# C.5 Doppler effect (AHL)

> IB Physics HL · Theme C: Wave behaviour
> Source: https://www.owlsprep.com/study/ib-physics-hl-u3-c-5-doppler-effect/

This sub-topic covers the Doppler effect, the shift in observed frequency caused by relative motion between a wave source and observer. You will learn formulas for sound and light, and common practical applications.

**Prerequisites:** [Wave frequency and wavelength](https://www.owlsprep.com/study/ib-physics-hl-u3-c-2-travelling-waves/); [Wave speed in a medium](https://www.owlsprep.com/study/ib-physics-hl-u3-c-2-travelling-waves/)

## Learning objectives

- Explain the physical origin of the Doppler effect for moving sources and observers
- Calculate frequency and wavelength shifts for non-relativistic motion
- Apply Doppler shift concepts to redshift, blueshift and practical problems
- Solve reflection problems requiring multiple sequential Doppler shifts

## 1. Physical Origin of the Doppler Effect

The Doppler effect describes the change in observed frequency of a wave when the source and observer move relative to one another. It occurs for all wave types, including sound, light, and water waves.

**Doppler Effect** — The change in the frequency of a wave observed by a detector moving relative to the wave source, caused by relative motion along the line connecting source and observer.

The physical cause of the shift differs for moving sources versus moving observers:
- A source moving towards an observer emits each new wavefront closer to the previous one, compressing wavelength in the direction of motion.
- An observer moving towards a stationary source encounters more wavefronts per unit time, because their speed relative to wavefronts is higher.

**Worked example:** A sound source moves towards a stationary observer. Explain why observed frequency is higher than source frequency.

1. When the source moves towards the observer, each successive wavefront is emitted from a position closer to the observer than the previous wavefront.
2. This reduces the distance between adjacent wavefronts, so the wavelength in the direction of the observer is shorter than the source wavelength.
3. The speed of sound in air is constant for a fixed medium, so $v = f' \lambda'$. A lower $\lambda'$ gives a higher observed frequency $f'$.

> **Exam tip:** Motion towards always increases observed frequency; motion away always decreases it. Use this rule to check your calculations.

## 2. Non-Relativistic Doppler Formulas

For relative speeds much less than the speed of light ($v \ll c$), which applies to almost all sound problems on IB exams, we use the standard non-relativistic Doppler formula:

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

**Non-relativistic Doppler Shift** — Formula for observed frequency for non-relativistic motion relative to a medium. $v_o$ is observer speed, $v_s$ is source speed relative to the medium.

*Notation:* f' = f \frac{v \pm v_o}{v \mp v_s}

> **Sign Convention Mnemonic**
>
> Towards = Top Add, Away = Bottom Add:
> - Observer moving towards source: add $v_o$ to the numerator
> - Source moving away from observer: add $v_s$ to the denominator

**Worked example:** A car horn emits 400 Hz and moves towards a stationary observer at 25 m/s. Speed of sound is 340 m/s. Calculate the observed frequency.

1. Identify values: $f = 400$ Hz, $v_s = 25$ m/s, $v_o = 0$, $v = 340$ m/s. Source moves towards observer, so subtract $v_s$ from the denominator.
2. $$f' = 400 \times \frac{340 + 0}{340 - 25}$$
3. $$f' = 400 \times \frac{340}{315} \approx 432 \text{ Hz}$$
4. Check: source moves towards observer, so frequency should be higher than 400 Hz. This matches our result.

*Calculator:* allowed

## 3. Doppler Effect for Light

Light does not require a medium, so the Doppler shift for light depends only on the relative speed between source and observer. For IB HL, we only use the non-relativistic approximation for $v \ll c$, given by:

$$\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$$

Where $\Delta \lambda = \lambda' - \lambda$ is the change in wavelength, $v$ is the relative speed along the line of sight, and $c$ is the speed of light. A receding source gives a positive $\Delta \lambda$ (longer wavelength, called redshift), while an approaching source gives a negative $\Delta \lambda$ (shorter wavelength, called blueshift).

**Redshift** — Doppler shift of light from a receding source that increases observed wavelength, shifting it towards the red end of the visible spectrum.

**Worked example:** A hydrogen spectral line from a distant galaxy has a rest wavelength of 656 nm, and is measured at 682 nm on Earth. Calculate the recessional speed of the galaxy.

