von Karman (Hungary-USA 1881-1963) Caltech 1930s gas dynamics hypersonic. Adolf Busemann (Germany 1901-1986) NASA Langley 1935 swept-wing supersonic. Hans Pabst von Ohain (Germany 1911-1998) and Frank Whittle (Great Britain 1907-1996) invented jet engine 1939-1941 Heinkel He-178 and Gloster Whittle E.28/39. Chuck Yeager (USA 1923-2020) USAF Bell X-1 October 14 1947 first supersonic flight Ma 1.06 record above Edwards Air Force Base California. Concorde Anglo-French Aerospatiale-British Aircraft Corporation 1976-2003 Ma 2.02 commercial cruise civil, retirement tragedy AF 4590 July 25 2000 Paris 113 victims.

Tupolev Tu-144 Soviet 1968-1978 first supersonic flight Ma 2.29 (commercial record before Concorde 1969). NASA X-15 Joseph Walker August 22 1963 Ma 6.7 record altitude 107.8 km North American Aviation Inglewood California. SR-71 Blackbird Lockheed Skunk Works Burbank California 1964-1998 Ma 3.32 cruise altitude 26000 m record, designer Kelly Johnson. Global aerospace industry: Boeing Seattle WA 737-787 commercial, Airbus Toulouse France A320-A380 commercial, Lockheed Martin Bethesda MD F-22 F-35 fighters, Northrop Grumman Falls Church VA B-2 stealth, McDonnell Douglas (acquired Boeing) F-15 F-18, General Dynamics Falls Church VA F-16 (now Lockheed). USA: NASA Glenn Cleveland OH propulsion research, NASA Ames Mountain View CA computational fluid dynamics, NASA Langley Hampton VA wind tunnels supersonic, NASA Marshall Huntsville AL launch vehicles, NASA Johnson Houston TX manned spaceflight, NASA Kennedy Cocoa Beach FL launch operations. SpaceX Hawthorne CA Falcon 9 9 Merlin engines Ma 25 first stage reentry, Starship Boca Chica Texas full reusable Mars architect Ma 25 reentry. Blue Origin Kent WA New Shepard suborbital Ma 3, ULA United Launch Alliance Atlas Delta. Boom Supersonic Denver Colorado XB-1 demonstrator 2024 Mach 1, Overture commercial 2029 Ma 1.7. Hermeus Atlanta Georgia Quarterhorse Mark 1 Mach 5 hypersonic test 2024. Stratolaunch Mojave CA carrier aircraft hypersonic vehicles testing.

Pentagon DARPA Defense Advanced Research Projects Agency Arlington VA Falcon HTV-2 hypersonic glide. Orbital Reef Sierra Space ULA station Mach 25 LEO. China: J-20 stealth fighter Ma 2.5, DF-17 hypersonic glide vehicle Ma 5+. Russia: Tu-160 White Swan strategic bomber Ma 2.05. American Institute Aeronautics Astronautics AIAA Reston VA professional society, NASA STI Scientific and Technical Information Hampton VA database, Federal Aviation Administration FAA Washington DC certification rules. Universities: MIT 16.20 Aerospace Vehicle Engineering Cambridge MA, Stanford AA210 Aeronautics Astronautics, Caltech Ae101 GALCIT Pasadena, Georgia Tech AE6450 Atlanta, U Michigan Aerospace Ann Arbor, Purdue West Lafayette IN.

