# Doppler effect

> Physics SL · IB Diploma Programme Physics Standard Level
> Source: https://www.owlsprep.com/study/ib-physics-sl-u3-doppler-effect/

This module explains non-relativistic Doppler shift for sound and light, standard IB formulae, sign conventions, and step-by-step exam problem solving workflows.

**Prerequisites:** [1D kinematics and relative motion](https://www.owlsprep.com/study/ib-physics-sl-u2-1d-kinematics/); [Wave properties: frequency, wavelength and wave speed](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-properties/)

## Learning objectives

- Define the Doppler effect for sound and non-relativistic electromagnetic waves
- Distinguish between physical mechanisms for Doppler shift from moving sources vs moving observers
- Apply the IB-approved sign convention to solve quantitative Doppler shift problems
- Identify standard exam applications of the Doppler effect including radar and redshift

## Physical Origin of the Doppler Effect

**Doppler Effect** — The apparent change in observed frequency of a wave caused by relative motion along the line connecting the source and observer. When the source and observer move closer, observed frequency increases; when they move apart, observed frequency decreases.

For a stationary source, wavefronts spread out evenly in all directions with constant separation equal to the rest wavelength. If the source moves towards the observer, each successive wavefront is emitted closer to the previous one, compressing the wavelength in the direction of travel. If the observer moves towards a stationary source, they encounter wavefronts at a faster rate than if they were at rest, even though the wavelength remains unchanged.

> **tip**
>
> You can always verify the direction of the shift without using formulae: if distance between source and observer is decreasing, $f_o > f_s$. If distance is increasing, $f_o < f_s$.

**Worked example:** A stationary fire truck emits a siren of rest frequency 1000 Hz. A pedestrian runs towards the siren at 5 m/s, while a second pedestrian runs away from the siren at 5 m/s. Compare the observed frequencies for the two pedestrians.

1. Step 1: Confirm the source is stationary, so wavelength is unchanged at $\lambda = v / f_s$
2. Step 2: The observer moving towards the source encounters wavefronts faster, so observed frequency is higher than 1000 Hz
3. Step 3: The observer moving away from the source encounters wavefronts slower, so observed frequency is lower than 1000 Hz
4. Step 4: The magnitude of the frequency shift is identical for both observers, only the direction of shift differs.

*Calculator:* forbidden

## Doppler Shift Formulae for Sound Waves

**Derivation:** Derive the Doppler formula for a moving source

*Starting from:* Stationary observer, source moving at speed $v_s$ towards the observer in a medium where sound travels at speed $v$

1. $$Time between successive wavefront emissions from source is $T_s = 1/f_s$$$
2. $$Distance source travels between emissions is $v_s T_s$$$
3. $$Compressed observed wavelength is $\lambda_o = \frac{v}{f_s} - v_s T_s = \frac{v - v_s}{f_s}$$$
4. $$Observed frequency is $f_o = \frac{v}{\lambda_o} = f_s \frac{v}{v - v_s}$$$

*Conclusion:* When the source moves towards the observer, the denominator is smaller than $v$, so $f_o > f_s$ as expected.

$$f_o = f_s \frac{v + v_o}{v - v_s}$$

> **mnemonic**
>
> Use the sign rule: O+ (add $v_o$) if observer moves towards source, S- (subtract $v_s$) if source moves towards observer.

**Worked example:** A car travels towards a stationary listener at 30 m/s, emitting a horn of frequency 250 Hz. The speed of sound in air is 340 m/s. Calculate the frequency observed by the listener.

1. Step 1: Identify known values: $f_s = 250$ Hz, $v_s = 30$ m/s, $v_o = 0$, $v = 340$ m/s
2. Step 2: Source moves towards observer, so use $-v_s$ in denominator, $+0$ in numerator
3. $$f_o = 250 \times \frac{340 + 0}{340 - 30} = 250 \times \frac{340}{310}$$
4. $$f_o \approx 274 \text{ Hz}$$

**Check your understanding**

Test your sign convention understanding:

1. If an observer moves away from a stationary sound source, what is the correct sign for $v_o$?

   - Positive
   - Negative
   - Zero
   - Equal to $v_s$

   *Why:* When moving away, the observer encounters wavefronts slower, so you subtract $v_o$ from $v$ in the numerator.

