A guitar string vibrates in its fundamental mode. At time t = 0, the midpoint (an antinode) is at maximum displacement. What is the displacement at a point along the string that happens to be a node?
AMaximum — nodes and antinodes reach maximum displacement simultaneously
BZero at t = 0 but oscillating — nodes pass through zero periodically like all other points
CZero always — nodes are perpetually at zero displacement regardless of the antinode's state
DHalf-maximum — nodes lag antinodes by a quarter cycle
A node is not a point that oscillates with smaller amplitude — it is a point of perpetual zero displacement. At a node, the incident and reflected waves are always exactly out of phase and always cancel completely. The node does not 'pause' at zero; it is permanently fixed there while the antinodes oscillate. This is what makes the standing wave pattern stationary in space: nodes and antinodes do not move, even as the wave breathes in and out.
Question 2 Multiple Choice
Why do guitar strings produce only specific pitches rather than vibrating at any arbitrary frequency?
AThe string's material limits which frequencies can propagate through it
BOnly frequencies whose half-wavelength fits an integer number of times between the fixed ends produce stable standing waves
CHigher frequencies require more energy than the string can sustain
DThe resonance condition is determined by string density alone, not length
The fixed endpoints must be nodes — they cannot move. This forces a geometric requirement: the wavelength must fit such that nodes land exactly at both ends. For a string of length L, this means L = n·(λ/2) for integer n, giving discrete allowed wavelengths λ = 2L/n and frequencies f = nv/(2L). Frequencies that don't satisfy this condition produce interference that cancels out over time rather than reinforcing. The string literally 'selects' its resonant frequencies through geometry.
Question 3 True / False
In a standing wave on a string, different points along the string reach their maximum displacement at different times.
TTrue
FFalse
Answer: False
This is the crucial difference between standing and traveling waves. In a traveling wave, the phase pattern moves — different points reach maximum displacement at different times. In a standing wave, the mathematical form is 2sin(kx)cos(ωt): the spatial factor sin(kx) is fixed, and the temporal factor cos(ωt) is the same for every point. Every point reaches maximum simultaneously (when cos(ωt) = ±1) and every point passes through zero simultaneously. The wave breathes in and out in perfect unison.
Question 4 True / False
Nodes in a standing wave are fixed positions in space that remain at zero displacement at all times, regardless of the wave's amplitude.
TTrue
FFalse
Answer: True
Nodes occur where the spatial factor sin(kx) = 0. Since this is purely a function of position, not time, a node is zero at all times — even when antinodes are at maximum displacement. Increasing the wave amplitude increases antinode displacement but does not move the nodes or change their zero-displacement character. This positional permanence is what makes the word 'standing' apt.
Question 5 Short Answer
What two physical ingredients must be present simultaneously to produce a standing wave, and why is each necessary?
Think about your answer, then reveal below.
Model answer: Two waves of equal frequency and amplitude traveling in opposite directions must coexist. Counter-propagation is necessary because standing waves arise from superposition of an incident wave and its reflection; without waves moving in both directions, you have only a traveling wave. Equal frequency is necessary because if frequencies differ, the interference pattern shifts in space rather than remaining fixed. Equal amplitude ensures complete cancellation at nodes; unequal amplitudes produce partial standing waves where nodes have non-zero minimum displacement.
The mathematical result sin(kx − ωt) + sin(kx + ωt) = 2sin(kx)cos(ωt) shows both requirements: identical k (frequency) and identical amplitude allow the product form to emerge. If either condition fails, the clean separation into a fixed spatial pattern times a pure oscillation breaks down, and a true standing-wave pattern cannot form.