Speed of Sound, Mach Number & Sound Barrier


This week, we have decided to address a number of common questions people have about the speed of sound. This explanation builds on a number of previous questions we have answered in the past, but will hopefully bring all the information together in a single place.

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: 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 Since there obviously is no such barrier, the term sound barrier is outdated and really should not be used any more.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 human speech, and continues in use. (learn 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. A procedure for calculating the temperature, as well as the density and pressure, using the standard atmosphere model was described in a previous question (learn more). Towards the end of that explanation, an equation for calculating the speed of sound based on temperature is also introduced. This equation is based on the more general form

where

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. Here at Aerospaceweb.org, we've provided an atmospheric properties calculator to simplify these calculations. 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.

where

where

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:

One of the most common questions we receive is fow fast is the speed of sound, and as was pointed out earlier, there is no single value to quote. 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

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. We first addressed this topic in a question regarding how fast is Mach 2 in miles per hour (learn more). 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.

Conclusion:

We hope that this answer has addressed any questions you may have about the speed of sound, the sound barrier, Mach numbers, and how to calculate these values. Please review the links given above for further information.
- answer by Greg Alexander, 1 June 2003

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