Doppler Effect Wavelength Behind Calculator

What is the Doppler Effect?

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.

Formula

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)

Calculation Steps

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

1. 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)
2. Apply the Doppler effect formula: \[ \lambda' = \lambda \cdot \frac{v_m - v_o}{v_m - v_s} \]
3. Substitute the known values: \[ \lambda' = 0.5 \text{ m} \cdot \frac{340 \text{ m/s} - 0 \text{ m/s}}{340 \text{ m/s} - 20 \text{ m/s}} \]
4. Simplify: \[ \lambda' = 0.5 \text{ m} \cdot \frac{340 \text{ m/s}}{320 \text{ m/s}} \]
5. Perform the final calculation: \[ \lambda' = 0.5 \text{ m} \times 1.0625 = 0.53125 \text{ m} \]

Example and Visual Representation

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)