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The Mach number is a ratio between the aircraft's speed (v) and the speed of sound (a). That is,
M = v/a
The Mach number is named for the Austrian physicist, Ernst Mach (1838-1916). Technically, as you can see, the Mach Number is not a speed but a speed ratio. However, it is used to indicate how fast one is going when compared to the speed of sound.
Scientifically, the speed at which sound travels through a gas depends on
1) the ratio of the specific heat at constant pressure to the constant volume, 2) the
temperature of the gas, and 3) the gas constant (pressure/density X temperature). This is
represented by the formula:
a = Square Root(g R T)
a = speed of sound
g = ratio of the specific heat at constant pressure to the specific heat at constant volume
R = universal gas constant
T = Temperature (Kelvin or Rankin)
Fortunately, in the earth's atmosphere (a gas) several of these variables are constant. In our atmosphere, g is a constant 1.4. R is a constant 1718 ft-lb/slug-degrees Rankin (in the English system of units) or 287 N-m/kg-degree Kelvin (in SI units). With g and R as constant values, this results in the speed of sound depending solely on the square root of the temperature of the atmosphere.
Since aircraft and engines are affected by atmospheric conditions and these conditions are rarely (if ever) the same, we use a "standard day atmosphere" to give a basis for determining aircraft performance characteristics. The temperature for this standard day is 59 degrees Fahrenheit (15 degrees Celsius) or 519 degrees Rankin (288 degrees Kelvin) at sea level Thus, the speed of sound at sea level on a standard day is:
a = SQRT[ (1.4) X (1718) X (519) ] = 1116 feet/second
To convert this to miles per hour use the formula 1 foot/second = 0.682 miles per hour (statute miles). 1116 X 0.682 = 761 miles per hour.
Explanation: How can you confirm 0.682 times the number of feet per second will equal provide the miles per hour equivalent? Let's do the math!
Convert 1 foot per second to feet per minute (1 ft/sec x 60 seconds/min) = 60 ft/min
Convert feet per minute to feet per hour (60 ft/min x 60 minutes/hr) = 3600 ft/hr
Thus, 1 foot/second = 3600 feet/hour. All that is required now is to convert feet into miles. One statue mile = 5280 feet. Thus we divide 3600 by 5280 and our answer is 0.682.
One method commonly used to prevent reinventing the wheel is to develop charts with ratios to mathematical equations. One chart available to F-15E aircrews is the "Standard Atmosphere Table." This table provides a "Speed of Sound ratio" column. This column provides the speed of sound (standard day data) ratio for any altitude based on the speed of sound at sea level of 761 MPH.
|Altitude in feet||Speed of Sound ratio|
Using the chart, on a standard day, the speed of sound at 10,000 feet is 761 x 0.9650 or 734 miles per hour. The ratio continues to get smaller until 37,000 feet, where it remains at 0.8671. Any idea why?
HINT: Remember what we stated earlier was the ONLY factor that affected the speed of sound in the earth's atmosphere? If you stated that the temperature of the atmosphere stopped decreasing you're correct! At 37,000 feet, the temperature is a balmy -69.7 degrees Fahrenheit (or -56.5 degrees Celsius).
While several supersonic aircraft like the F-15E are capable of flying faster than twice the speed of sound, they can only reach these speeds at very high altitudes where the air is thin and extremely cold. At sea level, supersonic aircraft are limited to speeds just above Mach 1 due to the atmosphere's temperature and density ("thicker" air that causes more drag on the aircraft).
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Updated: March 12, 2004