Calculations

  1. If the air temperature at 10 km altitude is -50 °C, what is the speed of sound in air there?

The speed of sound (v) in the air can be obtained by the formula where

  1. How does the speed of sound at this flight altitude compare to the speed of sound in air where you are on Earth’s surface?

Because of the decrease in air temperature with altitude, the speed of sound is often slower at greater altitudes—such as 10 km—than it is at the surface of the Earth. Because there is less air pressure at higher altitudes, air expands as it rises. This expansion causes adiabatic cooling or a decrease in temperature.

  1. Since the local “Mach Number” is the ratio of the speed of aircraft to the speed of sound in the surrounding air, what is the Mach Number for a subsonic Boeing 737 flying at a cruising smoothly (we hope!) speed of 800 km/h?

The formula for calculating the Mach Number is whereby,

Then, we convert 800km/h to m/s to be consistent with the speed of sound.

Thus, the March Number for a subsonic Boeing 737 is 0.7422.

  1. At the moment a plane is directly overhead, why does the direction from which the sound comes depend on the speed of the aircraft? For the example of the Boeing 737 at 10 km altitude, assume an average sound speed along the path from the plane to your ear and find the angle above the horizontal from which the sound comes at the moment the plane itself is overhead.

The direction from which sound comes when a plane is directly overhead depends on the aircraft’s speed because of the Doppler effect. The named effect is the shift in a wave’s wavelength or frequency as a function of an observer’s motion with respect to the wave source. Like a plane flying overhead, the sound waves get compressed for an approaching sound source, leading to a higher frequency (higher pitch). The waves get stretched as the plane moves away, resulting in a lower frequency (lower pitch).

The angle above the horizontal from which the sound comes at the moment the

the plane is directly overhead can be calculated using the formula:

Where,

For a stationary observer on the ground . Assuming the plane is

directly overhead, the formula simplifies to:

The speed of sound at a 10 km altitude is approximately 299.4 m/s and for the Boeing
737 flying at a cruising speed of 800 km/h , convert the speed to
meters per second:

Thus, the angle is 47.9 degrees.

  1. Jet engines typically emit sound over a wide spectrum of frequencies, with the maximum sound energy at 200 Hz. For the Boeing 737 flying at 800 km/hr, what is the change in the frequency (i.e., pitch) of the sound you would hear as the plane approaches, passes overhead, and then recedes. At what moment in its flight you would hear the same pitch that you would if it were not moving at all? Explain your answers, in your own words.

The sound I will hear as the airplane gets closer will have a higher frequency (pitch) than the frequency the aircraft emits. The sound waves are compressed because of the aircraft’s and my relative velocity, giving rise to a shorter wavelength and a higher perceived frequency. When the aircraft passes directly overhead, I will hear the actual frequency emitted by the aircraft, which is 200 Hz. It is because there is no relative motion between the plane and me along the line of sight, and the sound waves are not compressed or stretched. The sound I will hear as the aircraft recedes will have a lower frequency (pitch) than the frequency the aircraft emits. The sound waves are stretched because of the aircraft’s and my relative velocity, giving them a longer wavelength and a lower perceived frequency.

The moment in the aircraft’s flight when I would hear the same pitch if it were not moving is when the aircraft passes directly overhead. At this instant, the relative motion between the plane and me along the line of sight is zero, and the Doppler effect does not affect the sound waves.

  1. What would you hear, from a hypersonic missile or a meteor traveling at 6000 km/h that passes directly overhead at 10 km altitude? Where would it be on its flightpath at the moment that you first hear it?

Since the object’s speed (6000km/h=1667 m/s) is much greater than the speed of sound (299.4 m/s), it will create a sonic boom or a series of shock waves. When the object passes directly overhead, I would first hear the sonic boom created by the object as it approaches, followed by the sound of the object itself, and then another sonic boom as it moves away.

To determine the object’s location on its flight path when I first hear it, we must consider the distance between me and the object when the sound reaches me. Let us assume that the object is approaching from a distance and passing directly overhead. The sound created by the object travels at the speed of sound while the object itself travels at its hypersonic speed. The distance between me and the object when I first hear the sonic boom can be calculated using the following formula:

Rearranging the equation, we get:

Substituting the values:

During this time, the object would have traveled a distance equal to:

Therefore, when I first heard the sonic boom, the hypersonic missile or meteor would be approximately 12.19 km ahead of me on its flight path, having already passed directly overhead.

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