Dopple shift during super sonic chase pt2

DOPPLER SHIFT DURING SUPER SONIC CHASE PT2

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 supersonic military jet [source] is receding away from another supersonic jet [observer]. 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 two supersonic military jets, jet1 [source] moving at Mach 2 and jet2 [observer] moving at Mach 1.5. We’ll determine the apparent frequency of source [jet1] noise as registered by the observer in jet2 when jet1 is receding away from him. In other words, jet2 is chasing jet1.

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 [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]} (Eqn. 2)

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

The velocity of Jet1 V_s = Mach2

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

              = 2*343

              = 686 m/s (Eqn. 3)

The velocity of Jet2 V_o = Mach1.5

              = 1.5*speed of sound                                     {⸪ Mach1 = speed of sound}

              = 1.5*343

              = 514.5 m/s (Eqn. 4)

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

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

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

f’ = 1000*{[343 + 514.5]/[343 + 686]}

f’ = 833.33 Hz

This is the frequency of sound as registered by the observer in the supersonic military jet when another supersonic source is receding away from him. We observe that the apparent frequency is approximately more than two thirds of the original value. For a stationary observer, it will be only one third of the original value. So when the supersonic jet2 approaches the oncoming sound wave, it experiences a Doppler shift and registers the apparent frequency.

Difference in frequency = f’ – f₀

                                       = 833.33 – 1000

                                       = – 166.67 Hz

Negative sign indicates that apparent frequency is less than the original but magnitude is always positive.

CONCLUSION

We thus determined the apparent frequency as registered by the observer due to Doppler shift and concluded that the observer in jet2 will be able to hear more than two thirds the original value of jet1 sound.