Super sonic doppler shift in a military jet

SUPER SONIC DOPPLER SHIFT IN MILITARY JET

INTRODUCTION

We know that Doppler Effect or Doppler shift occurs between a source and observer when they are in relative motion with respect to each other. In this case we’ll determine the Doppler shift that occurs when a supersonic military jet is moving toward a stationary observer in a building. A supersonic military jet is a jet that moves faster than the speed of sound thereby leading to a sonic boom. A sonic boom is an explosion that occurs when any object travels faster than sound. Consider a military jet moving at a speed of Mach2 approaching an observer who is inside a building. We’ll determine the apparent frequency of the jet’s noise as registered by the observer.

ASSUMPTIONS

1. The atmospheric air has standard temperature and pressure conditions

·         Temperature T = 298 K or 25°C or 77°F

·         Pressure = 1 bar = 10⁵ N/m²

2. The effect of humidity on sound is negligible

3. The amplitude of sound is unity

4. The air molecules do not move with respect to source and observer

CALCULATION

The equation for Doppler shift is given by,

f’ = f₀*{[V ± V_o]/[V ± V_s]} (Eqn. 1) 

f₀ – Original frequency (Hz)

f’ – Apparent or observed frequency (Hz)

V – Velocity of Sound in air at standard temperature and pressure conditions (m/s) {V = 343 m/s}

V_o – Velocity of observer (m/s)

V_s – Velocity of Source [Jet] (m/s)

Since the observer is stationary,

V_o = 0 (Eqn. 2)

Substitute equation (2) in equation (1),

f’ = f₀*{[V]/[V – V_s]} (Eqn. 3)

The ‘–’ sign in the denominator of equation (3) indicates that the source is approaching the observer.

The velocity of jet V_s = Mach2

           = 2*speed of sound                                           {⸪ Mach1 = speed of sound}

           = 2*343

           = 686 m/s (Eqn. 4)

Frequency of jet exhaust noise f₀ = 1000 Hz (Eqn. 5)

Speed of sound in air V = 343 m/s (Eqn. 6)

Substitute equations (4), (5) and (6) in equation (3),

f’ = 1000*{343/[343 – 686]}

f’ = – 1000 Hz

This is the frequency of sound as registered by the stationary observer in a building when a supersonic military jet approaches him. The negative value of apparent frequency indicates that the object [jet] is traveling faster than sound waves. Therefore the observer will hear the same frequency but after the jet has passed as the sound waves take some more time to reach the observer. Apparently there is no Doppler shift since the observer is registering the exact frequency of sound as emitted by the source. The observer will hear sound of different frequency when the jet moves away from him which we’ll discuss in the next post.

Difference in frequency = f’ – f₀

                                       = – 1000 – 1000

                                       = – 2000 Hz

CONCLUSION

We thus determined the apparent frequency as registered by the observer due to Doppler shift and concluded that the observer will be able to hear the exact value of frequency of jet noise only after the supersonic jet has passed.