Speed of sound media table

• Dry air 0°C 1 atm: c=331.3 m/s = 1192 km/h = 643 knots
• Dry air 15°C 1 atm (ISA sea level): c=340.3 m/s = 1225 km/h = 761 mph
• Dry air 25°C 1 atm: c=346.3 m/s = 1247 km/h
• Air 11000 m -56.5°C (commercial cruise): c=295 m/s = 1062 km/h
• Air 30000 m -46.7°C (X-15): c=302 m/s = 1087 km/h
• Helium He 0°C: c=970 m/s (3x air, light gas gamma 1.67)
• Hydrogen H2 0°C: c=1270 m/s (4x air, lightest)
• Carbon dioxide CO2 0°C: c=259 m/s (heavy gas)
• Methane CH4 0°C: c=430 m/s (natural gas pipelines)
• Fresh water 20°C: c=1481 m/s (4.4x air)
• Seawater salinity 35: c=1531 m/s (sonar oceanography)
• Mercury 20°C: c=1450 m/s
• Ice 0°C: c=3980 m/s
• Steel: c=5960 m/s (longitudinal)
• Aluminum: c=6420 m/s (record metals)
• Concrete: c=3700 m/s
• Wood (along grain): c=3850 m/s
• Rubber: c=60 m/s (extremely slow)
• Incompressible Ma<0.3: negligible compressibility
• Subsonic 0.3<Ma<0.8: Prandtl-Glauert corrections
• Transonic 0.8<Ma<1.2: drag rise, local shock waves
• Supersonic 1.2<Ma<5: Mach cone, sonic boom
• Hypersonic 5<Ma<10: molecular dissociation
• Extreme hypersonic Ma>10: plasma, thermal ablation

Detailed numerical examples

Example 1 - Boeing 737 typical cruise. Boeing 737-800 commercial aircraft cruise altitude 11000 m (35000 ft), v=850 km/h=236 m/s. ISA atmosphere h=11000m: T=216.65 K=-56.5°C. c=sqrt(1.4*287.05*216.65)=295.1 m/s. Ma=236/295.1=0.80. SUBSONIC COMPRESSIBLE 0.3<Ma<0.8 borderline transonic. Stagnation T_t=216.65*(1+0.2*0.64)=244.4 K=-28.7°C. p_t/p=(1.128)^3.5=1.524 (52% pressure rise). Drag rise from Mach divergence Mc=0.82 Boeing design limit. Applications: commercial aircraft Boeing Airbus cruise 0.78-0.85 Ma.

Example 2 - Concorde historical supersonic. Concorde British-French 1976-2003 cruise altitude 18000 m, v=2179 km/h=605 m/s. ISA h=18000m T=216.65 K (lower stratosphere). c=295 m/s. Ma=605/295=2.05. SUPERSONIC 1.2<Ma<5. Mach cone mu=asin(1/2.05)=29.2°. Stagnation T_t=216.65*(1+0.2*4.2)=395 K=122°C aerodynamic heating. Nose fuselage anticipated 127°C cruise temperature due to adiabatic compression and viscous friction. Sonic boom 105 dB ground level prohibited overland flight USA Europe.

Example 3 - Sonar oceanography. US Navy Ohio-class SSBN submarine or Italy Todaro-class U212A: active sonar emits 10 kHz waves. v_perturbation=500 m/s impulse side-scan sonar. Seawater 20°C salinity 35 PSU: c=1531 m/s (Mackenzie 1981 precise formula). Ma=500/1531=0.327. SUBSONIC compressible. Application: marine oceanography NOAA Boulder CO research, Woods Hole Oceanographic Institution Falmouth MA, Scripps La Jolla CA, ASW Anti-Submarine Warfare US Navy NATO, bathymetric mapping NOAA mapping coastal data USA.

Example 4 - Helium gas in Saturn V J-2 stage rocket. NASA Apollo Saturn V rocket J-2 second upper stage uses H2-O2. Helium pressurization piping: v=600 m/s, T=20°C=293 K. Helium He gamma=1.67 R=2077: c=sqrt(1.67*2077*293)=1008 m/s. Ma=600/1008=0.595. SUBSONIC compressible. Application: vapor cycling pressurization tank, NASA Marshall Space Flight Center Huntsville AL. Modern SpaceX Starship methalox uses similar tank pressurization.

Example 5 - X-15 hypersonic record 1963. NASA X-15-3 pilot Joseph Walker August 22 1963 altitude record 107.8 km Karman line space. Typical cruise h=30000 m, v=7274 km/h=2020 m/s. ISA h=30000m T=226.5 K. c=sqrt(1.4*287*226.5)=302 m/s. Ma=2020/302=6.69. HYPERSONIC 5<Ma<10. Mach cone mu=asin(1/6.69)=8.6°. Stagnation T_t=226.5*(1+0.2*44.8)=2256 K=1983°C. Inconel-X superalloy nickel-chromium-iron heat shield 1100°C, ablation phenolic resin coating thermal protection. Designer North American Aviation collaboration with NASA Edwards Air Force Base California flight test center.

