Why do two waves reinforce or cancel?

Two waves travelling in the same direction along the same line add up point by point: at every position the resultant displacement is the algebraic sum of the two individual displacements. Where two crests meet the resultant grows, and where a crest meets a trough it shrinks or cancels completely. This experiment shows y₁ = A₁ sin(kx − ωt), y₂ = A₂ sin(kx − ωt + φ) and their sum y = y₁ + y₂, all animated in time on a shared x-axis.

Questions

When two waves have equal amplitudes and a phase difference of zero, how many times larger is the resultant amplitude than a single wave?

What happens to the resultant displacement when two waves of equal amplitude have a phase difference of π?

If one of the two amplitudes is set to zero, what is left of the resultant?

Predict

Write down your prediction first: as the phase difference grows from 0 to π, does the resultant amplitude fall steadily, or does it stay large and only drop sharply near π?

Try

Press Play and watch the three curves move together in the +x direction. Keep A₁ = A₂ = 1.0 m and φ = 0 and note the resultant amplitude. Then set φ to 180° (π) and watch the sum curve. Finally set A₂ to 0.

Observe

At φ = 0 the two waves coincide and the resultant amplitude is twice a single wave. At φ = π the two waves are opposite everywhere and the sum stays at zero: they cancel completely. With A₂ = 0 the sum is simply the first wave.

Explain

The superposition principle

At any position x and time t the resultant displacement is the algebraic sum of the individual displacements: y = y₁ + y₂. This experiment assumes two waves travelling in the same direction (+x), with the same wavelength and frequency in a linear medium, with no boundaries, no reflections and no damping.

The resultant amplitude from phasors

Treating A₁ and A₂ as two vectors with an angle φ between them gives the resultant amplitude A = √(A₁² + A₂² + 2A₁A₂ cos φ). For φ = 0 this is A₁ + A₂, for φ = π it is |A₁ − A₂|, and when A₁ = A₂ with φ = π it is exactly zero.

Instantaneous displacement and energy

Superposition says that displacements add; it does not say that energy adds. Where the phase difference is π and the amplitudes are equal, the displacement is always zero. This cancellation is a cancellation of instantaneous displacement, and it should not be read as a claim that energy is destroyed. Energy involves the square of the amplitude and needs a fuller analysis that this experiment does not attempt.

Check

With A₁ = A₂ = 1.0 m and φ = 0 the resultant amplitude should read 2.0 m; with φ = π the sum curve should lie on zero; and with A₂ = 0 the sum should match the first wave. Compare each case with the readout panel.

Explore further

If the two waves had slightly different wavelengths, would they still cancel everywhere? If they travelled in opposite directions instead, what pattern would appear in space?