E. Waves and Sound · Waves and SoundThe Doppler Effect and Sonic Booms
E2.5 E2.1 E1.1About 70 minutes
Learning goals- I can explain, with a wavefront diagram, why a moving source sounds higher in pitch as it approaches and lower as it moves away.
- I can calculate the frequency heard by a stationary observer from a moving source.
- I can explain how a shock wave and a sonic boom form, and calculate Mach number.
- I can describe technologies that use the Doppler effect.
Words to know
Doppler effect
The change in the observed frequency of a wave because the source (or observer) is moving.
observed (apparent) frequency f′
The frequency the listener actually receives.
Mach number
The speed of an object divided by the speed of sound in the surrounding air.
subsonic / supersonic
Slower than the speed of sound (Mach < 1) / faster than the speed of sound (Mach > 1).
shock wave
A cone-shaped region of very high pressure formed when a source moves at or faster than the waves it creates.
sonic boom
The loud bang heard when a supersonic object's shock wave passes over a listener.
Neeee-yowww
Stand beside a road as an ambulance races past with its siren on. As it comes toward you the siren sounds high. The instant it passes, the pitch drops: neeee-yowww. The siren itself never changed. What changed was the spacing of the waves reaching your ears. This is the Doppler effect, named after the Austrian physicist Christian Doppler, who described it in 1842.
Why the pitch changes
A stationary source sends out wavefronts that spread as circles centred on the source, evenly spaced: every listener hears the true frequency.
Now let the source move. Each new compression is made from a spot a little farther along. So:
- In front of the source, the wavefronts are squeezed together: shorter wavelength, more waves reach you each second, higher frequency (higher pitch).
- Behind the source, the wavefronts are spread apart: longer wavelength, lower frequency.
The speed of the sound in the air is unchanged; it depends only on the air. Only the wavelength (and so the frequency heard) changes.
The Doppler equation (moving source, stationary observer)
Let f be the frequency the source emits, v the speed of sound, and v_s the speed of the source. Then the frequency heard is
Approaching: f′ = f × v / (v − v_s)
Moving away: f′ = f × v / (v + v_s)
Check: approaching makes the bottom smaller, so f′ > f (higher pitch). ✓
Where does it come from? In one period T the source moves v_sT toward you, so the wavelength in front shrinks to λ′ = (v − v_s)T. Then f′ = v/λ′ = v/[(v − v_s)T] = f v/(v − v_s).
Breaking the sound barrier
What if the source moves as fast as the sound it makes? Every wavefront piles up at the front of the source, forming a wall of high pressure. If the source goes faster than sound, the wavefronts overlap to form a cone called a shock wave, trailing behind the object. When the cone sweeps over the ground, people hear a sudden sonic boom. The boom is not a one-time event at the moment the plane "breaks the sound barrier": the plane drags the cone along the whole time it flies supersonically.
We describe such speeds by the Mach number:
Mach number = v_object / v_sound
Mach 1 is the local speed of sound. Because sound is slower in cold air, Mach 1 at high altitude is a lower speed than at ground level. The crack of a whip is a small sonic boom: the tip of the whip moves faster than sound.
Using the Doppler effect
The Doppler effect happens with all waves, including light and radio waves, which leads to many technologies:
- Doppler ultrasound: ultrasound reflected from moving blood cells comes back at a slightly changed frequency, so doctors can measure blood flow in arteries and in a baby's heart.
- Radar speed guns and weather radar: radio waves reflected from a moving car or moving raindrops come back shifted in frequency, which reveals their speed (and, for storms, whether winds are rotating).
- Bats detect the motion of insects partly from Doppler shifts in their echoes.
- Astronomy: light from galaxies moving away from us is shifted toward longer (redder) wavelengths.
Worked exampleExample 1: An ambulance siren
An ambulance siren emits 1000 Hz and travels at 30 m/s. The speed of sound is 340 m/s. What frequency does a stationary listener hear (a) as it approaches, (b) as it moves away?
- (a) Approaching: f′ = f v/(v − v_s) = 1000 × 340/(340 − 30) = 1000 × 340/310 = 1097 Hz (about 1.10 × 10³ Hz).
- (b) Moving away: f′ = 1000 × 340/(340 + 30) = 1000 × 340/370 = 919 Hz.
- The pitch drops by about 178 Hz as it passes.
(a) ≈ 1097 Hz (b) ≈ 919 Hz
Worked exampleExample 2: How fast is the train?
A train horn emits 440 Hz. As the train approaches, a person at a level crossing hears 480 Hz. The speed of sound is 343 m/s. How fast is the train moving?
- Approaching: 480 = 440 × 343/(343 − v_s).
- Rearrange: 343 − v_s = 440 × 343 ÷ 480 = 314.4.
- v_s = 343 − 314.4 = 28.6 m/s (about 103 km/h).
About 29 m/s (≈ 103 km/h).
Worked exampleExample 3: Mach number
A fighter jet flies at 590 m/s at an altitude where the air is −50 °C. What is its Mach number?
- Speed of sound at −50 °C: v = 331 + 0.6(−50) = 301 m/s.
- Mach number = 590 ÷ 301 ≈ 1.96, so the jet is supersonic (about Mach 2).
Just for funRiddle: What did the stationary siren say to the speeding siren?
"Stop acting so high and mighty. Once you pass me, you'll sound way more down to earth."
Hands-on activityHear and measure the Doppler effect
You need: a phone with a tone-generator app, a long sock or a light cloth bag, string (about 1 m), and plenty of open space outdoors or in a large room. A grown-up should check that the area is clear before anything is swung. Make sure the phone is tied securely.
