Felix Baumgartner: The physics of supersonic falling

Felix Baumgartner exiting his balloon to begin his descent
October 14, 2012

CORRECTED ON 10/16/2012 WITH THANKS TO ERIC LANE WHO SPOTTED THE ERROR (I HAD READ THE NEWSPAPER WRONG AND THOUGHT THAT HIS VELOCITY WAS 833.9 FEET PER SECOND, NOT MILES PER HOUR. HIS COMMENT IS ATTACHED.) 


Twenty four miles (128,100 feet) above Earth, a man encapsulated in a suit much like the astronauts wore on the Moon stepped out of his capsule to begin a death-defying plunge. Whether you approve or not, it's difficult not to admire the courage, imagination, and training that leads up to such a feat!

According to the press, he reached a velocity of 833.9 miles per hour (roughly 371 meters per second) and a Mach number of 1.24. Using the definition of Mach number (M = v/c, where v is his velocity and c is the speed of sound), the speed of sound was 298 meters per second. 

The speed of sound in a perfect gas depends on the square root of the temperature (in degrees Kelvin). The sound speed in air at ambient sea level temperature is about 330 meters per second, and so he would have still been falling at supersonic velocities, though his Mach number would have been slightly less.

Now, here's a thought exercise: If he had jumped through a bottle of bubbly champagne, and attained the same velocity, his Mach number would have been as high as 15 or 30 because the speed of sound in bubbly mixtures is greatly reduced by the presence of bubbles. It could be as low as 10-20 meters per second. The physics behind this is that the sound speed depends on the compressibility and density. A bottle of champaign with tiny bubbles has the density of the liquid (~water) but the compressibility of the gas (CO2), and hence a significantly lower sound speed than pure water and also less than pure air at the same temperature.

Even more extreme (and this is not a suggested experiment!!), if he had attained this velocity in a pot of boiling water, his Mach number could have been of the order 300!! Boiling water has the compressibility of champagne, but an additional factor that reduces the sound speed to only a few meters per second--latent heat transfer as water changes phase (liquid-to-steam-and back to liquid) as the tiny pressure pulses of sound waves pass through it. Here's a reference to the calculations:

Kieffer, Susan Werner, Sound speed in liquid-gas mixtures: Water-air and water-steam, Journal of Geophysical Research, v. 82(20), page 2895-2904, 1977. 

(Yep, I did publish in 1977, and yep, I'm that old! But, it was one of my first papers! I was working on the dynamics of eruption of Old Faithful geyser and it's boiling water, as an analog for eruptions of volcanoes.) 

Here's the abstract from that paper:

The sound speed of a two-phase fluid, such as a magma-gas, water-air, or water-steam mixture, is dramatically different from the sound speed of either pure component. In numerous geologic situations the sound speed of such two-phase systems may be of interest: in the search for magma reservoirs, in seismic exploration of geothermal areas, in prediction of P wave velocity decreases prior to earthquakes, and in inversion of crustal and upper mantle seismic records. Probably most dramatically, fluid flow characteristics during eruptions of volcanoes and geysers are strongly dependent on the sound speed of erupting two-phase (or multiphase) fluids. In this paper the sound speeds of water, air, steam, water-air mixtures, and water-steam mixtures are calculated. It is demonstrated that sound speeds calculated from classical acoustic and fluid dynamics analyses agree with results obtained from finite amplitude ‘vaporization wave’ theory. To the extent that air and steam are represented as perfect gases with an adiabatic exponent γ, independent of temperature, their sound speeds vary in a simple manner directly with the square root of the absolute temperature. The sound speed of pure liquid water is a complex function of pressure and temperature and is given here to 8 kbar, 900°C. In pure water at all pressures the sound speed attains a maximum value near 100°C and decreases at higher temperatures; at high pressures the decrease is continuous, but at pressures below 1 kbar the sound speed reaches a minimum value in the vicinity of 500°–600°C, above which it again increases. The sound speed of a water-air mixture depends on the pressure, the void or mass fraction of air, the frequency of the sound wave, and, if surface tension effects are included, on bubble radius. The admixture of small volume fractions of air causes a dramatic lowering of the sound speed by nearly 3 orders of magnitude. The sound speeds of the pure liquid and gas end-members are nearly independent of pressure, but the sound speed of a mixture is highly dependent on pressure. Calculated values for water-air mixtures are in good agreement with measured values. The sound speed in a single-component two-phase system, such as a water-steam mixture, depends on whether or not equilibrium between the phases on the saturation curve is maintained. Heat and mass transfer which occur when equilibrium is maintained cause the sound speed to be much lower than under non-equilibrium conditions in which heat and mass transfer are absent. The sound speed in a water-steam mixture may be as low as 1 meter per second.