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The Speed Of Sound, Mach Numbers, & The Sound Barrier

Have you ever asked yourself. What is the speed of sound? How fast is Mach 1?

Basic Definitions:

Speed of sound: The speed of sound is a basic property of the atmosphere that changes with temperature. For a given set of conditions, the speed of sound defines the velocity at which sound waves travel through a substance, such as air. Scientists have devised a standard atmosphere model that defines typical values for the speed of sound that change with altitude. (learn more)

Mach number: The Mach number is a quantity that defines how quickly a vehicle travels with respect to the speed of sound.

The Mach number (M) is simply the ratio of the vehicle's velocity (V) divided by the speed of sound at that altitude (a).

For example, an aircraft flying at Mach 0.8 is traveling at 80% of the speed of sound while a missile cruising at Mach 3 is traveling at three times the speed of sound.

Subsonic: A vehicle that is traveling slower than the speed of sound (M<1) is said to be flying at subsonic speeds.

Supersonic: A vehicle that is traveling faster than the speed of sound (M>1) is said to be flying at supersonic speeds.

Sound barrier: The term sound barrier is often associated with supersonic flight. In particular, "breaking the sound barrier" is the process of accelerating through Mach 1 and going from subsonic to supersonic speeds. The term originated in the 1940s when researchers discovered a large increase in drag that seemed to indicate that an infinite amount of thrust would be needed to fly at the speed of sound. In other words, some believed that a physical barrier existed that would prevent an aircraft from ever being able to travel at supersonic speeds. Since there obviously is no such barrier, the term sound barrier is outdated and really should not be used any more. Nevertheless, it has become a popular part of the human language, and continues in use. (Read More)


Calculating the Speed of Sound:

As indicated above, the speed of sound is not a single value, but changes with altitude. To be more precise, the speed of sound (a) can be directly calculated based on the air temperature (T), and temperature is a function of altitude. . This equation is based on the more general form

a = speed of sound [ft/s or m/s]
g = specific heat ratio, which is usually equal to 1.4
R = specific gas constant, which equals 1716 ft-lb/slug/R in English units and 287 J/kg/K in Metric units
T = atmospheric temperature in degrees Rankine (R) in English units and degrees Kelvin (K) in Metric units

Once the speed of sound is known, the Mach number can be easily computed by dividing the airspeed of the vehicle by the speed of sound. Or conversely, the airspeed of the vehicle can be found by multiplying the speed of sound by the Mach number. To simplify these calculations we've provided an atmospheric properties calculator . The user simply enters an altitude and the calculator will provide the air temperature and speed of sound at that altitude. The user can also enter a velocity or a Mach number at that altitude and the calculator will compute the corresponding airspeeds.

However, it should be noted that the above methodology is based on the standard atmospheric model, which assumes a temperature at sea level of 60F (15C). For most engineering purposes, this model is sufficiently accurate for computing the speed of sound, and the change in speed due to a different temperature is small enough that it can be neglected. However, if one already knows the temperature at a given altitude and wishes to calculate a more precise value for the speed of sound, the following equations can also be used. The first is specific to English units while the second applies to the Metric System.

a = speed of sound [ft/s]
TF = atmospheric temperature in degrees Fahrenheit (F)

a = speed of sound [m/s]
TC =atmospheric temperature in degrees Celcius (C)

According to these equations, a 1F change in temperature produces a 1.08 ft/s change in the speed of sound, or a 1C change causes a 0.6 m/s change in the speed of sound. This difference is insignificant enough that we can usually ignore it and use the standard atmospheric model as is.


Values of the Speed of Sound:

 The speed of sound, also known as Mach 1, changes throughout the atmosphere based on the temperature at any given altitude. Probably the most important value to remember, however, is the speed of sound at sea level. Based on the standard atmospheric model, this value has been defined to be

  • 1,116.4 ft/s
  • 340.3 m/s
  • 761.2 mph
  • 1,225.1 km/h
  • 661.5 knots

If you were to reach Mach 1, or "break the sound barrier," at sea level, the above speed is how fast you would have to travel in order to do so.

Another question we often receive is how fast is the speed of sound at other altitudes besides sea level. While the most accurate method of computing these values would be to use the equations listed above or an atmospheric properties calculator, we realize that this method is not always the most convenient approach. In light of this fact, we have provided tables listing the speed of sound in both English and Metric units for altitudes ranging from below sea level to the edge of the atmosphere (learn more). These tables provide the accepted values for Mach 1 in small increments of altitude allowing the reader to observe how the speed of sound varies through different regions of the atmosphere.


Mach Number Examples:

A final question that comes up frequently is how fast is Mach 2, 3, 5, 10, or any other value besides Mach 1 at a given altitude. Here, we pointed out that the Mach number is a multiple of the speed of sound. Therefore, if you know the value of Mach 1 in miles per hour, feet per second, kilometers per hour, or any other unit of measurement at the altitude in question, you merely have to multiply that value by the desired Mach number to determine the speed in that particular unit.

For example, say we wanted to know the speed of a cruise missile traveling Mach 0.8 at sea level in knots. To solve the problem, we can use the speed of sound value listed above at sea level, given as 661.5 knots, and multiply it by 0.8. The answer turns out to be 529 knots.