- Play a steady 1000 Hz tone on the phone. Put the phone in the sock and tie the sock firmly to the string.
- Stand well away from people and furniture. Swing the sock in a horizontal circle at about head height or below, while a partner stands at least 3 m away and listens.
- Your partner should hear the pitch rise and fall each time around. Ask them to describe it.
- Simpler version: have someone walk or cycle quickly past you holding the phone, or listen to cars passing on a street from the sidewalk.
- Estimate the sock's speed: time 10 turns, measure the circle's radius r, and use v_s = 2πr ÷ (time per turn). Calculate the highest and lowest frequencies your partner should hear.
What to look for: with r = 1.0 m and one turn every 0.5 s, v_s ≈ 12.6 m/s, so the pitch changes between about 1038 Hz and 965 Hz: a small but clearly noticeable wobble.
Your turn
1What is the Doppler effect?
The change in the observed frequency of a wave because the source (or observer) is moving.
2As a car with a horn drives away from you, is the pitch you hear higher or lower than the true pitch? Why?
Lower: the wavefronts behind the car are spread out, so the wavelength reaching you is longer and fewer waves arrive each second.
3Does the speed of the sound in air change when the source moves?
No; the speed depends only on the air. The wavelength and observed frequency change.
4A 500 Hz source approaches you at 20 m/s (v = 340 m/s). Find f′.
f′ = 500 × 340/320 = 531 Hz.
5The same source moves away at 20 m/s. Find f′.
f′ = 500 × 340/360 = 472 Hz.
6A plane flies at 680 m/s where sound travels at 340 m/s. Find its Mach number.
7Give two technologies that use the Doppler effect.
Sample: Doppler ultrasound for blood flow, radar speed guns, weather radar.
More practice: warm-up
1In front of a moving source, are the wavefronts closer together or farther apart?
The loud bang heard when the shock wave from a supersonic object passes over you.
Moving at exactly the speed of sound.
4In f′ = f v/(v − v_s), what does v_s stand for?
5A stationary source and a stationary listener: is there a Doppler shift?
More practice: on your own
1A police siren emits 1200 Hz while the car moves toward you at 25 m/s on a 0 °C day. Find the frequency you hear.
v = 331 m/s. f′ = 1200 × 331/(331 − 25) = 1200 × 331/306 ≈ 1298 Hz (≈ 1.30 × 10³ Hz).
2A train whistle is heard at 380 Hz as the train moves away at 30 m/s (v = 340 m/s). What frequency does the whistle actually emit?
380 = f × 340/370 → f = 380 × 370/340 ≈ 414 Hz.
3A race car passes a microphone. The highest pitch recorded is 1150 Hz and the lowest is 850 Hz. Using v = 340 m/s, find the car's speed. (Hint: divide the two Doppler equations.)
f_high/f_low = (v + v_s)/(v − v_s) = 1150/850 = 1.353. So 340 + v_s = 1.353(340 − v_s) → 340 + v_s = 460 − 1.353v_s → 2.353v_s = 120 → v_s ≈ 51 m/s.
4Explain why a jet flying at Mach 1.2 at a constant altitude produces a sonic boom that is heard by people all along its flight path, not just at the moment it first passes Mach 1.
The shock-wave cone travels with the plane the whole time it is supersonic, so it sweeps over the ground along the entire path; each person hears the boom when the cone passes them.
5Why is Mach 1 a lower speed at high altitude than at sea level?
The air at high altitude is much colder, and sound is slower in colder air (v = 331 + 0.6T).
Challenge
1A bat flies toward a wall at 5.0 m/s, emitting 40.0 kHz. Treat the wall as a stationary listener. What frequency reaches the wall? (v = 343 m/s.) The echo returning to the bat is shifted again; explain why the bat hears an even higher frequency.
f′ = 40.0 × 343/(343 − 5.0) = 40.0 × 343/338 ≈ 40.6 kHz reaches the wall. The wall reflects 40.6 kHz back, and the bat is moving toward the reflected waves, so it meets them more often than once per period: the frequency it hears is shifted up again (to about 41.2 kHz).
2Derive the moving-away equation f′ = f v/(v + v_s) using wavelengths.
In one period T, the source moves v_sT away from the listener while the previous wavefront moves vT toward the listener. So the wavelength behind the source is λ′ = vT + v_sT = (v + v_s)T. Then f′ = v/λ′ = v/[(v + v_s)T] = f v/(v + v_s).
3What happens to f′ = f v/(v − v_s) as v_s approaches v? Interpret this physically.
The denominator approaches zero, so f′ grows without limit. Physically, the wavefronts in front of the source pile up on top of each other (zero wavelength) and form a shock wave: the sound barrier.
Quiz
1. As a sound source moves toward a stationary listener, the listener hears…
2. Which quantity does not change for a moving source?
3. A 600 Hz source moves away at 40 m/s (v = 340 m/s). The observed frequency is…
4. A shock wave forms when the source…
5. A jet at 510 m/s in air where sound travels at 340 m/s has a Mach number of…
6. Doppler ultrasound is used in medicine to…
Big idea: expectation E2.5 asks students to analyse the conditions that produce the Doppler effect for a sound source moving toward or away from a stationary observer, and to solve problems with it. Shock waves and technologies extend the idea (E1.1).
How to help: Listen together to traffic, trains or a passing ambulance and talk about the pitch change. The "phone in a sock" demo is memorable; supervise it and keep people well clear.
Common difficulties:
- Using the wrong sign. Remember: approaching → minus → bigger f′.
- Thinking the sound gets louder or faster. Loudness increases as it gets closer, but the Doppler effect is about frequency.
- Thinking a sonic boom happens only once. The cone follows the plane continuously.