The Doppler effect is a phenomenon observed when there is relative motion between a wave source and an observer. It causes a change in the observed frequency or wavelength of the wave. When applied to sound waves, it explains why the pitch of a siren changes as it passes by. For light waves, it's used to measure the speed of distant stars and galaxies.

The formula for calculating the observed wavelength behind a moving source is:

\[ \lambda' = \lambda \cdot \frac{v_m - v_o}{v_m - v_s} \]

Where:

- \( \lambda' \) is the observed wavelength behind the source
- \( \lambda \) is the original wavelength emitted by the source
- \( v_m \) is the velocity of the wave in the medium
- \( v_o \) is the velocity of the observer (positive if moving away from the source)
- \( v_s \) is the velocity of the source (positive if moving away from the observer)

Let's calculate the observed wavelength behind a moving source:

- Given:
- Original wavelength (\( \lambda \)) = 0.5 m
- Velocity of the wave in the medium (\( v_m \)) = 340 m/s (speed of sound in air)
- Velocity of the source (\( v_s \)) = 20 m/s (moving away from the observer)
- Velocity of the observer (\( v_o \)) = 0 m/s (stationary)

- Apply the Doppler effect formula: \[ \lambda' = \lambda \cdot \frac{v_m - v_o}{v_m - v_s} \]
- Substitute the known values: \[ \lambda' = 0.5 \text{ m} \cdot \frac{340 \text{ m/s} - 0 \text{ m/s}}{340 \text{ m/s} - 20 \text{ m/s}} \]
- Simplify: \[ \lambda' = 0.5 \text{ m} \cdot \frac{340 \text{ m/s}}{320 \text{ m/s}} \]
- Perform the final calculation: \[ \lambda' = 0.5 \text{ m} \times 1.0625 = 0.53125 \text{ m} \]

Let's visualize the Doppler effect for a moving source and stationary observer:

This diagram illustrates:

- The source (red circle) moving away from the stationary observer (yellow circle)
- The wave propagation (green curve) showing elongated wavelengths behind the source
- The source velocity (\( v_s \)) represented by the red arrow
- The direction of motion (blue text)

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