Doppler Effect Wavelength Calculator

What is the Doppler Effect?

The Doppler effect is a change in the observed frequency of a wave when there is relative motion between the source of the wave and the observer. This phenomenon affects all types of waves, including sound waves and electromagnetic waves. It was named after Austrian physicist Christian Doppler, who described it in 1842.

Formula

The formula for the observed wavelength in the Doppler effect is:

\[ \lambda_o = \lambda_s \left(\frac{v \pm v_o}{v \mp v_s}\right) \]

Where:

• \( \lambda_o \) is the observed wavelength
• \( \lambda_s \) is the source wavelength
• \( v \) is the velocity of the wave in the medium
• \( v_o \) is the velocity of the observer relative to the medium
• \( v_s \) is the velocity of the source relative to the medium

Note: Use the top sign if the source and observer are moving towards each other, and the bottom sign if they are moving apart.

Calculation Steps

Let's calculate the observed wavelength for a moving source and stationary observer:

1. Given:
   • Source wavelength (\( \lambda_s \)) = 0.5 m
   • Wave velocity in medium (\( v \)) = 340 m/s (speed of sound in air)
   • Source velocity (\( v_s \)) = 20 m/s (moving towards the observer)
   • Observer velocity (\( v_o \)) = 0 m/s (stationary)
2. Apply the Doppler effect formula: \[ \lambda_o = \lambda_s \left(\frac{v - v_o}{v + v_s}\right) \]
3. Substitute the known values: \[ \lambda_o = 0.5 \text{ m} \left(\frac{340 \text{ m/s} - 0 \text{ m/s}}{340 \text{ m/s} + 20 \text{ m/s}}\right) \]
4. Simplify: \[ \lambda_o = 0.5 \text{ m} \left(\frac{340 \text{ m/s}}{360 \text{ m/s}}\right) \]
5. Perform the final calculation: \[ \lambda_o = 0.5 \text{ m} \times 0.9444 \approx 0.4722 \text{ m} \]

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 towards the stationary observer (yellow circle)
• The wave propagation (green curve) showing compressed wavelengths in front of the source
• The source velocity (\( v_s \)) represented by the red arrow
• The direction of motion (blue text)