Vibrations of Strings and Pipes

Listen to a professional sitar player at a morning mehfil in Pune. They pluck the open baajh taar and a note rings out — say, middle C, 262 Hz. But lean closer and you can hear something richer behind the fundamental: a second, softer tone one octave higher (524 Hz), a still fainter one at 786 Hz, another at 1048 Hz. The pure 262 Hz pluck is accompanied by an entire ladder of higher frequencies. That ladder is what gives the sitar its unmistakable voice and distinguishes it from a flute, a violin, or a ghatam struck to the same pitch.

Walk across to a flautist playing a bansuri, adjusting the length of the air column inside the bamboo by covering and uncovering holes. The same ladder of frequencies appears, only now it is a column of air that is vibrating instead of a string. The air inside the bansuri chooses exactly which frequencies to resonate at, and those frequencies depend — in a way the physics of this article will make precise — on the length of the bamboo and on whether each end is open or closed.

The remarkable fact is that a finite length of string, or a finite length of air column in a pipe, rings only at certain discrete frequencies. A continuous medium, free to vibrate any way it likes, somehow picks out a countable set of "allowed" frequencies f_1, f_2, f_3, \ldots — arranged in an arithmetic sequence — and ignores everything in between. That choice is what separates music from noise, and what lets an instrument maker in Miraj or Kolkata craft a sitar whose seven main strings each ring out a specific pitch of the Indian classical scale.

This article explains why. It uses the theory of standing waves and normal modes — which you should read first if you have not yet — and applies it separately to strings under tension and to air columns in pipes. The results are two tight little formulas — f_n = nv/(2L) and f_n = nv/(4L) — that together describe almost every acoustic instrument ever built.

You will need the wave speed formulas from speed of waves on a string (v = \sqrt{T/\mu}) and sound waves — nature and propagation (v = \sqrt{\gamma P/\rho}). Both are used below.

Why a finite medium can only ring at certain frequencies

Start with the sitar string. It is fixed at the bridge on one end (the jawari) and at the nut on the other; the two fixed ends are separated by a length L. When you pluck the middle, two travelling waves run out in opposite directions from your finger. Each reflects from a fixed end. The reflected waves meet the continuing pluck wave and superpose, and — after a fraction of a second of transient settling — the string falls into a standing wave pattern: a fixed shape in which each point of the string oscillates up and down but the pattern itself does not travel.

The fixed ends force a hard constraint. Each fixed end cannot move. At the nut and at the bridge, the string's displacement must be zero at every instant. That is a displacement node at each end. Only those standing-wave shapes that have a node at x = 0 and a node at x = L are compatible with the string being tied down there. No other shapes are allowed — they would violate the constraint at the endpoints.

Write the standing wave as y(x, t) = 2A\sin(kx)\cos(\omega t) (the form derived in standing waves and normal modes from the superposition of two counter-travelling sine waves). The boundary conditions are:

• At x = 0: y(0, t) = 2A\sin(0)\cos(\omega t) = 0. Automatically satisfied by the \sin(kx) factor — this is why the sine-form is the right one for this problem.
• At x = L: y(L, t) = 2A\sin(kL)\cos(\omega t) = 0 at every t requires \sin(kL) = 0, which forces

Why: \sin(\theta) = 0 only at \theta = 0, \pi, 2\pi, 3\pi, \ldots. n = 0 would mean k = 0, \omega = 0 — no wave at all — so you start the count at n = 1.

The allowed wavelengths and frequencies follow immediately. From k = 2\pi/\lambda and v = f\lambda = \omega/k:

Why: the boundary conditions quantise k to a discrete set k_n = n\pi/L; everything else — wavelength, frequency — is fixed once k is fixed, because v is set by the string's tension and mass density.

The lowest frequency (n = 1) is the fundamental; the higher ones (n = 2, 3, \ldots) are the harmonics or overtones. In Indian musical theory, the fundamental is the shadja (Sa), the second harmonic is the Sa an octave higher, the third is the Pa a fifth above that, and so on — the whole raga framework is built on top of the harmonic series delivered by each string.

Three useful facts fall straight out of equation (1):

1. The harmonics are an arithmetic series: f_1, 2f_1, 3f_1, \ldots. The spacing between consecutive harmonics is always f_1 = v/(2L), which is itself the fundamental. That is what makes the sound "musical" — the human auditory system interprets any set of frequencies in this arithmetic-series pattern as a single tone with a timbre, not as separate pitches.

