Add wave displacements and interpret cancellation

Add contributions at one place and time

When a medium responds linearly, two overlapping wave displacements add algebraically. Resultant displacement = right-wave contribution + left-wave contribution, evaluated at the same position and time. If one contribution is +0.030 m and the other is −0.020 m, the result is +0.010 m. Adding their magnitudes would lose the direction information.

The panels show two mathematical contributions and their resultant on identical scales. They are not three separate strings. The tracked material coordinate is fixed at x = 0.5 m. Both contributions have amplitude 0.050 m, wavelength 2 m and frequency 1 Hz, but travel in opposite directions at 2 m/s. Their sum produces a standing pattern.

Compare reinforcement with a flat instant

Start at phase offset 0° and time 0 s. At the tracked point, both contributions are +0.050 m, giving +0.100 m. This is constructive interference at that point and instant. The resultant may exceed either component’s amplitude without violating the addition rule.

Move time to 0.25 s. The resultant graph is now flat across the viewing window. At the tracked point both contributions happen to cross zero, but at many other positions they are equal and opposite. For example, at x = 0 the right contribution is −0.050 m and the left contribution is +0.050 m. Their sum is zero.

A flat displacement snapshot does not mean the medium is at rest. At x = 0.5 m and t = 0.25 s, the velocity contributions are each approximately −0.314159 m/s, so the resultant transverse velocity is −0.628319 m/s. The point is crossing equilibrium at its greatest speed. At t = 0.50 s, its displacement is −0.100 m and its velocity is zero again.

Use the ledger to distinguish adding displacements from adding velocities. Both operations are valid here, but one cannot infer the second result merely from the first. A zero value and a zero rate of change are different claims.

Change phase without inventing a collision

The phase control shifts the left wave’s cycle relative to its specified formula. It selects a different steady pattern; it does not simulate changing a source while a wave is already in flight. At 90° and time zero, the tracked point has a right contribution of +0.050 m and a left contribution of zero, giving +0.050 m. Its resultant velocity is approximately −0.314159 m/s.

The phase label is not a universal statement that the two waves are 90° apart at every time: because they propagate oppositely, their relative phase at a point also evolves with time. In this model the right contribution uses sin(kx − omega t), while the left uses sin(kx + omega t + phi). The control changes phi only.

These prescribed waves pass through one another according to linear superposition. They do not bounce off each other or permanently erase one another when the sum vanishes. Strongly nonlinear media need a different model; adding two arbitrary disturbances is not always a physically accurate rule.

Cancellation is not an energy measurement

Zero displacement at an instant does not establish that energy has disappeared. In the zero-phase example, points away from nodes are moving when the resultant passes through its flat configuration. Their kinetic energy would require material mass information to calculate. The browser supplies no material density, tension, source work or energy flux, so it does not calculate an energy budget.

Likewise, do not add component intensities as though they were signed displacements. Energy-related quantities generally depend on squares and can include interference terms. This lesson verifies displacement and velocity addition, not an energy-loss claim.

Transfer to an unfamiliar overlap

At one point two contributions are +0.040 m and −0.040 m, with transverse velocity contributions +0.10 m/s and +0.20 m/s. Predict the resultant displacement and velocity, then decide whether the point is a persistent node.

Check the superposition transfer

The displacement is zero and the velocity is +0.30 m/s. The point is crossing equilibrium, so this observation cannot identify it as a persistent node. A node must retain zero resultant displacement throughout the motion, not merely at one instant.

OpenStax interference of waves introduces linear superposition. The browser’s component formulas and numerical cases are original prescribed examples, not measurements of a particular medium.