Wave Interference and Standing Waves
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletSuperposition Bench · flatten two pulses to nothing and watch both survive, hold the nodes still, then ask a closed pipe for its second harmonicSuperposition adds signed displacements point by point where two waves overlap, and the waves then pass through each other unchanged in shape, height and direction, so even complete cancellation is temporary. Two waves of the same frequency travelling in opposite directions in a confined region make a standing wave whose pattern does not travel: nodes stay fixed and the wave speed comes from $v = f\lambda$ applied to the traveling waves that build it. The allowed wavelengths follow from the boundary conditions, $\lambda = 2L/n$ for a string or an open pipe and $\lambda = 4L/n$ with odd $n$ only for a pipe closed at one end. Beats occur at the difference of the two frequencies.
Six errors dominate. Having overlapping pulses collide and rebound, or cancel permanently. Adding displacement magnitudes without signs, so a crest meeting a trough gives the sum rather than the difference. Describing a standing wave as travelling, and timing the pattern to find the wave speed. Naming the harmonic by counting bumps without matching the boundary conditions. Reusing the string formula for a pipe closed at one end, which produces harmonics that cannot exist. And reporting the beat frequency as the average or the sum of the two frequencies instead of their difference.
The work
3 ways in · any order
Lesson
Wave Interference and Standing Waves
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Superposes with signs and lets the pulses pass through, keeps the standing pattern stationary while its constituent waves travel, reads the harmonic off the boundary conditions, and takes beats as a difference.
Diagnostic
10-item topic check
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Ten items spanning the failure modes of this topic: rebounding or permanently cancelling pulses, adding displacement magnitudes without signs, a standing wave that travels, harmonics counted off bumps, one formula reused for every pipe, and beats taken as an average. Take it cold to find which one is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.