2. Mode n has exactly n - 1 internal nodes between x = 0 and x = L, located at x = mL/n for m = 1, 2, \ldots, n - 1. A sitar player can enforce a higher harmonic by lightly touching the string at L/2 (which forces a node there, killing the fundamental and the odd harmonics and leaving the second harmonic to dominate) — a technique that produces the high, pure "harmonic" tone heard at the climax of a raga.

3. The fundamental is controlled by three dials: the length L, the tension T (which enters through v = \sqrt{T/\mu}), and the mass density \mu. Sitar tuning adjusts T by turning the peg; a guitar player selects L by pressing a string against a fret; the instrument maker chooses \mu by picking a string material and gauge. The bassline of a sarangi, deep and growling, comes from heavier (larger \mu) strings than the sitar; the bright high pitch of a santoor hammer string comes from a thinner (smaller \mu) and more tensioned wire.

Watch the fundamental and second harmonic

Pipes — same idea, two boundary conditions

Swap the string for a column of air inside a pipe — the bore of a bansuri, or a plastic tube, or the air inside a conch-shell ( shankha ). Instead of a transverse displacement y(x, t), the oscillating quantity is the longitudinal air-parcel displacement s(x, t) (or equivalently the pressure variation p(x, t)) that you met in sound waves — nature and propagation. The wave speed is the speed of sound v = \sqrt{\gamma P/\rho}.

A wave sent down the pipe reflects from each end, and after transients die out, a standing wave is set up — provided the frequency matches a normal mode. Which frequencies do? That depends on the boundary conditions, and the boundary conditions depend on whether each end is open or closed.

The two boundary rules for a pipe

• Closed end (pipe capped, or a wall): air parcels cannot move through the wall. So the displacement is zero at the closed end — a displacement node. Because pressure and displacement are a quarter wavelength out of phase, this means the pressure is maximum there — a pressure antinode.

• Open end (free atmosphere): the air just outside the pipe is at atmospheric pressure, and it takes no force to push the air parcels out into that reservoir. So the pressure variation is zero at the open end — a pressure node. Equivalently, because of the quarter-wavelength phase shift, the displacement is maximum — a displacement antinode.

The last rule is not perfectly exact: the pressure node does not form at the geometric opening but slightly beyond it, because some of the wave's energy leaks into the outside air before the pressure can fully relax to atmospheric. This shift is called the end correction, and the exact mathematical treatment (due to Rayleigh, and refined by Levine and Schwinger) gives an effective length extension

for a pipe of inner radius R opening into free space. For a bansuri bore of radius 8 mm, e \approx 5 mm per open end — small but not negligible. You will meet it quantitatively below.

Open pipe: both ends open

An open pipe (open at x = 0 and at x = L) has pressure nodes at both ends. If you write the pressure standing wave as p(x, t) = p_0^{\text{amp}}\cos(kx - \phi)\cos(\omega t), the endpoint conditions p(0, t) = 0 and p(L, t) = 0 force \cos(-\phi) = 0 (so \phi = \pi/2) and hence p(x, t) = p_0^{\text{amp}}\sin(kx)\cos(\omega t). The condition at x = L then requires \sin(kL) = 0:

which is exactly the string condition. So an open pipe supports every harmonic:

The standing wave in an open pipe has displacement antinodes at the two ends (the air parcels at each end swing in and out of the pipe freely) and a displacement node in the middle for n = 1, at L/4 and 3L/4 for n = 2, and so on. A bansuri with all holes closed is effectively an open pipe from the blowing hole to the open far end — both act as pressure nodes — and you hear the full harmonic series f_1, 2f_1, 3f_1, \ldots when a note is played. That full harmonic content is why the bansuri sounds bright and full.

Closed pipe: one end closed, one end open

Now close one end of the pipe, say x = 0. The boundary conditions become:

• Pressure antinode at x = 0 (closed end): \partial p/\partial x = 0 at x = 0, or equivalently p(x, t) has maximum amplitude there — written as p(x, t) = p_0^{\text{amp}}\cos(kx)\cos(\omega t), which already satisfies the closed-end condition.
• Pressure node at x = L (open end): \cos(kL) = 0, which requires

Re-index with n_{\text{odd}} = 1, 3, 5, \ldots:

The allowed frequencies are

Only odd harmonics: f_1 = v/(4L), f_3 = 3v/(4L), f_5 = 5v/(4L), and so on. The even harmonics (n = 2, 4, 6) are absent. The fundamental of a closed pipe is at v/(4L) — half the fundamental of an open pipe of the same physical length. Close one end of a bansuri (say, by putting your palm over the far hole) and the pitch drops an octave. A shankha — the conch blown during temple aarti — is approximately a closed-end resonator (the closed end is the narrow shell tip), which is why its rich low note sounds so much deeper than a straight open pipe of comparable length.

