DOPPLER SHIFT UNDERWATER
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 toward a stationary submarine
underwater. Consider a dolphin emitting ultrasound moving at a speed of 8.3 mph
[13.28 kmph] and approaching an observer who is inside a stationary submarine.
We’ll determine the apparent frequency of dolphin as registered by the observer
in the submarine.
ASSUMPTIONS
1. Water
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 pressure is negligible
3. The
amplitude of sound is unity
4. The
water molecules do not move with respect to source and observer
CALCULATION
The
equation for Doppler shift is given by,
f’
= f0*{[V ± Vo]/[V ± Vs]} (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}
Vo
– Velocity of observer (m/s)
Vs
– Velocity of Source [Dolphin] (m/s)
Since the observer is
stationary,
Vo = 0 (Eq. 2)
Substitute equation (2)
in equation (1),
f’ = f0*{[V]/[V
– Vs]} (Eq. 3)
The ‘–’ sign in the
denominator of equation (3) indicates that the source is approaching the
observer.
The velocity of Dolphin
Vo = 8.3 mph
= 3.68 m/s (Eq. 4)
Frequency of Dolphin’s
sound f0 = 10000 Hz (Eq. 5)
Speed of sound in water
V = 1531 m/s (Eq. 6)
Substitute equations
(4), (5) and (6) in equation (3),
f’ = 10000*{1531/[1531
– 3.68]}
f’
= 10024.09 Hz
This is the frequency
of sound as registered by the stationary observer in the submarine when a
dolphin emitting ultrasound approaches him.
Difference in frequency
= f’ – f0
=
10024.09 – 10000
= 24.09
Hz
CONCLUSION
We thus determined the
apparent frequency as registered by the observer due to Doppler shift.
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