Can adding sound make things quieter?

Align two waves and hear how addition can bring quiet.

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Preparing the experiment…

0:00Start with sound pressure
How can a headphone add sound and make things quieter? Start with the blue waveform: it shows pressure varying at your ear, not air traveling along a drawn curve. At one point, pressure changes add together.
0:16Two waves in step
Add the orange wave in step with the blue one. Peak meets peak and trough meets trough, so the result below grows. Merely adding another sound does not produce noise cancellation; its timing matters.
0:30Shift the timing
Slide the orange waveform gradually. Its frequency stays the same, but its peaks arrive at different moments. That relative shift is phase difference. Watch the lower curve respond continuously as the timing changes.
0:45Exactly opposite
Align a blue peak with an orange trough. One raises pressure by exactly as much as the other lowers it. Their sum is zero. At this idealized listening point, this particular frequency is canceled.
1:01Amplitude must match too
Keep the opposite phase but reduce the orange amplitude. It no longer offsets the full blue wave, so a residual returns. Cancellation requires both suitable timing and a suitable amplitude, not just an inverted waveform.
1:17Restore the match
Bring the orange amplitude back toward the blue one and watch the residual shrink. Headphones must also estimate what remains near the ear and adjust their output. In a changing sound field, one fixed setting is not enough.
1:33Arrive two milliseconds late
What if processing and propagation make the cancellation sound arrive late? Add two milliseconds here. Peaks and troughs drift out of alignment, and cancellation worsens. A tiny delay can occupy a substantial fraction of a wave cycle.
1:50Change the frequency
Keep the delay and raise the frequency. The peaks are closer together, so the same time difference represents more phase rotation. The result varies with frequency; tuning one tone does not remove every sound at once.
2:05Not silence everywhere
We have been observing one location. Move elsewhere and the two sounds travel different distances, changing their phase relationship. Headphone cancellation is designed around the ear; it does not silence an entire room at once.
2:22How headphones keep up
A real headphone collects microphone signals, estimates an appropriate response and drives its speakers continuously. Our two sine waves isolate the clearest part of the idea. The full controller must also handle fit changes and complicated noise.
2:39Why steady low tones help
Return to the lower frequency and remove the extra delay. Its longer, steadier cycle makes the compensation easier to understand. Sudden and complex high-frequency sounds are harder, while the earcups also provide passive isolation.
2:55Check three ingredients
Align the waves one last time. If cancellation deteriorates, consider amplitude, phase and delay together. Superposition is straightforward; the difficult part is delivering the right wave at the right place and the right moment.

Your ear receives the sum at one place

Sound makes air pressure oscillate around its resting value. For the small amplitudes used here, pressure variations from different sources add. If one raises pressure while the other lowers it by the same amount, the total becomes smaller.

Blue is ambient sound, orange is the canceling sound, and green is their sum at the observation point. Equal amplitudes and a 180° phase difference align every peak with a trough. Under ideal conditions, the total pressure variation is zero.

Enable the quiet listening example and move phase toward 0°. The sound grows as peaks meet peaks. Adding sound can make things quieter or louder, depending on how and where the waves meet.

Noise-canceling headphones measure noise and create an appropriate sound near your ear. They control the result near the eardrum, not everywhere in a room.

Why does a tiny delay break the match?

Match the wave sizes and align opposite peaks and troughs. Watch their sum.
Add a little delay. How does the alignment change?
Restore the timing and change only the cancelling wave’s size. Why can sound remain? Start listening only if you choose, at a quiet volume.

Real noise is more than a sine wave

Further controls and numerical checks

Start with 160 Hz, 180° phase and amplitude 1. Increase delay from 0 to 1 ms, then raise frequency. The same timing error now causes a larger phase error because each cycle is shorter.

phase error = 360° × frequency × delay in seconds

Change only the canceling amplitude. Even perfectly opposite phases leave a residual when their amplitudes differ. Good cancellation needs both the right timing and the right size.

At 500 Hz, what does a 1 ms delay do to a signal that was intended to be 180° out of phase?

Why is steady low-frequency noise easier to handle?

Real noise contains many frequencies. Head movement, fit and the surroundings change acoustic paths. Headphones must estimate these paths and adjust quickly. High frequencies tolerate smaller spatial and timing errors, while steady low-frequency noise is easier to predict.

Real systems use techniques such as adaptive filtering. Simply inverting a microphone recording is not enough. Passive isolation contributes too, especially at higher frequencies.

Pressure level and energy

The readout is the residual amplitude of equal-frequency tones at one point. Relative pressure-level change is 20 log10(residual / original). Zero amplitude gives mathematical negative infinity, not unlimited real-world noise reduction.

Local cancellation does not erase all energy in the sound field. Other locations, energy exchange with sources and absorption belong to the full analysis. This experiment isolates linear superposition at one point. Narration is separate from the synthetic listening example.

Your ear receives the sum at one place

Why does a tiny delay break the match?

Real noise is more than a sine wave

Further reading & sources

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