Your target variable is δf, the magnitude of the difference in frequency between the waves emitted from the sonar device and the waves received by the device after reflecting off the whale. write an expression for δf in terms of the relevant frequencies using the subscript notation introduced above.

  f´ = f (v ±v₀) / (v ⁻₊ v_{s})

Explanation:

This is a doppler effect exercise that is the change of frequency by the relative speed of the emitter and receiver of sound

For the first part, the sound comes out of the sounding device, which is generally removed, so the speed of the emitter is vs = 0 and the receiver (the whale) is moving, so the equation is

               f ”= f ((v ± v₀) / v)

where the sign plus (+) corresponds to the observer moving towards the source that is at rest, f is the frequency emitted by the probe and f” the frequency received by the whale.

In the second part, the waves are reflected by the whale, which becomes the emitter and the receiver that is the probe remains at rest, so the expression is

               d´ = d ”(v / (v ⁻₊ ))

where the least succor for the source moving towards the observer,

Therefore, the frequency received by the probe (f´) is

              f´ = f ((v ±-v₀) / v) (v / v⁻₊ v_{s}))

              f´ = f (v ±v₀) / (v ⁻₊ v_{s})

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