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:

$${\lambda }^{\prime }=\lambda \cdot \frac{v_m-v_o}{v_m-v_s}$$

Where:

• ${\lambda }^{\prime }$ 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: $${\lambda }^{\prime }=\lambda \cdot \frac{v_m-v_o}{v_m-v_s}$$
3. Substitute the known values: $${\lambda }^{\prime }=0.5\text{ m}\cdot \frac{340\text{ m/s}-0\text{ m/s}}{340\text{ m/s}-20\text{ m/s}}$$
4. Simplify: $${\lambda }^{\prime }=0.5\text{ m}\cdot \frac{340\text{ m/s}}{320\text{ m/s}}$$
5. Perform the final calculation: $${\lambda }^{\prime }=0.5\text{ m}\times 1.0625=0.53125\text{ m}$$

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)