Yet another example is provided above when someone asks, "what is the speed of an aircraft traveling at Mach 3 at an altitude of 30,000 feet?" If we take another look at the Mach 1 vs. altitude tables already discussed, we see that the speed of sound at 30,000 ft is 678.2 miles per hour. All we have to do is multiply this value by 3 to determine the speed of a vehicle traveling Mach 3 at 30,000 ft in miles per hour. The answer is 2,035 mph.

Let us now consider an example of the opposite problem. About 8 1/2 minutes into the flight of a Space Shuttle, the vehicle's main engines are disengaged. At that point in its trajectory, the Shuttle is traveling about 7,000 meters per second at an altitude of 110,000 m. What is the Shuttle's Mach number? If we again look at the Mach 1 vs. altitude tables, we see that the speed of sound at 110,000 m is 300.7 m/s. When we divide 7,000 m/s by 300.7 m/s, we find that the Space Shuttle is traveling at Mach 23.3, or 23.3 times the speed of sound at that altitude.

by Greg Alexander,

Sonic Booms


Ernst Mach and Mach Number


Having discussed the term Mach number above and how it relates to the speed of sound. The term is named in honor of Ernst Mach, a great 19th century scientist from Austria. Ernst Mach was a well-known figure of his day who made his mark in a variety of fields, most notably in physics and philosophy. Mach was actually the first person in history to develop a method for visualizing the flow passing over an object at supersonic speeds. He was also the first to understand the fundamental principles that govern supersonic flow and their impact on aerodynamics.

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Ernst Mach

Ernst Mach was born on 18 February 1838 in the Austrian town of Turas. His father studied philosophy and classical literature while his mother was a musician and poet. The family lived on an isolated farm where Mach's father raised silkworms and tutored young Ernst in Latin, Greek, history, algebra, and geometry. Though Mach had been a rather average student through grade school and high school, his intellectual talents emerged while attending the University of Vienna. There, he studied mathematics, physics, philosophy, and history, receiving his PhD in physics in 1860 for his thesis "On Electrical Discharge and Induction." Mach was actually offered a position as a professor of surgery at the University of Salzburg, but he instead took a position as a professor of physics at the University of Graz in 1864. Mach later moved to the University of Prague in 1867 where he taught experimental physics for the next 28 years.

Mach's great contributions to understanding supersonic aerodynamics came in the revolutionary paper "Photographische Fixierung der durch Projektile in der Luft eingeleiten Vorgange" that he presented to the Academy of Sciences in Vienna in 1887. In this paper, Mach published the very first photograph of the shock waves formed by a bullet traveling faster than the speed of sound. The photo, shown below, illustrates the strong shock wave formed by the nose of the bullet, the weaker shock wave created at the aft end of the bullet, and the turbulent wake downstream of the bullet's base. Also visible in the photo are two vertical lines made by the trip wires that triggered the camera as the projectile passed by.

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Ernst Mach's photo of a bullet in supersonic flight

What makes Mach's achievement all the more remarkable was the technique he used to take the historic photograph. He employed an innovative approach called the shadowgraph. A typical shadowgraph experiment is illustrated below. In this technique, light is passed through an airflow and reflected onto a screen or film plate. Since shock waves create changes in the temperature and density of the airflow, the light waves are bent, or refracted, as they pass through the shock waves. These refracted light patterns create shadows that can be seen on the screen.

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How a shadowgraph works

Not only was Mach was able to make the invisible shock waves visible, but it is even more amazing that he was able to photograph the phenomenon. His experiments required split-second timing in an age before computers or electronics were available. Mach's shadowgraph technique and a related method called Schlieren photography are still widely used to observe supersonic flowfields even today.

Yet Mach's contributions to supersonic aerodynamics were not limited to experimental methods alone. He was also the first physicist to understand the basic characteristics of supersonic flow. We know today that one of the most important variables affecting aerodynamic behavior is the speed of the air flow over a body (V) relative to the speed of sound (a). Mach was the first to recognize that dependency. He was also the first to note the sudden and discontinuous changes in the behavior of an airflow when the ratio V/a goes from being less than 1 to greater than 1. Today, we call this ratio the Mach number (M):

The Mach number (M) is simply the ratio of the vehicle's velocity (V) divided by the speed of sound at that altitude (a).

However, Mach himself did not coin the phrase. The term was first publicized in 1929 when Swiss engineer Jakob Ackeret (1898-1981) named the variable in honor of Mach during a lecture at the Eidgenossiche Technishe Hochschule in Zurich. The term did not appear in English publications until 1932, 16 years after Mach's death on 19 February 1916.

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Jakob Ackeret, creator of the Mach number

Mach's life spanned 78 years, and his contributions to human knowledge during that time are impressive. In addition to his writings and lectures on supersonic flow, Mach's studies also included physical optics, the history of science, mechanics, philosophy, the origins of relativity theory, physiology, thermodynamics, the sugar cycle in grapes, the physics of music, and classical literature. Mach also wrote about world affairs, including a prescient commentary on the "absurdity committed by the statesman who regards the individual as existing solely for the purpose of the state" that was about a century ahead of its time. Mach received severe criticism for his statement by no less than Vladimir Lenin, future leader of the Soviet Union. Mach's studies of supersonic aerodynamics are still taught to aeronautical engineers, but he is even more widely known in philosophy classrooms where his thoughts on epistemology--the study of knowledge--are still discussed.

 by Jeff Scott,






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