How to solve Doppler effect problems step by step?

The Doppler effect is one of the key phenomena in physics, describing changes in the frequency of waves perceived by an observer when the wave source moves relative to them. It can be observed in sound waves, light waves, and many other fields of science. Understanding this phenomenon requires knowledge of basic formulas and a methodical approach to solving problems.

Basics of the Doppler Effect

The Doppler effect refers to the change in the frequency of a wave perceived by an observer due to the motion of the source or the observer. When the source approaches the observer, the frequency increases. In contrast, when the source moves away, the frequency decreases. This phenomenon is applied in meteorology, medicine (e.g., Doppler ultrasound), and astronomy.

General Formula for the Doppler Effect

The formula used for Doppler effect calculations is as follows:

Where:

• f' – frequency perceived by the observer,
• f – frequency emitted by the source,
• v – speed of the wave in the medium (e.g., sound in air),
• v_o – speed of the observer relative to the medium,
• v_s – speed of the source relative to the medium.

The signs in the numerator and denominator depend on the direction of motion. Use a positive sign for the observer’s velocity and a negative sign for the source’s velocity when approaching.

Doppler Effect Problems – Practical Examples

Solving problems involving the Doppler effect requires careful analysis of the question and the correct application of formulas. Let’s look at two examples.

Example 1: Approaching Train

Problem: A train moves at a speed of 30 m/s and emits a sound with a frequency of 500 Hz. An observer stands on the platform. The speed of sound in air is 340 m/s. Calculate the frequency perceived by the observer when the train:

1. Approaches,
2. Moves away.

Solution:

Use the formula:

• For approaching: f' = 500 \cdot \frac{340}{340 - 30} = 546.88 \text{ Hz}
• For moving away: f' = 500 \cdot \frac{340}{340 + 30} = 455.45 \text{ Hz}

The observer perceives a higher frequency when the source approaches and a lower frequency when it moves away.

Example 2: Doppler Shift in Astronomy

Problem: A star emits light with a frequency of 6 × 10¹⁴ Hz. It moves at a speed of 5 × 10⁶ m/s towards the observer. Calculate the frequency perceived by the observer, assuming the speed of light is 3 × 10⁸ m/s.

Solution:

Use the formula for electromagnetic waves:

The observer perceives a slightly higher frequency than the one emitted by the star.

Common Mistakes in Problem Solving

When solving Doppler effect problems, pay attention to the following key points:

• Accurately determine the direction of motion of the observer and the source.
• Correctly apply the signs in the numerator and denominator of the formula.
• Ensure proper unit conversions, especially for velocity and frequency.

Common mistakes include:

1. Confusing the observer’s velocity with the source’s velocity.
2. Failing to convert units (e.g., km/h to m/s).
3. Incorrect simplification of the formula.

Applications of the Doppler Effect

The Doppler effect has applications in various fields. Here are a few examples:

• Medicine: Doppler ultrasound is used to assess blood flow in blood vessels.
• Astronomy: Doppler shifts help determine the speed of stars and galaxies.
• Radar: Police use Doppler-based radars to measure vehicle speed.

Tips for Learners

Effective problem solving requires good organization. To improve your skills, consider the following:

• Practice with a variety of examples.
• Write down all calculation steps to avoid errors.
• Apply formulas in the context of specific situations to understand them better.

Conclusion

The Doppler effect is a fundamental concept in physics with wide-ranging applications. Understanding it requires careful attention to formulas and principles, which are best mastered through practice. Whether applied in astronomy, medicine, or everyday technology, it provides valuable insights into wave behavior.