The Doppler effect for a receding source is a phenomenon in which the observed frequency of a wave decreases as the source moves away from the observer. This effect is named after Austrian physicist Christian Doppler, who described it in 1842. In the case of a receding source, the wavelengths are stretched out, resulting in a lower perceived frequency.

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

\[ f' = f \cdot \frac{c}{c + v} \]

Where:

- \( f' \) is the observed frequency (in Hz)
- \( f \) is the original frequency emitted by the source (in Hz)
- \( c \) is the speed of sound in the medium (in m/s)
- \( v \) is the speed of the source receding from the observer (in m/s)

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

- Given:
- Original frequency (\( f \)) = 440 Hz
- Speed of sound (\( c \)) = 343 m/s
- Source speed (\( v \)) = 20 m/s

- Apply the Doppler effect formula: \[ f' = f \cdot \frac{c}{c + v} \]
- Substitute the known values: \[ f' = 440 \text{ Hz} \cdot \frac{343 \text{ m/s}}{343 \text{ m/s} + 20 \text{ m/s}} \]
- Perform the calculation: \[ f' = 440 \text{ Hz} \cdot \frac{343 \text{ m/s}}{363 \text{ m/s}} \approx 415.43 \text{ Hz} \]

Let's visualize the Doppler effect for a receding source:

This diagram illustrates:

- The sound source (blue circle) moving away from the observer (red dot)
- The sound waves (green curves) stretched out behind the moving source
- The observer perceiving a lower frequency due to the stretched waves

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