Questions: Resonance in Strings with Fixed Ends

Question 1

A guitarist presses a string down at a fret, shortening the vibrating length from L to L/2 while keeping tension unchanged. What happens to the resonant frequencies?

• A. Only the fundamental frequency changes; the harmonics remain the same
• B. All resonant frequencies double, since fₙ = nv/(2L) and L is halved
• C. All resonant frequencies halve, because less string is available to vibrate
• D. The resonant frequencies are unchanged because the same string and tension are used

Answer: (B)

The resonant frequencies are fₙ = nv/(2L). Wave speed v = √(T/μ) depends on tension and string density, neither of which changes when you press a fret. Only L changes — it halves. Since fₙ is proportional to 1/L, halving L doubles every harmonic: f₁ becomes 2f₁, f₂ becomes 2f₂, and so on. This is how fretting works: all harmonics shift up by the same factor, so the entire pitch rises by one octave.

Question 2

Why can only specific frequencies create standing waves on a string fixed at both ends, rather than any arbitrary frequency?

• A. Most frequencies dissipate too quickly in the string material to build up amplitude
• B. The boundary condition — zero displacement at both fixed ends — requires the string length to equal an integer number of half-wavelengths
• C. Wave speed on the string changes with frequency, filtering out most values
• D. The string acts as a low-pass filter that only transmits frequencies below a cutoff

Answer: (B)

The fixed endpoints are hard constraints: the string cannot move there. Any standing wave must therefore have a node at each end. Only wavelengths satisfying L = nλ/2 (equivalently, λₙ = 2L/n) produce a pattern with nodes at both ends simultaneously. Any other wavelength would require non-zero displacement at a fixed end — an impossibility. The resonant condition is purely geometric: it is not about damping, dispersion, or filtering, but about which spatial patterns are compatible with the boundary.

Question 3

Increasing the tension in a guitar string raises the frequencies of all its harmonics by the same multiplicative factor.

• T. True
• F. False

Answer: True

Wave speed v = √(T/μ) increases with tension T, and all harmonic frequencies fₙ = nv/(2L) are proportional to v. Doubling the tension multiplies v by √2, which multiplies every harmonic by the same factor √2. This is why tuning a string raises (or lowers) its pitch uniformly across all harmonics rather than distorting the harmonic relationships — the timbre (ratio of harmonics) is preserved while the pitch shifts.

Question 4

A string fixed at both ends can sustain a standing wave at any frequency, provided the driving amplitude is small enough to avoid nonlinear effects.

• T. True
• F. False

Answer: False

Resonance in a fixed-fixed string is not a matter of amplitude — it is a geometric constraint. Only frequencies corresponding to integer multiples of the fundamental (fₙ = nv/2L) produce standing waves, because only these frequencies yield wavelengths where exactly n half-wavelengths fit in length L, placing nodes at both fixed ends. At any other driving frequency, the reflections from the two ends interfere destructively and no standing wave builds up, regardless of how small the amplitude is.

Question 5

A guitarist plucks a string near the bridge and hears a bright, cutting tone; plucking near the middle produces a rounder, warmer sound. Explain this difference in terms of the resonance condition and harmonic content.

Model answer: Plucking at a point excites harmonics whose antinodes are at or near that point, and suppresses harmonics whose nodes fall there. The middle of the string is a node for all even harmonics (n = 2, 4, 6...) and an antinode for odd harmonics (n = 1, 3, 5...). Plucking near the middle therefore strongly excites the fundamental and odd harmonics while suppressing even harmonics — producing a round, hollow tone. Plucking near the bridge, which is close to a node for all harmonics, excites higher harmonics more strongly and the fundamental less, producing a bright, cutting sound rich in high-frequency overtones.

This is a direct application of the resonance condition: the pluck point determines the initial displacement shape of the string, which can be decomposed into contributions from each harmonic. A point that is a node for a given harmonic contributes nothing to that harmonic's excitation. Guitar and other stringed instrument players exploit this continuously — playing position (sul tasto near the fingerboard vs. sul ponticello near the bridge) is one of the primary timbre controls available to them.