Sound That Erases Sound
On a long flight, the engines fill the cabin with a deep, steady roar for hours. Put on noise-cancelling headphones, press one button, and most of the roar is gone.
Here is the strange part. The headphones don't beat the roar with thicker padding. To make the roar quieter, the headphones add sound: a tiny speaker next to your ear plays something on purpose.
Make your guess. What does the speaker play to make the engine roar quieter?
- A calm sound, like rain, to cover up the roar
- A copy of the roar, turned upside down
- A very high whistle that numbs your ear
Show answer
B. A copy of the roar, turned upside down
The speaker plays a copy of the roar, turned upside down. Wherever the roar pushes on the air, the copy pulls, and the push and the pull cancel. Rain sounds would only cover the roar, a different trick called masking: the roar would still reach your ear.
Try the trick before any physics. Can you make a hum disappear?
Erase the hum
Your goal: drag the blue dot until the bottom line goes flat. Then turn your volume down, press Play sound, and drag the blue dot away from silence and back.
The winning blue wave is the grey hum turned upside down. Engineers call a wave like the blue one anti-noise.
Notice how picky the match is. A blue wave a little too small, or a little too late, leaves part of the hum behind. But why should two sounds cancel at all? The answer starts with what a sound is.
Sound is squeezed air
A speaker makes sound by moving a cone back and forth. Each push forward squeezes the air in front of the cone a little. Each pull back stretches the air a little thinner.
An animation of a speaker making sound. The speaker cone pushes and pulls on layers of air, and bunched and spread-out layers travel away from the cone, with a pressure wave below that moves along with them. The animation plays by itself. Every few seconds it pauses with one squeeze and one stretch labelled, then runs again.
Text version
A speaker cone on the left moves back and forth. In front of it, thin vertical layers of air sit side by side. Each push of the cone bunches the nearest layers together, and each pull spreads them apart. The bunches and gaps travel to the right, away from the cone, while each layer only rocks back and forth around its own resting spot. One layer is drawn bold, with a tick mark under its resting spot, to show this. Below the layers, a line graph shows the air pressure along the same line. The graph is highest where the layers bunch, lowest where they spread, and slides to the right with them. At each pause, the label squeezed sits above a bunch and a crest of the graph, and the label stretched sits above a gap and a dip of the graph.
The squeezes travel away from the cone at about 343 meters per second, the speed of sound. When the squeezes reach you, they push your eardrum in and out, and you hear a sound.
A steady hum is the same push and pull, repeated. The grey wave in the game is a graph of the air pressure over time: up means squeezed, down means stretched. So what happens when two sounds push on the same air at the same moment?
Two sounds, one air
The air does the simplest thing possible: the air adds the two pressures. A squeeze plus a squeeze makes a bigger squeeze. A squeeze plus an equal stretch makes no change at all.
Waves adding up like this is called interference. Adding explains why the game started at 200%: an exact copy of the hum doubled the hum. Flipping the copy turned every squeeze into a stretch, and the sum became zero.
Two ways to flip a wave (play the game first)
Two moves in the game reach silence. Drag the blue crest straight down into a dip, and the blue wave is flipped. Or slide the blue wave sideways by half a wave, so each blue crest lands on a grey dip. For a steady hum, both moves make the same upside-down wave.
Flipping a recording is easy. But headphones can't know what the engine will do next. How can headphones make a flipped copy of a sound while the sound is still arriving?
Listen, flip, play
Noise-cancelling headphones run a fast loop. A microphone on the outside of the ear cup hears the noise. A chip flips the signal and sends the flipped signal to the speaker, which plays the flipped copy into your ear.
Many headphones add a second microphone inside the ear cup, right next to your ear. Mic 2 hears whatever sound the flip missed, and the chip keeps adjusting until mic 2 hears as little as possible. Some headphones use only one of the two microphones; many use both.
The idea is older than you might think. In the 1930s a German physicist, Paul Lueg, patented cancelling sound with sound, but the electronics of his day were far too slow to use it on the go. In 1978, an engineer named Amar Bose sat on a flight, annoyed that the airline headphones couldn't beat the engine noise. By the company's own telling, Bose worked out the math for noise-cancelling headphones before the plane landed.
In 1986, two prototype Bose headsets flew on Voyager, the first plane to circle the world without stopping or refuelling. The pilots, Dick Rutan and Jeana Yeager, wore the headsets for the whole nine-day flight.
The loop has one weak spot. The flipped copy must hit your ear at exactly the right moment, and every step of the loop takes a little time. Which sounds does a tiny delay hurt most?
Suppose the flipped copy arrives just 0.1 milliseconds late. Which sound gets through the most?
- A low engine hum
- A voice
- Both the same: a delay of 0.1 ms is 0.1 ms for every sound
Show answer
B. A voice
The voice gets through more. A voice is higher than an engine hum: the pushes and pulls of a voice come faster, so 0.1 ms is a bigger piece of each wave. Try it below.
Always a little late
Your goal: slide the pitch from the engine hum up to the whistle, and watch how much sound gets through.
An engine hum might shake the air about 100 times a second, so one wave lasts 10 ms, and 0.1 ms is only 1% of a wave. A voice shakes the air hundreds to thousands of times a second. At 1,000 times a second, the same 0.1 ms is a tenth of a wave, and more than half of the sound gets through.