A hands-on summary

A useful side-by-side for open versus closed pipes of the same length L:

Notice a subtle point in the last row of the table: the spacing between consecutive resonances is v/(2L) for both kinds of pipe. In an open pipe, that spacing is the fundamental itself; in a closed pipe, the spacing is twice the fundamental (because you skip every even integer). This will matter for the resonance-tube experiment in a moment — the spacing, not the fundamental, is what you measure, and it gives you v directly.

End correction — the antinode sticks out of the pipe

The boundary condition "pressure node at the open end" is an idealisation. In reality the air pressure does not relax to atmospheric instantaneously at the geometric opening of the pipe; it continues to oscillate a little way into the external air before dissolving into the ambient. The effective location of the pressure node is therefore slightly outside the physical opening. The amount of overshoot is the end correction e.

For a pipe of inner radius R opening into a semi-infinite free region (the ambient atmosphere), Rayleigh's calculation gives approximately

often rounded to 0.6\,R for everyday estimates. For an unflanged opening (plain pipe into still air) the value is closer to 0.6\,R; for a flanged opening (the pipe ends in a large flat plate, like an organ flute rank) it is closer to 0.8\,R.

For a pipe closed at one end and open at the other, the effective acoustic length is

with the correction applied only at the open end. For a pipe open at both ends, the correction applies at each end:

So the corrected formulas are:

Open pipe:

Closed pipe:

For a bansuri with bore radius R = 8 mm and length L = 40 cm played as an open pipe, e \approx 5 mm per end, so L_{\text{eff}} = 410 mm — about 2.5% larger than L. The corrected fundamental is

a little over two percent flatter than the naïve 343/(2 \times 0.400) = 429 Hz. A real flute maker tunes by reaming the bore to compensate — instrument making is applied end-correction theory.

There is a clever way to eliminate the end correction from a measurement of v: take the difference of two consecutive resonance lengths. The end correction enters the effective length as a constant e, so it cancels. This is the central idea of the resonance-tube experiment, below.

Explore how the fundamental depends on pipe length

The resonance-tube experiment — measuring v from a tuning fork

This is a standard JEE-relevant experiment and a small masterpiece of how careful design sidesteps a systematic error.

Apparatus. A vertical glass tube is clamped upright, open at the top. The lower end dips into a reservoir of water so that the effective length of the air column inside the tube is whatever stretch of tube is above the water line. A tuning fork of known frequency f is struck and held horizontally above the open top. The water level is raised or lowered (by adjusting the reservoir height) until the column rings — resonance is heard as a sudden increase in loudness, like a miniature organ pipe. Two successive resonance lengths L_1 < L_2 are recorded.

Theory. The air column above the water is a closed pipe of length L (plus end correction e). Its resonance frequencies are f_n = n v/[4(L + e)] for n = 1, 3, 5, \ldots.

If the fork has frequency f and the first resonance is at length L_1 with n = 1:

If the second resonance (at the same fork frequency) occurs at length L_2 with n = 3:

Equations (3) and (4) both contain the unknown e. But their difference eliminates it:

Subtracting:

Why: the end correction e enters each resonance length as a fixed additive constant, so it cancels when you take the difference. Once you know the fork's frequency (stamped on its base, typically 256 Hz or 512 Hz for a school experiment) and you have measured L_1 and L_2 with a metre ruler, v follows from a single multiplication.

Extracting the end correction too. If you want e as well, use either of equations (3) or (4) with v from (5). For example, from (3):

In a typical careful measurement with a 512 Hz fork in a 2.5 cm diameter tube, L_1 \approx 0.155 m and L_2 \approx 0.490 m, giving v \approx 2 \times 512 \times 0.335 \approx 343 m/s — in agreement with the value for dry air at 20 °C to three significant figures.

Worked examples

Common confusions

• "Only the fundamental is present; the harmonics are separate sounds." No — when you pluck a string, every harmonic is excited simultaneously, with an amplitude that depends on where and how you plucked. The shape of the pluck determines the relative mix of harmonics; that mix is the Fourier decomposition of the initial shape. What you hear as "a sitar tone" is the fundamental plus a characteristic blend of harmonics. Change the plucking position (from near the middle to near the bridge) and the mix of harmonics changes — the pitch stays the same but the timbre changes.

