Superposition

Linear sum of wave disturbances at the same point of a medium. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Superposition explains what happens when several waves reach the same point of a medium. Instead of choosing one wave and discarding the other, the medium shows one disturbance obtained by adding the instantaneous contributions.

This principle supports interference, standing waves, diffraction, sound beats, and many measurement techniques. Its power lies in a simple idea: in a linear medium, each wave passes through the others without losing its identity, while the local detector observes only the sum.

🟢 Essential level

When two pulses cross on a string, the string cannot occupy two different positions at once. At each point it takes a single position: the sum of what each pulse would produce separately. If both push upward, displacement increases; if one pushes upward and the other downward, they may cancel.

The central quantity is $y$. It is obtained from $y_1$ and $y_2$, which are signed disturbances. That detail is decisive: a downward disturbance is not added as a positive amount, but as an opposite contribution. This is why two waves can cross and, for an instant, leave the medium almost at equilibrium.

The essential reading is local: one specific point of the medium and one specific instant are being observed. The same pair of waves can reinforce in one region and compensate in another, because disturbance signs change with position and time.

[!TIP] Before speaking about intensity or energy, always ask what is being added: displacements, pressures, fields, or intensities. In linear superposition, disturbances are added.

🔵 Formal level

Basic formulas

The most direct form of the principle is point-by-point addition of disturbances:

Linear superposition of disturbances

This relation must be read locally. $y_1$ and $y_2$ must be evaluated at the same point and at the same instant. If one wave is measured at a crest and the other somewhere else, the sum does not represent a real physical state of the medium. The sign of each disturbance contains local phase information.

Real student questions

Why do waves not collide like objects?
Because in the linear regime they do not exclude one another. The medium can respond to the sum of disturbances and, after crossing, each pulse keeps propagating.

Can I always add amplitudes?
No. Only in compatible-phase cases. In general, instantaneous disturbances are added, and the resultant amplitude is inferred later.

If the sum is zero, is there no wave?
There is local cancellation at that point and instant. The individual waves still exist as contributions and may reappear separated after crossing.

Why does intensity grow so much when amplitude increases?
Because the average energy reading often depends on the square of the resultant amplitude. That is why a small amplitude reinforcement can be very noticeable.

Final synthesis

Superposition states that, in linear media, the observable disturbance is the local algebraic sum of individual disturbances. Understanding it requires adding signed values first, interpreting amplitude and intensity afterward, and always remembering the limit of linear validity.