Can two nonzero waves add to zero?
Predict the sum of equal waves with phase difference π.
- Select the interference experiment.
- Change relative phase from 0 to π ≈ 3.14 rad (180°).
- Compare both component waves and their sum.
What to measure
At a point inspect y₁, y₂ and y₁+y₂; measure resultant amplitude.
The idea underneath
Linear displacements add. Equal contributions reinforce in phase and cancel in antiphase. Cancellation at a point does not destroy energy; spatial interference redistributes it. This principle connects to noise cancellation and antenna arrays. A phase-offset trace is a local slice, not a complete spatial radiation model.
y=y₁+y₂; A_result=2A|cos(Δφ/2)|; Δφ=2πΔr/λ+φ₀A is each amplitude in m; Δφ in rad; path difference Δr and wavelength λ in m.
Build the argument
- Write a travelling disturbance as y = A sin(kx − ωt + φ), with k = 2π/λ and ω = 2πf.
- Keeping phase constant gives v = ω/k = fλ. Add displacements, not amplitudes, when phases differ.
- Two waves travelling in opposite directions form nodes when their sum is zero for all times.
Worked example 1
For A=0.2 m and Δφ=0, resultant amplitude is 0.4 m.
Worked example 2
At Δφ=π equal contributions cancel. At Δφ=π/2 the resultant is 0.283 m.
Applications
The same language describes musical instruments, communications, seismic waves and interference-based measurement. Frequency content reveals structure hidden in a time trace.
Common misconception
The moving pattern is not the motion of a piece of the medium. Interference redistributes energy. A DFT has finite resolution and a Nyquist limit.
A further investigation
Why do 0.2 m and 0.1 m antiphase waves not cancel completely?
Check your understanding
Equal in-phase waves produce amplitude…
- 2A
- 0
- A/2
Explanation
Displacements add. Unequal antiphase amplitudes leave |0.2−0.1|=0.1 m.
Model limits
Analytic linear educational model, not a numerical fluid solver. No damping, dispersion or nonlinear coupling.