A string of length L is fixed at both ends and vibrates in the third harmonic (n = 3). Which of the following correctly describes this mode?
AWavelength = 2L/3, with 3 antinodes and 4 nodes
BWavelength = 2L/3, with 2 antinodes and 3 nodes
CWavelength = 2L/3, with 3 antinodes and 2 nodes
DWavelength = L, with 3 antinodes and 4 nodes
For a fixed-fixed string, λₙ = 2L/n, so λ₃ = 2L/3. The nth harmonic has n antinodes (points of maximum displacement) and n+1 nodes (including the two fixed endpoints). For n = 3: 3 antinodes and 4 nodes (2 endpoints + 2 interior nodes). A common error is forgetting to count the fixed endpoints as nodes.
Question 2 True / False
A standing wave on a vibrating string transports energy from one end to the other, just like a traveling wave.
TTrue
FFalse
Answer: False
Standing waves have zero net energy transport. They are formed by two traveling waves of equal amplitude moving in opposite directions; their energy fluxes cancel exactly. The nodes — points of permanently zero displacement — cannot transmit energy past them, which is inconsistent with net energy flow. Energy is instead stored locally, oscillating between kinetic and potential forms at each point.
Question 3 Short Answer
Why do only specific discrete frequencies produce standing waves on a fixed-ended string, rather than any arbitrary frequency?
Think about your answer, then reveal below.
Model answer: The boundary conditions require nodes at both fixed ends. This means an integer number of half-wavelengths must fit exactly in the string length L, restricting allowed wavelengths to λₙ = 2L/n and allowed frequencies to fₙ = nv/(2L) for integer n.
If the frequency doesn't satisfy the boundary conditions, the wave reflected from each end arrives out of phase with the incident wave. The result is destructive interference that prevents a stable pattern from forming. Only when a whole number of half-wavelengths spans L does the reflected wave reinforce the incident wave coherently at every point, locking in the standing wave pattern. This quantization of allowed modes is a boundary value problem — the same mathematical structure that appears in quantum mechanics when a particle is confined to a box.