The Doppler effect with a receding receiver is a phenomenon in which the observed frequency of a wave decreases as the receiver moves away from the source. This effect is commonly experienced with sound waves, but it applies to all types of waves, including light. When the receiver is moving away from the source, the wavelengths appear to be stretched out, resulting in a lower observed frequency.

The formula for the Doppler effect with a receding receiver is:

\[ f_o = f_s \left(\frac{v - v_r}{v - v_s}\right) \]

Where:

- \( f_o \) is the observed frequency (Hz)
- \( f_s \) is the source frequency (Hz)
- \( v \) is the speed of the wave in the medium (m/s)
- \( v_r \) is the velocity of the receiver (m/s, positive when receding)
- \( v_s \) is the velocity of the source (m/s, positive when receding)

Let's calculate the observed frequency for a receding receiver:

- Given:
- Source frequency (\( f_s \)) = 440 Hz
- Speed of sound (\( v \)) = 343 m/s
- Receiver velocity (\( v_r \)) = 20 m/s (receding)
- Source velocity (\( v_s \)) = 0 m/s (stationary)

- Apply the Doppler effect formula: \[ f_o = f_s \left(\frac{v - v_r}{v - v_s}\right) \]
- Substitute the known values: \[ f_o = 440 \text{ Hz} \left(\frac{343 \text{ m/s} - 20 \text{ m/s}}{343 \text{ m/s} - 0 \text{ m/s}}\right) \]
- Simplify: \[ f_o = 440 \text{ Hz} \left(\frac{323 \text{ m/s}}{343 \text{ m/s}}\right) \]
- Perform the final calculation: \[ f_o = 440 \text{ Hz} \times 0.9417 \approx 414.35 \text{ Hz} \]

Let's visualize the Doppler effect with a receding receiver:

This diagram illustrates:

- The stationary source (blue circle) emitting sound waves
- The receiver (red circle) moving away from the source
- The sound waves (green curves) being stretched out as the receiver recedes
- The receiver velocity (\( v_r \)) represented by a red arrow

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