Standing Waves & Interference
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
When waves overlap, their displacements add (superposition). Crest on crest builds up (constructive); crest on trough cancels (destructive). Waves bouncing back and forth on a string or in a pipe make standing waves, but only at certain frequencies.
1. Standing waves on a string
2. Pipes and beats
Beats: two sounds with slightly different frequencies drift in and out of step, so the loudness pulses at fbeat = |f1 − f2|. Musicians tune by adjusting until the beats disappear.
Two equal and opposite pulses passing through each other cancel for an instant, then continue on unchanged.
3. Example, then try it
A guitar string is 0.65 m long, and waves travel along it at 260 m/s. Find its first three harmonics.
f1 = v/(2L) = 260 ÷ 1.3 = 200 Hz.
Then f2 = 400 Hz and f3 = 600 Hz.
A pipe is 0.50 m long, and sound travels at 340 m/s. Find its fundamental frequency if it’s open at both ends, and if it’s closed at one end. The open pipe is sounded with a 342 Hz tuning fork. What beat frequency is heard?
Open–open: λ = 2L = 1.0 m, so f = 340 ÷ 1.0 = 340 Hz.
Closed at one end: λ = 4L = 2.0 m, so f = 340 ÷ 2.0 = 170 Hz. Its next harmonic is 3 × 170 = 510 Hz.
Beats: |342 − 340| = 2 Hz: two pulses of loudness every second.
Answer: 340 Hz open, 170 Hz closed; 2 Hz beats.
4. Common mistakes
A string fixed at both ends vibrates in its third harmonic. How many nodes does it have, counting the ends?
Three loops need four nodes: one at each end and two in between.
Ready to practice?
4 questions on Standing Waves & Interference, each with a hint and an explanation for every wrong answer.