Physics · Oscillations and waves · Grade 11

The dutar string and sound

Why does the pitch rise when a string is shortened or tightened?

The situation

A dutar player turns a peg to tighten the string — the pitch rises. Pressing a finger on a fret shortens the string, and the note goes higher still. Yet the same note sounds ringing if plucked near the bridge and soft and muffled if plucked in the middle of the string. How can one string vibrate at many frequencies at once, and what determines the pitch of a note?

Topic

Sound waves. Dutar string: string length and pitch of the note

Key ideas

Standing waves
Harmonics
Pitch

Formulas

f_{n} = \frac{n}{2 L} \sqrt{\frac{T}{μ}}

Frequency of the standing waves on a string: $n = 1$ — the fundamental, $n \geq 2$ — overtones (harmonics).

v = \sqrt{\frac{T}{μ}}

Wave speed on a string: $T$ — tension, $μ$ — mass per 1 m of the string.

λ_{n} = \frac{2 L}{n}

Wavelength of the standing wave on a string: a node at each end.

b_{n} \propto \frac{\sin (n π x_{0} / L)}{n^{2}}

Amplitude of the $n$ th harmonic when the string is pulled aside at the point $x_{0}$ and released. Harmonics with a node at $x_{0}$ are not excited.

λ_{a i r} = \frac{c}{f}, c \approx 343 m/s

Wavelength of the sound wave in air (20 °C).

Theory

Standing wave

Waves from a plucked string travel in both directions, reflect at the bridge and at the fret, and add together (interference). Only vibrations for which a whole number of half-wavelengths fits on the string survive: $L = n λ / 2$ . The points that do not move are nodes; the places that vibrate most are antinodes. A standing wave does not carry energy along the string: no energy passes through a node, and in each segment it swaps back and forth between kinetic and potential forms. Use the “View” menu to look at each harmonic separately.

What determines the pitch of a note?

$f_{1} = \frac{1}{2 L} \sqrt{T / μ}$ : if you increase the tension 4 times, the frequency rises 2 times. If you halve the string length (fret 12), the frequency also rises 2 times — the note goes up by one octave. A heavier (thicker) string gives a lower sound. In the model the frets are in equal temperament: each fret raises the frequency by a factor of $2^{1 / 12} \approx 1.0595$ . On a real dutar the frets are tied to the neck with thread.

Timbre depends on where you pluck

The string is pulled into a triangular shape and released. If we break this shape down into a sum of sine waves, the amplitude of the $n$ th harmonic is proportional to $\sin (n π x_{0} / L) / n^{2}$ . If it is plucked in the middle, all the even harmonics have a node in the middle — they are not excited at all, and the sound is soft. If it is plucked near the bridge, many harmonics are excited — the sound is ringing. In slow motion you can see the “kink” running along the string.

From string to ear

A thin string hardly moves the air by itself: through the bridge it makes the soundboard vibrate, and the soundboard sends a sound wave into the air. The frequency does not change, but the wavelength does: the speed on the string is $\sqrt{T / μ}$ , while in air it is 343 m/s. The wave on the string is transverse (the string vibrates perpendicular to the direction of the wave), whereas sound in air is longitudinal (the air is compressed and rarefied along the direction of the wave) and spreads from the instrument as a spherical wave. On an ideal string the speed does not depend on frequency, so the harmonics are exact whole-number multiples; phase and group velocities, plane waves and the flow of wave energy are covered in the “Waves on a rope and a spring” experiment. The model is simplified: the string is perfectly flexible, the decay times of the harmonics are approximate, and the sound is synthesised from the harmonics of the force on the bridge ( $\propto n b_{n}$ ).

Did you know?

The word “dutar” comes from Persian: “du” means two and “tor” means string, so it is a “two-stringed” instrument.
A4 = 440 Hz (the note A of the first octave) is the tuning frequency set by the international standard ISO 16. Musical instruments and tuners are based on this note.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Swing — Grade 10
Spring pendulum — Grade 10
Waves on a rope and a spring — Grade 10
Sound: speed, pitch, ultrasound — Grade 10
Waves and the Doppler effect — Grade 11
Bridge resonance — Grade 11
Coupled oscillations — Grade 11
Nonlinear oscillations — Year 2
Fourier: the spectrum of sound — Year 3