Questions: Acoustic Resonance in Strings and Tension

Question 1

A guitar string of length L resonates at fundamental frequency f₁. A guitarist frets the string at the midpoint, halving the vibrating length while keeping tension and mass density unchanged. The new fundamental frequency is...

• A. 2f₁ — halving the length doubles the fundamental frequency
• B. f₁/2 — a shorter string vibrates more slowly
• C. 4f₁ — frequency scales as 1/L²
• D. f₁√2 — frequency scales as 1/√L

Answer: (A)

The fundamental frequency is f₁ = (1/2L)√(T/μ). Halving L gives f = (1/(2·L/2))√(T/μ) = (1/L)√(T/μ) = 2f₁. Frequency is inversely proportional to length (not length squared or √L), so halving the length doubles the frequency — exactly one octave higher. This is the physical basis for how fretting works on stringed instruments.

Question 2

Why are the resonant frequencies of a fixed string spaced at integer multiples of the fundamental (f₁, 2f₁, 3f₁, ...)?

• A. Because each harmonic requires an integer number of half-wavelengths to fit between the fixed endpoints, and frequency is inversely proportional to wavelength at fixed wave speed
• B. Because string tension increases proportionally with each successive harmonic, raising the frequency by equal steps
• C. Because higher harmonics travel faster through the string, raising their frequency above the fundamental
• D. Because each harmonic corresponds to a different linear mass density along the string

Answer: (A)

Fixed endpoints must be nodes, so the string length L must equal n·(λ/2) for integer n. This gives λₙ = 2L/n. Since f = v/λ, the resonant frequencies are fₙ = nv/(2L) = nf₁. The harmonic spacing is a direct consequence of the boundary constraint and the fixed wave speed — not changes in tension or density, which are properties of the string, not of the mode number.

Question 3

Increasing the tension of a guitar string raises all of its resonant frequencies — fundamental and harmonics alike.

• T. True
• F. False

Answer: True

True. The wave speed on the string is v = √(T/μ), so increasing tension T increases v. Since all resonant frequencies are fₙ = nv/(2L), every harmonic scales proportionally with v — the entire harmonic series shifts upward. A guitarist tightening a tuning peg raises the fundamental and all overtones simultaneously, maintaining the integer-multiple harmonic structure while shifting the overall pitch.

Question 4

The second harmonic of a string has twice the wavelength of the fundamental.

• T. True
• F. False

Answer: False

False — it has half the wavelength. The fundamental (n=1) has wavelength λ₁ = 2L: one half-wavelength fits across the string. The second harmonic (n=2) has two half-wavelengths fitting across L, so λ₂ = L = λ₁/2. The second harmonic has half the wavelength and twice the frequency of the fundamental. A common confusion is to associate 'second harmonic' with 'twice as large' — but wavelength and frequency move in opposite directions.

Question 5

Explain why a string fixed at both ends cannot resonate at an arbitrary driving frequency — why only certain discrete frequencies produce standing waves.

Model answer: A string fixed at both ends must have displacement nodes (zero motion) at both endpoints, since neither end can move. A stable standing wave requires the reflected waves from each end to arrive back in phase with the original wave. This happens only when the round-trip distance (2L) equals an integer multiple of the wavelength — equivalently, when an integer number of half-wavelengths fits exactly between the endpoints: L = nλ/2. At any other driving frequency, reflections arrive out of phase and destructively interfere, preventing any sustained standing wave from building up.

These discrete, allowed wavelengths λₙ = 2L/n directly determine the resonant frequencies fₙ = nv/(2L). The harmonic series is not a coincidence — it is the set of frequencies whose wavelengths are compatible with the boundary conditions. This same logic applies to any resonating system with two boundaries: open or closed organ pipes, quantum mechanical particle-in-a-box, microwave cavities.