Harmonics & Musical Sources

A standing wave survives only where the boundary conditions are met, and for a uniform string clamped at both ends or a uniform air column the allowed wavelengths come out in the ratio 1 : 1/2 : 1/3 : …, so the frequencies come out as fₙ = n f₁. That integer relation is what makes the radiated sound periodic: add sinusoids at f₁, 2f₁ and 3f₁ with any amplitudes and phases you like and the sum still repeats exactly every T₁ = 1/f₁. A repeating waveform is what the ear reports as a pitch. Nothing guarantees the relation. A circular drumhead's modes sit at 1, 1.59, 2.14, 2.30, … times its lowest, ratios fixed by the zeros of Bessel functions rather than by integers, so the sum never repeats and the drum has colour but only a vague pitch. Standing waves and resonance came earlier; the question here is which set of frequencies a real musical source ends up with, and how loud each member of the set is.

Clamping both ends forces a displacement node at each, so a whole number of half wavelengths must fit: λₙ = 2L/n and fₙ = n v/(2L) with n = 1, 2, 3, …. The speed is the string's own, v = √(F_T/μ), so f₁ = (1/2L)√(F_T/μ) and every tuning control follows from that one line. A guitar's top E sounds 329.6 Hz over a 0.650 m scale, so v = 2Lf₁ = 428 m s⁻¹, and a string with μ = 0.40 g m⁻¹ needs F_T = μv² = 73 N to sit there. Fretting at the twelfth fret halves L and doubles every fₙ: one octave. Raising the tension by 1% raises the pitch by only 0.5%, because f ∝ √F_T — which is why a fine tuner has to move so little. Quadrupling μ halves the pitch, and is how a bass string stays short instead of long and slack. Every n is allowed, so the ladder runs 1 : 2 : 3 : 4 : 5, whose successive steps are an octave, a fifth, a fourth and a major third.

In a sound wave an open end is very nearly a pressure node, and therefore a displacement antinode; a closed end is a displacement node and a pressure antinode. Sketch the displacement wave and the two cases separate at once. A pipe open at both ends needs an antinode at each, giving λₙ = 2L/n and fₙ = n v/(2L) with n = 1, 2, 3, …: for L = 0.500 m and v = 343 m s⁻¹ that is 343, 686, 1029 and 1372 Hz. Stop one end and the fit becomes a node at one end and an antinode at the other, which takes an odd number of quarter wavelengths: λₙ = 4L/n and fₙ = n v/(4L) with n = 1, 3, 5, …. The same 0.500 m pipe now gives 171.5, 514.5 and 857.5 Hz. Two things changed together: the fundamental dropped an octave, so a stopped pipe sounds as low as an open pipe of twice the length, and every even harmonic vanished. A clarinet is close to this case. It overblows to a twelfth, 3f₁, not an octave, and its lowest note near 147 Hz implies an effective length 343/(4 × 147) ≈ 0.58 m.

The pressure node does not sit exactly at the mouth of a pipe. Air just outside is dragged along with the column, so the tube behaves as though it ran slightly further, by about 0.6a at an unflanged end of bore radius a. Take L = 0.500 m and a = 0.020 m. Open at both ends there are two corrections, so L(eff) = 0.524 m and f₁ = 343/(2 × 0.524) = 327 Hz rather than 343 Hz — 4.6% flat, or 81 cents, close to a semitone. Stopped, only one correction applies: L(eff) = 0.512 m and f₁ = 167 Hz. Any pipe that is not thin compared with its length has to be measured, not merely cut to a formula. Temperature shifts the whole spectrum as well, since v ≈ 331√(1 + T/273) m s⁻¹ in air: warming a wind instrument from 15 °C to 25 °C sharpens it by 1.7%, near 30 cents, while a string, whose speed depends on tension and not on the air, does not follow.

The boundary conditions hand you a list of allowed frequencies and say nothing about their amplitudes. Those come from how the source is started. Release a string from a shape y(x,0) and expand that shape in the modes: y(x,0) = Σ Aₙ sin(nπx/L), with Aₙ = (2/L) ∫₀^L y(x,0) sin(nπx/L) dx. Pluck it into a triangle whose corner sits at x = pL and the integral returns Aₙ ∝ sin(nπp)/n². Two audible consequences follow. The 1/n² makes a pluck low-harmonic-heavy, and the sine factor punches holes in the series: at p = 1/5 the amplitudes run 1, 0.40, 0.18, 0.063, 0, 0.028, and the fifth harmonic is gone because the pluck point is one of its nodes. Plucking at the midpoint kills every even harmonic. Plucking near the bridge makes p small, so sin(nπp) ≈ nπp and the fall flattens to 1/n, leaving many more strong harmonics — the same string at the same pitch, sounding thin and bright.

Plot amplitude against frequency and a sustained note becomes a line spectrum: one line at each n f₁, its height set by the excitation and by how well the body radiates that frequency. Those heights are the main physical content of timbre, and a clarinet's near-missing even harmonics stand out in the plot before you hear them. The steady spectrum is not the whole story. Cut the first 50 ms from a recorded piano note and it becomes hard to name, because the attack — hammer noise, breath noise, bow scratch, the moment before the modes settle — carries much of the identity. Real sources also drift from exact harmonicity: a stiff piano string follows fₙ = n f₁√(1 + Bn²), and with B = 4 × 10⁻⁴ the eighth partial lands 1.3%, or 22 cents, sharp of 8f₁, which is why piano octaves are tuned stretched. Pitch even survives losing f₁: partials at 200, 300 and 400 Hz still repeat every 10 ms and are heard at 100 Hz, which is how a small loudspeaker suggests a bass note it cannot radiate.