Harmonics on a String
f_n = nv/2L — first 6 harmonics.
Key Notes
String fixed at both ends: standing-wave modes have wavelengths λ_n = 2L/n, frequencies f_n = nf₁ (n = 1, 2, 3, …).
Fundamental (n = 1): one antinode in middle, nodes at both ends. Lowest frequency.
n-th harmonic: n antinodes, n+1 nodes.
All integer multiples of f₁ are PRESENT — string can produce 1st, 2nd, 3rd, 4th, … harmonics. (Difference from closed pipe.)
Frequency: f_n = nv/(2L) = (n/2L)·√(T/μ).
Higher harmonics give a 'richer' sound (timbre depends on harmonic mix).
Plucking position determines WHICH harmonics get excited. Plucking center excites odd harmonics; plucking off-center excites both.
Tuning: change T (peg) or L (finger) or μ (different string).
Formulas
Allowed wavelengths
Length L = n·λ/2 ⇒ integer half-wavelengths fit.
Allowed frequencies
n-th harmonic.
Fundamental
Lowest mode; n = 1.
Harmonic intervals (musical)
Each harmonic is an integer multiple of fundamental.
Important Points
String allows ALL integer harmonics: f₁, 2f₁, 3f₁, 4f₁, … — different from closed pipe (only odd).
Octave higher = double frequency = 2nd harmonic.
Doubling tension increases each f_n by √2.
Halving the length doubles each f_n.
Timbre (sound color) is determined by RELATIVE STRENGTHS of harmonics — same f₁ but different mixes sound different (violin vs flute).
Plucking a string at one end favours high harmonics; plucking at middle favours fundamental.
Harmonics on a String notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.