A submarine with a cruising speed of 15.0 m/s uses a sonar (Sound Navigation and Ranging) device operating at 15000 Hz. The speed of sound in sea water at the submarine's depth is 1551 m/s. Calculate the frequency of the echo a sonar operator would receive from a torpedo coming directly towards the submarine at a speed of 30.0 m/s.
The Doppler effect means in its essence the fact that for a moving wave (sound in this case) source and for a moving receiver the frequency of the waves is shifted backwards or forwards depending on the relative movement of source-receiver.
Imagine you have a static source that emits waves which arrives to a static receiver. Let the frequency of the waves be `F_0` in this case. If the source remains static and the receiver moves towards the source with speed `v_r` more wave fronts will hit the receiver in the same time interval. (The reverse is true, when the receiver stays still and the source moves towards the receiver with speed `v_s` )
This means that the shift in the original frequency will be proportional to the difference in the speeds if the receiver and source:
`Delta(F) = F_0 *(Delta(v))/c = F_0*(V_r-V_s)/c`
where c is the speed of waves in the medium.
In the case of the problem the submarine and the torpedo are coming one towards the other it means
`Delta(v) = 15-(-30) =15+30 =45 m/s`
and `c =1551 m/s`
The back frequency is shifted by
`Delta(F) = 15000*45/1551 =435.2 Hz`
The frequency of the echo will be:
`F = F_0 +Delta(F) = 15000+435.2 =15435.2 Hz`
Answer: The frequency of the echo is 15435.2 Hz