Noise-canceling headphones do not hide the world. They argue with it. A microphone catches the pressure wave coming in, a chip flips that wave upside down, and the speaker fires the mirror image into your ear before the noise arrives.
The figures play audio and render live 3D. Find a comfortable volume, then start pressing play and dragging the waves around.
When something vibrates, it shoves the air next to it. That air shoves the air next to it, and a ripple of pressure races outward. The surface below is that pressure, drawn as height: the peaks are air squeezed tight, the troughs are air pulled thin. Spin it around, change the pitch, and press play to hear the same wave your eyes are watching.
Here is the fact that makes everything else work. When two sound waves arrive at the same place, the air does not pick one. It adds them. At every point, the push from one wave and the push from the other combine into a single push.
If two waves rise and fall together they reinforce, and you get a taller wave. But if one rises exactly as the other falls, they fight. Line them up perfectly opposite, same shape, opposite direction, and they cancel completely. The surface goes dead flat. Silence. Drag the phase to 180° and listen to the sound vanish.
That is the entire trick. A noise-canceling headphone is a timing machine bolted to the side of your head. A tiny microphone outside hears the noise first. A chip flips it, turning every push into a pull. The inside speaker plays that flipped pressure wave beside your eardrum, and the two pressures add to nearly nothing.
Below is a droning engine hum. Press play, then switch cancellation on. Watch the outside microphone pick up the incoming wave, the signal race through the chip, and the eardrum surface collapse toward flat as the drone falls away.
If this worked on everything, the world would be silent on demand. It does not, and the reason is timing. To cancel a wave you have to play its mirror image at exactly the right moment. Get the timing wrong by even a fraction of the wave's length and your anti-noise stops subtracting and starts adding.
Low sounds, like an engine, have long slow waves, so there is plenty of time to measure them and fire back in step. They cancel beautifully. High sounds, like a voice, have tiny fast waves. By the time the chip has measured one and computed its opposite, the wave has already moved on, and the anti-noise lands in the wrong place. That is why your headphones mute the plane but not the baby two rows back.
Real headphones usually run two microphones, a feedforward mic outside that hears noise before it arrives and a feedback mic inside that listens to whatever is left and corrects the system's own mistakes. Good ones blend both. But none of it is magic. It is the same humble fact you heard a few figures ago, that two equal and opposite waves add up to nothing, executed very fast and very precisely a few millimeters from your ear.