• "Closed pipes have no even harmonics because even-numbered modes are forbidden by some deep law." They are not forbidden by any deep law — they simply do not satisfy the boundary conditions. The "even" modes would require a pressure node at the closed end, but there is only air there, not free atmosphere — it cannot relax to atmospheric pressure. Those would-be modes fail at the closed end, and only the odd-labelled ones survive.

• "The end correction is a fudge factor." It is a real, calculable, measurable quantity. Rayleigh derived it over a century ago; Levine and Schwinger computed it to higher accuracy in 1948. For a thin unflanged pipe, e = 0.6133\,R; for a flanged pipe, 0.8216\,R. Real instrument makers — bansuri craftsmen in Varanasi, organ-pipe makers worldwide — know the effective length of their pipes to the millimetre.

• "A shankha is a closed pipe." Approximately, yes: the narrow tip acts as a closed end (air cannot escape there) and the wide bell-shaped mouth as an open end. But the shankha's bore varies wildly along its length — it is not a uniform cylinder — so the simple f_n = nv/(4L) formula is an order-of-magnitude estimate, not an exact answer. The real frequencies are set by solving a more general wave equation with variable cross-section, a calculation organ-pipe theorists routinely do.

• "Pipes can't vibrate faster than a certain frequency." There is no upper limit on the allowed harmonics in principle — f_n = nv/(2L) grows without bound as n \to \infty. In practice, very high harmonics are weakly excited and strongly damped by viscous losses at the pipe walls, so the spectrum of a real instrument falls off at high n. But the modes themselves exist, and in the limit of high n and short wavelengths, the one-dimensional pipe model starts to break down because transverse modes of the bore become important (which is why a narrow bansuri sounds more "pure" than a wide tuba — fewer cross-modes).

• "Thinner strings give higher pitch." Yes, if tension is fixed, because v = \sqrt{T/\mu} rises when \mu falls, and f_1 \propto v. But in real instruments, the thinner higher-pitched strings are usually also tensioned harder — so the effect doubles. A sitar's bronze jora taar is much thinner than its steel baajh taar, and also more tensioned; both effects raise its pitch.

A small aside — why instruments sound different at the same pitch

Every pitched instrument delivers the harmonic series f_1, 2f_1, 3f_1, \ldots (open-ended cases) or f_1, 3f_1, 5f_1, \ldots (closed). But the relative amplitudes of those harmonics — the spectrum — are different for different instruments, and different even for the same instrument played in different ways. That spectrum is what the ear identifies as timbre.

A bansuri, played with all holes closed, emits a nearly pure fundamental with weak higher harmonics — which is why its tone sounds soft and round. A shehnai (reed oboe-type), by contrast, emits strong odd harmonics well above the fundamental — its characteristic nasal, piercing sound is exactly what a strong 3rd, 5th, and 7th harmonic content sounds like. A sitar's brass jawari bridge is deliberately shaped to flatten out against the string during each oscillation, which emphasises very high harmonics and gives the sitar its famous buzzing, shimmering quality. A tabla is stranger still — its struck membrane was acoustically tuned by 17th-century instrument makers to have harmonic overtones (not the anharmonic overtones of an unadorned drum), which is why a tabla can play a definite pitch where a Western kettledrum cannot. C.V. Raman explained the tabla's secret in 1920.

The common thread: the allowed set of frequencies is the same across all these instruments (a harmonic series from equations 1 or 2 or their variants), but the amplitudes depend on how the instrument is excited and damped. This article tells you where the frequencies come from. The amplitudes are a deeper story — Fourier analysis combined with the shape of the plucking function, the reed's nonlinear dynamics, and so on.

Where this leads next

• Standing Waves and Normal Modes — the mathematical foundation that this article specialised to strings and pipes. Read for the general theory of superposition of counter-travelling waves.
• Sound Waves — Nature and Propagation — where the speed of sound v = \sqrt{\gamma P/\rho} comes from; used throughout this article.
• Doppler Effect — how a moving source or moving observer changes the frequency heard. The ambulance passing at speed, the train's whistle as it roars past the Howrah platform.
• Principle of Superposition — the linearity assumption that lets you decompose any pluck into a sum of normal modes; the Fourier decomposition that underlies timbre.
• Intensity and Loudness of Sound — from the pressure amplitudes of each harmonic to the perceived loudness in decibels.