Doppler shift for sonar reflecting off approaching whale

[betamu]

Homework Statement

The sound source of a ship’s sonar system operates at
a frequency of 18.0 kHz. The speed of sound in water (assumed
to be at a uniform 20°C) is 1482 m/s. What is the difference
in frequency between the directly radiated waves and the waves
reflected from a whale traveling directly toward the ship at
4.95 m/s ? The ship is at rest in the water.

Homework Equations

f_ship = (v+v_ship/v+v_whale)f_whale

The Attempt at a Solution

To find the frequency of the reflected waves, I used the above equation with v_ship=0 and v_whale=-4.95m/s. Found the reflected frequency to be 18060.32Hz, which means a 60Hz difference. My textbook says it should be double that, 120Hz.

 

[Cutter Ketch]

What you have determined is the frequency shift experienced by the whale if the whale were the receiver. Bouncing off the whale and returning to the ship doubles the effect. It’s as if the whale is now an emitter at the Doppler shifted frequency and the ship hears that frequency Doppler shifted again

By the way, I think in trying to do it in one step you’ve put f0 on the wrong side of the equation

 

[neilparker62]

Step 1: Whale is listener and listener is moving
Step 2: Whale is source and source is moving.