Can adding sound make things quieter?

Start with what you see

At one point in a medium, the combined disturbance is the sum of the arriving waves. A positive displacement and an equally large negative displacement sum to zero. This describes a local result, not the destruction of all sound everywhere.

What is happening?

Superposition adds disturbances point by point. Cancellation requires the right amplitude and timing at the observation point. Real rooms and broadband noise are more complicated.

Find the quiet point

Change the phase of the second wave.

Together

Matching peaks reinforce each other.

Partly shifted

The combined amplitude changes.

Opposite

Equal opposite disturbances cancel in this ideal model.

Equal frequency and amplitude; linear superposition. The optional sound is a synthetic tone, not a recording or a headphone test.

Try a concrete example

Imagine two identical ripples meeting: where both raise the surface, the rise is larger; where one raises it and the other lowers it equally, the displacements cancel. Move to another point and the timing can be different. This is why silence at one position does not imply silence throughout a room. A useful prediction always names the observation point, the waves being compared and the assumptions about their timing.