Limits and considerations

1. Formula c=sqrt(gamma*R*T) valid only for ideal gas: real gas Z>1 at high pressure corrects density.
2. ISA International Standard Atmosphere idealized: real deviations ±15°C summer-winter, geography.
3. Mach cone mu=asin(1/Ma) valid only for steady supersonic point objects. Real shapes more complex.
4. Ma<0.3 incompressible: simplifying assumption, correct Bernoulli incompressible Euler equations.
5. Transonic 0.8<Ma<1.2 complex physics: local wing shocks, nonlinear drag rise, critical design.
6. Hypersonic Ma>5: air molecular dissociation, real gas effects, equilibrium chemistry, NOT perfect gas.
7. Stagnation properties assume adiabatic isentropic flow: real losses through shock, heat transfer.
8. Solids c_sound longitudinal vs transverse different (c_long > c_trans). Wood crystal anisotropy.
9. Liquids low compressibility: bulk modulus K=rho*c^2, water 2.2 GPa.
10. Human acoustic vibrations 20-20000 Hz: c defined for longitudinal waves, not transverse.

Common errors

1. Confusing c_sound air with fixed value 340 m/s: depends on temperature altitude humidity.
2. Neglecting compressibility effects for Ma>0.3: incompressible valid only Ma<0.3.
3. Calculating Ma with relative ground velocities without considering atmospheric wind speed.
4. Confusing Mach number Ma with Reynolds Re: Ma compressibility, Re viscosity.
5. Neglecting stagnation heating for Ma>3: thermal shields fundamental design.
6. Assuming shock waves for Ma1.
7. Computing Mach cone for Ma<1: angle undefined (subsonic perturbations precede).
8. Confusing solid c_sound longitudinal with transverse: c_long>c_trans typically factor 2.
9. Neglecting ISA atmosphere: assuming T=15°C constant at all altitudes wrong 15-20°C.
10. Assuming ideal gas Ma>5 hypersonic: N2 O2 dissociation, real gas equilibrium chemistry.

Numbered applications

1. Commercial aircraft: Boeing 737-787 Seattle WA, Airbus A320-A380 Toulouse France, cruise Ma 0.78-0.85.
2. Military fighters: Lockheed F-22 F-35 Bethesda MD, Eurofighter Typhoon (UK Germany Italy Spain), Dassault Rafael France.
3. Historical supersonic: Concorde 1976-2003 Aerospatiale-BAC Ma 2, Tu-144 Soviet 1968-1978, SR-71 Lockheed Ma 3.3.
4. USA aerospace: Boeing Seattle WA, Lockheed Bethesda MD, Northrop Grumman Falls Church VA, General Dynamics Falls Church VA.
5. NASA centers: Langley Hampton VA wind tunnels supersonic, Glenn Cleveland propulsion, Marshall Huntsville AL launch vehicles.
6. SpaceX Boca Chica Texas: Falcon 9 stage 1 reentry Ma 25, Starship Mars architect reusable, Hawthorne CA HQ.
7. Hypersonic startups: Hermeus Atlanta Mach 5 Quarterhorse 2024, Stratolaunch Mojave CA carrier hypersonic.
8. Sonar oceanography: NOAA Boulder CO, Woods Hole Falmouth MA, Scripps La Jolla CA marine research.
9. Industrial turbomachinery: GE Aerospace Cincinnati OH, Pratt Whitney East Hartford CT, Rolls-Royce Indianapolis IN turbines.
10. Educational teaching: MIT 16.20 Aerospace Vehicle Engineering, Caltech Ae101 GALCIT, Stanford AA210, Georgia Tech Atlanta.

Support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary aerodynamics/acoustics. For professional engineering Mach E 1887 'Photographische Fixierung der durch Projectile in der Luft eingeleiteten Vorgange' Sitzungsber Akademie Wissenschaften Vienna, Anderson JD 'Modern Compressible Flow' 2003 McGraw-Hill, Liepmann HW Roshko A 'Elements of Gasdynamics' 1957 Wiley, NASA NASA-TM-X-1049 'X-15 Research Results' 1965, ISA International Standard Atmosphere ICAO 1993.

Updated July 2026