Fit Half-Wavelengths Between the Fixed Ends
A string fixed at both ends supports only standing waves in which a whole number of half-wavelengths fits the length. From that follow its harmonics, the effect of changing tension or length, and the way to read standing-wave equations.
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Harmonics, length and tension
For a string fixed at both ends, fₙ = n v/(2L) with v = sqrt(T/μ), so consecutive resonant frequencies differ by the fundamental: resonances at 315 Hz and 420 Hz with nothing between them mean f1 = 105 Hz.
Since f is proportional to sqrt(T), a 2% change in tension changes the frequency and the wave speed by 1%, not 2%, leaves the wavelength of the fundamental (2L) unchanged, and a 15 Hz change corresponds to an original frequency of 1500 Hz.
Reading standing-wave equations
the shortest string with fixed ends is λ/2 = 0.5 m. For y = A sin(240πt) cos(4πx/5) with x in cm, ω = 240π and k = 4π/5 cm⁻¹, so v = ω/k = 300 cm/s = 3 m/s, and with μ = 4 g/cm = 0.4 kg/m the tension is μ v² = 3.6 N. If a wave y = a cos(ωt - kx) is superposed with another so that x = 0 is a node, the second wave is the reflection with a π phase change, y = -a cos(ωt + kx).
Reflections, nodes and plucking
A node 10 cm from the fixed end of a string, next to the node at the end itself, means λ/2 = 10 cm, so λ = 20 cm and v = 100 Hz x 0.20 m = 20 m/s. To produce the second harmonic of a string of length ℓ, pluck at an antinode (ℓ/4) and touch at the node (ℓ/2). Sound reflected from a wall forms standing waves with the incident sound, and the intensity minima are λ/2 apart: for 170 Hz in air at 340 m/s, λ = 2 m and the minima are 1 m apart. If a figure shows a string vibrating in p loops, its frequency is f = p v / (2L): a 1 m string of 0.5 g/m under 20 N has v = 200 m/s, so two loops give 200 Hz.
OlympIQ exam lens
- ✓Fixed string: fₙ = n v/(2L); consecutive resonances differ by the fundamental
- ✓v = sqrt(T/μ); frequency is proportional to sqrt(T) and a 2% tension change gives a 1% frequency change
- ✓Divided string: 1/n = 1/n1 + 1/n2 + 1/n3
- ✓Standing wave y = f(x) g(t): λ comes from k = 2π/λ, nodes are λ/2 apart
- ✓Reflection from a fixed end or wall reverses the phase and forms nodes at λ/2 spacing
Do not take the distance between a node and the nearest antinode as λ/2. It is λ/4; the distance between adjacent nodes (or adjacent antinodes) is λ/2.
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