Doppler shift underwater Pt4

DOPPLER SHIFT UNDERWATER PT4

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 dolphin is moving away from a stationary submarine which constantly emits sonar sound waves underwater. Consider a dolphin moving at a speed of 8.3 mph [13.28 kmph], receding away from an observer who is inside a stationary submarine that emits sound waves. We’ll determine the apparent frequency of submarine as registered by the moving dolphin.

ASSUMPTIONS

Water has standard temperature and pressure conditions

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

·         Pressure = 1 bar = 10⁵ N/m²

The effect of pressure is negligible

The amplitude of sound is unity

The water 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 water at standard temperature and pressure conditions {V = 1531 m/s}

V_o – Velocity of observer [Dolphin] (m/s)

V_s – Velocity of Source [Submarine] (m/s)

Since the source is stationary,

V_s = 0 (Eq.2)

Substitute equation (2) in equation (1),

f’ = f₀*{[V – V_o]/[V]} (Eq. 3)

The ‘–’ sign in the numerator of equation (3) indicates that the observer is receding away from the source.

The velocity of Dolphin V_o = 8.3 mph

                                            = 3.68 m/s (Eq. 4)

Frequency of Submarine’s sound f₀ = 500 Hz (Eq. 5)

Speed of sound in water V = 1531 m/s (Eq. 6)

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

f’ = 500*{1531 – 3.68/1531}

f’ = 498.79 Hz

This is the frequency of sound as registered by the moving observer [Dolphin] when it moves away from the submarine emitting sound waves.

Difference in frequency = f’ – f₀

                                       = 498.79 – 500

                                       = – 1.21 Hz

This is a relatively small change in frequency. It is because of the slow speed of Dolphin. The negative sign indicates that apparent frequency is less than original but the magnitude is always positive.

CONCLUSION

We thus determined the apparent frequency as registered by the observer due to Doppler shift.