1. Calculate $\Delta \lambda = 682 - 656 = 26$ nm, $\lambda = 656$ nm, $c = 3.0 \times 10^8$ m/s.
2. $$v = c \frac{\Delta \lambda}{\lambda}$$
3. $$v = 3.0 \times 10^8 \times \frac{26}{656} \approx 1.2 \times 10^7 \text{ m/s}$$
4. Check: wavelength increased, so the galaxy is moving away from Earth, which matches our positive speed result.

> **Exam tip:** Always confirm that a longer wavelength corresponds to a receding source, and shorter wavelength corresponds to an approaching source.

*Calculator:* allowed

## 4. Practical Applications and Reflected Waves

Common practical applications of the Doppler effect tested in IB exams include:

- Radar speed guns: measure vehicle speed via frequency shift of reflected radio waves
- Ultrasound: measure blood flow via shift from reflected waves off moving blood cells
- Astronomy: measure galaxy recessional speed to study cosmic expansion

For reflected waves off a moving object, you must calculate two sequential Doppler shifts: first the moving object acts as a moving observer, then as a moving source emitting the reflected wave.

**Worked example:** A 5.0 MHz ultrasound wave is reflected off blood moving towards the probe at 0.50 m/s. Speed of ultrasound in tissue is 1500 m/s. Calculate the frequency shift.

1. First step: blood acts as a moving observer towards the stationary source.
2. $$f' = f \frac{v + v_{\text{blood}}}{v} = 5.0 \times 10^6 \times \frac{1500 + 0.5}{1500} = 5001667 \text{ Hz}$$
3. Second step: blood acts as a moving source emitting frequency $f'$ towards the stationary probe.
4. $$f'' = f' \frac{v}{v - v_{\text{blood}}} = 5001667 \times \frac{1500}{1500 - 0.5} \approx 5003333 \text{ Hz}$$
5. $$\Delta f = f'' - f = 3300 \text{ Hz} = 3.3 \text{ kHz}$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Adding $v_s$ to the denominator when a source moves towards an observer
  - Why it fails: Sign convention is often misremembered, leading to the opposite frequency shift
  - Correct: Use the mnemonic: towards = subtract $v_s$ from the denominator, add $v_o$ to the numerator; away = reverse the signs
- **Wrong:** Using separate source and observer speed formulas for light, with medium-dependent terms
  - Why it fails: Light does not travel through a medium, so the formula only depends on relative speed
  - Correct: Use $\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$ for light, where $v$ is the relative speed along the line of sight
- **Wrong:** Claiming redshift means higher observed frequency
  - Why it fails: Redshift is defined as a shift to longer wavelength, which corresponds to lower frequency
  - Correct: Remember: redshift = longer wavelength, lower frequency; blueshift = shorter wavelength, higher frequency
- **Wrong:** Calculating only one Doppler shift for a reflected wave off a moving object
  - Why it fails: Most problems only require one shift, so it is easy to miss the second shift for reflections
  - Correct: Always calculate two sequential shifts: one for the moving object as observer, one as moving source
- **Wrong:** Calculating a Doppler shift for motion perpendicular to the line of sight
  - Why it fails: The Doppler effect only arises from motion along the line connecting source and observer
  - Correct: If there is no component of velocity along the line of sight, the Doppler shift is zero

## Cheatsheet

| Scenario | Formula | Rule of Thumb |
| --- | --- | --- |
| Moving observer, stationary source | $f' = f \frac{v \pm v_o}{v}$ | Towards: +, Away: - |
| Moving source, stationary observer | $f' = f \frac{v}{v \mp v_s}$ | Towards: -, Away: + |
| Non-relativistic light | $\frac{\Delta \lambda}{\lambda} = \frac{v}{c}$ | Receding: +v = redshift |
| Reflected wave off moving object | Two sequential shifts | First observer, then source |

## What's next

Mastery of the Doppler effect is critical for IB Physics HL exams, as it regularly appears in both Paper 1 calculation and Paper 2 extended response questions. This core wave phenomenon connects fundamental wave properties to real-world applications in medicine, engineering and astronomy, and builds on your understanding of basic wave behaviour. The concepts you learn here provide a foundation for astrophysics topics like cosmic expansion, and for relativity, where relativistic Doppler shifts are introduced for high-speed motion. Practising sign convention and reflected wave problems will ensure you earn full marks on this common exam topic.

- [C.2 Travelling waves](https://www.owlsprep.com/study/ib-physics-hl-u3-c-2-travelling-waves/)
- [C.6 Interference (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u3-c-6-interference/)
- [C.7 Diffraction and resolution (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u3-c-7-diffraction-and-resolution/)

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