Doppler effect – Scitura

What is it about?

When an ambulance drives past, its siren sounds higher as it approaches and lower as it moves away. This is the Doppler effect.

Each wave front is a circle centred where the source was when it emitted it. A source moving with $v_Q$ advances by $v_Q \cdot T$ per period, so the fronts are closer together ahead and further apart behind:

\lambda_a = \frac{c - v_Q}{f_Q}, \qquad \lambda_b = \frac{c + v_Q}{f_Q}, \qquad f_B = f_Q \cdot \frac{c}{c \mp v_Q}.

If instead the observer moves towards a source at rest, the wavelength stays the same but the fronts are met more often: $f_B = f_Q \cdot \frac{c + v_B}{c}$ . In general $f_B = f_Q \cdot \frac{c + v_{B,r}}{c - v_{Q,r}}$ with the velocity components along the line joining them. Standing beside the road, this component changes smoothly – the pitch glides.

For $v_Q = c$ the fronts pile up (sound barrier). For $v_Q > c$ they form the Mach cone with $\sin \alpha = \frac{c}{v_Q}$ ; where it passes, a sonic boom is heard.

Only every $n$ th wave front is drawn (shown in the picture), so the brackets measure $n \cdot \lambda$ .

Try it

Press “Restart” and watch the rings and the graph $f_B(t)$ . Why is it silent at first?
Drag the observer close to the road and away again. How does the curve change? Next to the road
Press “Play sound” and restart, with and without slow motion.
Let the observer move towards a source at rest. Moving observer
Increase $v_Q$ to $340\,\tfrac{\text{m}}{\text{s}}$ and beyond. Mach 2

Tasks

Task 1: Siren

An ambulance travels at $34\,\tfrac{\text{m}}{\text{s}}$ with a $440\,\text{Hz}$ tone. What does an observer at the road hear before and after it passes? Load task

Show solution

$440\,\text{Hz} \cdot \frac{340}{306} \approx 488.9\,\text{Hz}$ and $440\,\text{Hz} \cdot \frac{340}{374} = 400\,\text{Hz}$ . Show in the simulation

Task 2: Who is moving?

A cyclist moves at $34\,\tfrac{\text{m}}{\text{s}}$ towards a $440\,\text{Hz}$ source at rest. What does he hear? Compare with task 1.

Show solution

$440\,\text{Hz} \cdot \frac{374}{340} = 484\,\text{Hz}$ , slightly less than $488.9\,\text{Hz}$ : a moving source compresses the waves themselves, a moving observer only meets them more often. Show

Task 3: Sonic boom

A plane flies at Mach 2. Find the half-angle of the Mach cone.

Show solution

$\sin\alpha = \frac{1}{2}$ , so $\alpha = 30^\circ$ . Show

Ideas for the classroom

Use: Waves and sound in year 10 or upper secondary. A nice start is the everyday experience of a siren – first with sound and without numbers (val=0).
Model: Air at rest, uniform straight motion; the emission time of the sound arriving now is computed exactly. Vehicle and person are drawn larger than to scale. Sound is off by default and quiet.
Common stumbling blocks:
- “It gets louder, so it sounds higher.” – Loudness and pitch are different quantities.
- “The pitch jumps when it passes.” – Only right on the road; at a distance it glides.
- “Moving source and moving observer are the same.” – Not for sound, because the air is a preferred frame.