*Calculator:* allowed

## Doppler Effect for Electromagnetic Waves

For electromagnetic waves like light, no propagation medium exists, so the Doppler shift only depends on the relative radial speed $v_r$ between source and observer. At IB SL, you only need the non-relativistic approximation that applies when $v_r \ll c$, the speed of light in vacuum.

$$\frac{\Delta f}{f_s} \approx \frac{v_r}{c}, \quad \frac{\Delta \lambda}{\lambda_s} \approx \frac{v_r}{c}$$

> **info**
>
> If the source moves away from the observer, light is shifted to longer (redder) wavelengths, called redshift. If the source moves towards the observer, light is shifted to shorter (bluer) wavelengths, called blueshift.

**Worked example:** A distant galaxy recedes from Earth at 1.2% of the speed of light. Calculate the fractional Doppler shift of its visible 500 nm spectral line.

1. Step 1: Identify $v_r = 0.012 c$, source moves away so shift is positive for wavelength
2. $$\frac{\Delta \lambda}{\lambda_s} = \frac{v_r}{c} = 0.012$$
3. $$\Delta \lambda = 0.012 \times 500 = 6 \text{ nm}$$

*Calculator:* allowed

## Standard IB Exam Applications

- Radar speed guns: Use the double Doppler shift for reflected microwaves to calculate the speed of moving vehicles
- Ultrasound medical imaging: Measure blood flow speed using Doppler shift of reflected sound waves
- Astronomical redshift: Calculate recession speed of distant stars and galaxies from shifted spectral lines

**Exam command terms**

IB exam questions use specific command terms for Doppler effect tasks:

- **Explain the Doppler effect** — You must refer to changing wavelength for moving source, or changing encounter rate for moving observer *(2 mark definition question)*

- **Calculate the Doppler shift** — You are expected to use the full sound formula or the EM approximate formula as specified in the question

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Swapping the sign convention for $v_s$ and using $v + v_s$ in the denominator when the source moves towards the observer
  - Why it fails: This incorrectly increases the denominator, leading to a lower observed frequency than the rest frequency, which contradicts the physical compression of wavelength
  - Correct: Always apply the rule: source moving towards observer reduces wavelength, so subtract $v_s$ from $v$ in the denominator
- **Wrong:** Using the full sound Doppler formula for light wave problems
  - Why it fails: Light has no propagation medium, so separate speeds for source and observer are undefined, leading to incorrect results
  - Correct: Use the $\Delta f / f = v_r / c$ non-relativistic approximation for all EM wave Doppler problems at SL
- **Wrong:** Forgetting that radar speed guns produce a double Doppler shift (source to moving object, then moving object to detector)
  - Why it fails: This leads to calculating half the actual speed of the target object
  - Correct: Multiply the relative speed by 2 in the $\Delta f / f$ expression for reflected wave problems
- **Wrong:** Stating that Doppler shift is caused by a change in the wave's speed in the medium for a moving source
  - Why it fails: Wave speed in a stationary medium is constant, independent of source motion, so this is a common mark-deducting misconception
  - Correct: Explain that the shift comes from compression or stretching of the wavelength between emitted wavefronts
- **Wrong:** Using relativistic Doppler shift equations for sound problems
  - Why it fails: IB SL syllabus explicitly only requires non-relativistic formulae for sound, and no Lorentz transform derivation is expected
  - Correct: Stick strictly to the IB provided formula sheet expressions for all exam calculations

## Cheatsheet

| Scenario | IB Approved Formula | Key Condition |
| --- | --- | --- |
| Observer moving towards stationary sound source | $f_o = f_s \frac{v + v_o}{v}$ | Wavelength unchanged, observer encounters wavefronts faster |
| Observer moving away from stationary sound source | $f_o = f_s \frac{v - v_o}{v}$ | Wavelength unchanged, observer encounters wavefronts slower |
| Source moving towards stationary sound observer | $f_o = f_s \frac{v}{v - v_s}$ | Wavelength compressed in direction of travel |
| Source moving away from stationary sound observer | $f_o = f_s \frac{v}{v + v_s}$ | Wavelength stretched in direction of travel |
| Non-relativistic EM Doppler shift | $\frac{\Delta f}{f_s} = \frac{v_r}{c}$ | Only valid for $v_r \ll c$ |

## What's next

Mastering the Doppler effect gives you a critical foundation for IB SL wave behaviour and astrophysics topics, which together make up ~20% of your total exam marks. You will now be able to solve all standard Doppler shift multiple choice and paper 2 calculation questions, and correctly explain the physical mechanism to earn full method marks. This concept directly extends to cosmological redshift, one of the key pieces of evidence for the expanding universe covered in the astrophysics option. It also connects to wave interference and standing wave problems where relative motion of wave sources can produce beat frequencies. Follow the links below to continue your progress through related unit topics.

- [Wave Superposition and Interference](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-interference/)

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