Sonic Doppler shift in Stealth Jet

SONIC DOPPLER SHIFT IN STEALTH 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 one sonic military jet is chasing a non-sonic jet. A sonic military jet is a jet that moves at the speed of sound while a non-sonic jet moves at speeds less than that of sound. Consider two military jets, jet1 [source] moving at 343 m/s and jet2 [observer] moving at a speed of 300 m/s. We’ll determine the apparent frequency of source jet noise as registered by the observer in jet2 when jet1 is chasing him.

A Stealth Jet

ASSUMPTIONS

1    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    2. The effect of humidity on sound is negligible

3    3. The amplitude of sound is unity

4    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]} (Eq. 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 [Jet2] (m/s)

V_s – Velocity of Source [Jet1] (m/s)

The Doppler shift equation for this case is,

f’ = f₀*{[V – V_o]/[V – V_s]} (Eq. 2)                                                                                                   

The ‘–’ sign in the numerator of equation (2) indicates that the observer is moving away from the source while the ‘–’ sign in the denominator indicates that the source is moving toward the observer.

The velocity of jet1 [Source] V_s = 343 m/s (Eq. 3)                                                         

The velocity of jet2 [Observer] V_o = 343 m/s (Eq. 4)                                                          

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

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

Substitute equations (3), (4), (5) and (6) in equation (2),

f’ = 1000*{[343 – 300]/[343 – 343]}

f’ = ∞ Hz

This is the frequency of source noise as registered by observer [Jet2] in the jet when the source [Jet1] is chasing him. The ‘∞’ value of frequency indicates that time is zero. It means that the source is moving with the sound waves and hence the observer will never hear anything as frequency does not exist for him although he may be able to see the source.

Difference in frequency = f’ – f₀

                                       = ∞ – 1000

                                       = ∞ Hz

APPLICATION

Stealth aircrafts traveling at sonic speeds will be totally quiet with respect to the observer who is being followed due to the Doppler Effect. The fugitive or the enemy plane will never hear anything about an aircraft following them. This helps the stealth jet to make the move and quickly evade the enemy jet.

CONCLUSION

We thus determined the apparent frequency of source noise as registered by the observer due to Doppler shift and concluded that the observer will be able to hear nothing when he is being chased by the source.