Doppler Effect Formula: How to Calculate Frequency Shift

📚 What is the Doppler Effect?

The Doppler Effect (or Doppler shift) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It's commonly heard with sound waves (like sirens) but also applies to light and other electromagnetic waves.

📜 A Brief History

The Doppler Effect is named after Austrian physicist Christian Doppler, who described the phenomenon in 1842. He first proposed it in the context of light waves to explain the colors of binary stars. While initially unconfirmed, it was later verified for sound waves by Buys Ballot in 1845.

✨ Key Principles of the Doppler Effect

• 🌊 Wave Source: The object emitting the waves (e.g., a siren on an ambulance).
• 👂 Observer: The person or instrument detecting the waves.
• 🚗 Relative Motion: The movement of the source and/or observer relative to each other.
• 📈 Frequency Shift: The change in the perceived frequency of the wave.

🧮 The Doppler Effect Formula

The formula for calculating the observed frequency ($f'$) is:

$\displaystyle f' = f \frac{v \pm v_o}{v \pm v_s}$

Where:

• 📊 $f'$ = Observed frequency
• 📶 $f$ = Source frequency
• 💨 $v$ = Speed of wave in the medium (e.g., speed of sound in air)
• 🏃 $v_o$ = Speed of the observer relative to the medium; positive if the observer is moving towards the source, and negative if moving away.
• 📢 $v_s$ = Speed of the source relative to the medium; positive if the source is moving away from the observer, and negative if moving towards.

📝 Practical Examples

Example 1: Approaching Ambulance

An ambulance siren emits a sound at a frequency of 800 Hz. You are standing still, and the ambulance is approaching you at 25 m/s. The speed of sound is 343 m/s. What is the frequency you hear?

$\displaystyle f' = 800 \frac{343 + 0}{343 - 25} = 800 \frac{343}{318} ≈ 862.26 \text{ Hz}$

Example 2: Moving Away Train

A train whistle emits a sound at a frequency of 440 Hz. You are standing still, and the train is moving away from you at 30 m/s. The speed of sound is 343 m/s. What is the frequency you hear?

$\displaystyle f' = 440 \frac{343 + 0}{343 + 30} = 440 \frac{343}{373} ≈ 404.72 \text{ Hz}$

🚗 Real-World Applications

• 🚨 Law Enforcement: Radar guns use the Doppler Effect to measure the speed of vehicles.
• 🩺 Medical Imaging: Doppler ultrasound measures blood flow.
• 🔭 Astronomy: Astronomers use the Doppler Effect to determine the velocities of stars and galaxies.
• 🌦️ Meteorology: Weather radar uses the Doppler Effect to track the movement of storms.

🔑 Conclusion

The Doppler Effect is a fundamental concept in physics with numerous practical applications. Understanding the formula and its components allows us to calculate and predict frequency shifts in various scenarios. Whether it's the sound of a siren or the light from distant stars, the Doppler Effect helps us interpret the world around us.