When a linear system supports several waves at the same place and time, their disturbances add. This is the principle of superposition.

If two waves produce disturbances $y_1$ and $y_2$, the combined disturbance is

$$y=y_1+y_2.$$

Waves pass through one another

Two pulses travelling toward each other can overlap temporarily. During the overlap their displacements add, and afterward each pulse continues propagating according to the same underlying wave dynamics.

Superposition does not mean the waves permanently merge.

Reinforcement and cancellation

If two disturbances have the same sign at a point, the combined amplitude is larger. If they have opposite signs, they partially or completely cancel.

For example,

$$A+(-A)=0$$

at a point gives complete instantaneous cancellation.

Linear approximation

Superposition is exact for ideal linear wave equations. Real systems can become nonlinear at sufficiently large amplitudes, in which case overlapping waves may interact in more complicated ways.

Interference is the systematic pattern produced when superposing waves with controlled phase relationships.