# Doppler effect for a moving sound source

> Physics · CIE A-Level 9702
> Source: https://www.owlsprep.com/study/cie-9702-u7-doppler-effect-for-a-moving/

We cover the physical origin of Doppler shift for a moving sound source, the formal frequency formula derivation, exam-focused problem solving, and required mark scheme reasoning.

**Prerequisites:** [Wave properties: frequency, wavelength, wave speed relationship](https://www.owlsprep.com/study/cie-9702-u7-properties-of-waves/); [Basic 1D relative motion kinematics](https://www.owlsprep.com/study/cie-9702-u2-relative-motion/)

## Learning objectives

- Define the Doppler effect for sound emitted by a moving source
- Derive the observed frequency formula for a source moving towards or away from a stationary observer
- Solve quantitative problems involving source speed, speed of sound, emitted and observed frequency
- Explain mark scheme-approved physical reasoning for frequency shift in exam answers

## Physical Origin of the Shift

When a sound source moves towards a stationary observer, each successive wavefront is emitted at a position closer to the observer than the previous one. This compresses the spacing between wavefronts, reducing the observed wavelength and increasing the detected frequency.

If the source moves away from the observer, each new wavefront is emitted further away than the last, stretching the wavelength and lowering the observed frequency.

**Doppler Effect for Moving Sound Source** — The apparent shift in observed sound frequency caused by compression or rarefaction of emitted wavefronts as the source travels along the line connecting it to a stationary observer

> **info**
>
> Transverse motion (source moving perpendicular to the line of sight) produces no measurable Doppler shift for sound, as wavefront spacing remains unchanged.

**Check your understanding**

Test your basic understanding:

1. A siren moves directly towards you. What happens to the pitch you hear?

   - Increases
   - Decreases
   - Stays identical

   *Why:* Compressed wavefronts raise the observed frequency, so pitch increases.

2. The source moves perpendicular to your position. What happens to observed frequency?

   - Sharply increases
   - No measurable shift
   - Sharply decreases

   *Why:* No relative motion along the line connecting source and observer means no wavelength change.

## Derivation of the Frequency Formula

**Derivation:** Derive observed frequency for a moving sound source

*Starting from:* Speed of sound in still air = $v$, source speed = $v_s$, source emitted frequency = $f_0$

1. Time period of emitted wave: $T = 1/f_0$
2. In one time period, the source travels a distance $v_s T = v_s / f_0$ towards the observer
3. Original unshifted wavelength: $\lambda_0 = v / f_0$
4. New compressed wavelength for approaching source: $\lambda_o = \lambda_0 - v_s / f_0 = (v - v_s)/f_0$
5. Observed frequency: $f_o = v / \lambda_o = f_0 \times v / (v - v_s)$
6. For receding source, add $v_s / f_0$ to original wavelength to get $f_o = f_0 \times v / (v + v_s)$

*Conclusion:* The sign in the denominator depends only on direction of travel: minus for approaching, plus for receding.

**Worked example:** A siren emitting a 500 Hz tone moves towards a stationary observer at 20 m/s. Speed of sound in air is 340 m/s. Calculate the observed frequency.

1. Confirm the source is approaching the observer, so use the minus sign in the denominator
2. $$f_o = 500 \times \frac{340}{340 - 20}$$
3. Simplify denominator: 340 - 20 = 320 m/s
4. $$f_o = 500 \times 1.0625 = 531.25 \text{ Hz}$$
5. Round to 3 significant figures: 531 Hz

## Exam-Focused Problem Solving

**Exam command terms**

CIE 9702 uses specific command terms for these questions, note their required responses:

- **Show that** — You must substitute all given values explicitly, no skipped arithmetic steps *(Show the observed frequency is approximately 530 Hz)*

- **Calculate** — Give final answer to 2 or 3 significant figures, include the Hz unit

- **Explain** — Reference wavefront compression/rarefaction, do not only state frequency increases or decreases

> **tip**
>
> 1 independent mark is almost always awarded for selecting the correct + or - sign in the denominator, so double check direction of travel before substituting values.

**Worked example:** A train moving away from a station platform at 35 m/s emits a 1200 Hz whistle. Speed of sound is 330 m/s. Find the observed frequency for a passenger standing on the platform.

1. Confirm the source is receding, so use the plus sign in the denominator
2. $$f_o = 1200 \times \frac{330}{330 + 35}$$
3. Simplify denominator: 330 + 35 = 365 m/s
4. $$f_o = 1200 \times 0.9041 = 1085 \text{ Hz}$$
5. Round to 3 significant figures: 1090 Hz

## Common Sound Doppler Applications

The moving source Doppler effect for sound is used in multiple real-world systems that appear in CIE context questions.

- Ultrasound medical scanners that measure blood flow velocity via frequency shift of reflected sound waves
- Speed measurement devices for moving vehicles that use audible sound wave reflection
- Emergency siren pitch change that alerts drivers to approaching or receding emergency vehicles

> **note**
>
> Unlike light Doppler shift, sound Doppler shift depends on which entity is moving relative to the air medium, so you cannot use the relativistic red shift formula for sound problems.

## Common pitfalls

- **Wrong:** Mixing moving source and moving observer formulas
  - Why it fails: The two cases have different physical derivations, so swapping them gives incorrect frequency shift magnitude
  - Correct: Explicitly confirm if the source or observer is moving before selecting your formula
- **Wrong:** Using $v + v_s$ in the denominator for an approaching source
  - Why it fails: An approaching source compresses wavelength, so the denominator must be smaller than $v$ to produce a higher observed frequency
  - Correct: Use $v - v_s$ for approaching sources, $v + v_s$ for receding sources
- **Wrong:** Stating frequency shift comes from a change in speed of sound
  - Why it fails: Speed of sound in still air is constant for fixed temperature, shift comes only from changed wavefront spacing
  - Correct: Explicitly reference compressed or stretched wavelength in all explanation answers
- **Wrong:** Giving final answers to 1 significant figure
  - Why it fails: CIE mark schemes require 2 or 3 significant figures for all calculation questions, 1 s.f. loses the final accuracy mark
  - Correct: Round all final frequency answers to 3 significant figures unless specified otherwise
- **Wrong:** Forgetting the Hz unit for final frequency answers
  - Why it fails: Missing units lose 1 independent mark in almost all 9702 calculation questions
  - Correct: Add the Hz unit immediately after calculating your final value

## Cheatsheet

| Scenario | Formula | Key Exam Note |
| --- | --- | --- |
| Source moving towards stationary observer | $f_o = f_0 \frac{v}{v - v_s}$ | $f_o > f_0$, denominator smaller than $v$ |
| Source moving away from stationary observer | $f_o = f_0 \frac{v}{v + v_s}$ | $f_o < f_0$, denominator larger than $v$ |
| Standard speed of sound value | $v = 330 - 340$ m/s | Use the value given in the question, not a memorised one |

## What's next

Mastering the moving source Doppler effect is a core requirement for CIE 9702 A Level Physics, as this concept forms the foundation for extended A2 topics including electromagnetic Doppler red shift used in astrophysics, and combined source-observer relative motion problems that regularly appear on Paper 4. You will also apply your understanding of frequency shift to practical data analysis questions that test your ability to calculate unknown source speed from measured frequency values. Ensure you can fully distinguish between moving source and moving observer scenarios before progressing, as mixing these two cases is the single most common cause of lost marks on exam papers.

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