Push the pitch high enough, and a late flipped copy even makes the sound louder. So headphones turn the cancelling down for high sounds. A voice also changes from moment to moment, so the loop is always catching up. A steady hum is the easy case: the hum is low, and the hum never surprises the chip.
So the electronics handle the low hum. What stops the high sounds?
Two kinds of quiet
Guess first: which sound gets through a thick ear cushion more easily?
- A low hum
- A high voice
Show answer
A. A low hum
The low hum. Long, slow waves shake the cushion and the cup as a whole and leak through. A tight seal and padding block short, fast waves much better.
Blocking sound with padding and a tight seal is called passive noise cancelling. Playing anti-noise is active noise cancelling, or ANC. Good headphones use both, and each one covers the other one's weak spot.
| Sound | Examples | What stops it best |
|---|---|---|
| Low and steady | Engines, air conditioners, train rumble | Active: anti-noise |
| High or sudden | Voices, keyboard clicks, a dog barking | Passive: the seal and padding |
The game showed how much a delay costs at each pitch. Can one formula predict the game's number?
The one-line formula
Call the pitch \(f\) (waves per second) and the delay \(d\) (in seconds). When the flipped copy arrives \(d\) late, the sound left at your ear, as a fraction of the original sound, is:
For a hum at \(f = 100\) with \(d = 0.0001\) s, the formula gives \(2\sin(0.01\pi) \approx 0.06\), or 6%. For a voice at \(f = 1000\), the same formula gives 0.62, or 62%.
Show the derivation (try it yourself first)
- The sound from outside. Give the pressure wave a size of 1:
\[ p(t) = \sin(2\pi f t) \]2. The flipped copy, \(d\) late. Flipping puts a minus sign in front; being late replaces \(t\) with \(t - d\):
\[ q(t) = -\sin\bigl(2\pi f (t - d)\bigr) \]3. The air adds the two.
\[ p(t) + q(t) = \sin(2\pi f t) - \sin\bigl(2\pi f (t-d)\bigr) \]4. Use the identity \(\sin A - \sin B = 2\cos\frac{A+B}{2}\,\sin\frac{A-B}{2}\). Here \(\frac{A-B}{2} = \pi f d\), so:
\[ p(t) + q(t) = 2\sin(\pi f d)\,\cos\bigl(2\pi f t - \pi f d\bigr) \]5. Read off the size. The cosine swings between \(-1\) and \(1\), so the leftover wave has size \(2\,|\sin(\pi f d)|\).
- Small delays. For a small angle, \(\sin x \approx x\), so the sound left is about \(2\pi f d\). Double the pitch, and the leftover doubles.
- The worst case. When the delay is half a wave (\(f d = \tfrac12\)), the sine is 1 and the sound doubles: the late, flipped copy has become an exact copy. With \(d = 0.1\) ms, that happens at 5,000 waves per second, the right end of the slider.
- The first game. A copy of size \(a\), shifted by an angle \(\varphi\), leaves a wave of size \(\sqrt{1 + a^2 + 2a\cos\varphi}\). The size is zero only when \(a = 1\) and \(\varphi = 180^\circ\), or when the copy is flipped (\(a = -1\)) with no shift.
- In decibels. A leftover fraction \(r\) is \(20\log_{10} r\) decibels. A leftover of 6% is about \(-24\) dB.
The roar, explained
Now you can explain the button on your headphones. A sound is squeezed and stretched air. When two sounds meet, the air adds them, so a flipped copy of a hum adds up to silence. You did that yourself with one blue dot.
A loop of microphone, chip and speaker makes the flipped copy while the noise is still arriving. The copy is always a little late, which barely matters for a slow, low hum and matters a lot for a fast, high voice. So the engine roar fades while the flight attendant's voice comes through, with the cushion blocking much of the rest.
Where it shows up
- Planes, trains and offices: a steady rumble is exactly the low, predictable sound the loop handles best.
- Cars: many newer cars play anti-noise through their own speakers to cut engine and road boom.
- Transparency mode: earbuds use the outside microphone the other way round, playing the world back into your ear so you can hear a voice without taking the earbuds out.
Where to go next
A hum and a flipped hum opened a lot of doors. Here are the questions worth chasing next:
- If any two waves add up, can any sound be split back into simple waves?
- Why can headphones cancel a fridge hum at your ear, but no speaker can silence a whole room?
- How does a phone call remove the noise behind your voice without making anti-noise?
- What exactly is pitch, and why does a voice shake the air faster than an engine?
- Why are recording studios built with heavy walls, not just foam?
What to learn next: frequency and pitch first, then the Fourier transform, which splits any sound into pure tones, the way a prism splits light into colors. Wave interference and sound are part of AP Physics 2 and most high school physics courses. Search for "active noise control" and "zones of quiet" to see how far the idea goes.
To see the parts inside real earbuds, watch Branch Education take noise-cancelling headphones apart:
Ask Phy
- Why does a flipped copy that is half a wave late make a sound louder? Show me with a drawing.
- How do noise-cancelling headphones decide how much to cancel at each pitch?
- Explain the Fourier transform using the engine hum and the voice from this page.