Super sonic doppler shift in military jet pt3

SUPER SONIC DOPPLER SHIFT IN A MILTARY JET PT3

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 an observer in a supersonic military jet is moving toward a stationary source 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 [Observer] moving at a speed of Mach2 approaching a stationary source which is inside the building. We’ll determine the apparent frequency of the source 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 [Jet] (m/s)

V_s – Velocity of Source (m/s)

Since the source is stationary,

V_s = 0 (Eqn. 2)           

Substitute equation (2) in equation (1),

f’ = f₀*{[V + V_o]/V} (Eqn. 3)

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

The velocity of jet V_o = Mach2

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

           = 2*343

           = 686 m/s (Eqn. 4)  

Frequency of stationary source 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 + 686]/343}

f’ = 3000 Hz

This is the frequency of sound as registered by the observer in the supersonic military jet when it approaches the stationary source. We observe that the apparent frequency is thrice the original value. This means that the observer will register thrice the original frequency. Since the observer is traveling really fast toward the source, he receives sound waves quickly compared to a stationary observer. The greater the observer’s velocity the greater is the apparent frequency. However the observer will hear sound of different frequency when he moves away from the source which we’ll discuss in the next post.

Difference in frequency = f’ – f₀

                                       = 3000 – 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 thrice the original value.