Standing waves and musical instruments

A tube closed at one end and 67.2 cm long sounds 128 Hz on a 22 °C day, a little below the C under middle C (130.8 Hz).

Waves that stay in place

Tie a rope to a wall and shake the other end steadily. Waves run to the wall, reflect and come back, and the outgoing and returning waves overlap. At most shaking rates the result is a mess. At certain rates the rope settles into a steady pattern of loops that seem to stand still: some points never move, and others swing widely. OpenStax calls this a standing wave, formed by two identical waves moving in opposite directions. The points that do not move are nodes, and the points of largest motion are antinodes. Neighbouring nodes are half a wavelength apart.

Strings: the harmonic series

A guitar or violin string is held fixed at both ends, so both ends must be nodes. Only standing waves that fit a whole number of half wavelengths between the ends can last. The longest is one loop, with λ₁ = 2L for a string of length L. Its frequency is the fundamental, f₁ = v/2L, where v is the wave speed on the string. The next fit has two loops, λ₂ = L and f₂ = 2f₁; then three loops and f₃ = 3f₁. These are the harmonics, and OpenStax notes that tightening the string raises v and with it every one of these frequencies. Pressing a string against a fret shortens L, which also raises the note.

Air columns: open and closed pipes

Wind instruments use standing waves in a column of air. Blow across a bottle and the air inside rings at its own natural frequency. At a closed end the air cannot move, so it is a node of the air’s motion; at an open end the air moves most freely, so it is an antinode. In a tube closed at one end the lowest pattern has a node at the closed end and an antinode at the open end, a quarter of a wavelength: λ = 4L and f₁ = v/4L. OpenStax shows that only odd harmonics fit, f_n = n v/4L with n = 1, 3, 5, …. A tube open at both ends has antinodes at both ends, so λ₁ = 2L and all harmonics appear, f_n = n v/2L.

The OpenStax worked example puts numbers to this. On a 22 °C day sound travels at about 344 m/s. OpenStax calls 128 Hz the C below middle C, although with A tuned to 440 Hz that C is about 130.8 Hz. A closed tube with a 128 Hz fundamental must be L = v/4f₁ = 344 ÷ (4 × 128) = 0.672 m long. Its fourth overtone is the ninth harmonic, 9 × 128 Hz = 1.15 kHz. Because the speed of sound depends on temperature, so does the pitch of a pipe, which is why OpenStax mentions that musicians bring wind instruments to room temperature before playing with others.

Why instruments sound different

A real instrument rarely sounds only its fundamental. The string or air column vibrates in several harmonics at once, and the mix is what we hear as the tone colour, or timbre. OpenStax gives the example of middle C on a trumpet and on a clarinet: the fundamental is the same, but the overtones and their strengths differ, so you can tell the instruments apart with your eyes closed. The body of a guitar or violin adds its own resonances that strengthen some overtones. Even the human voice works this way: the throat and mouth act as a resonating tube whose shape we change to make different vowels.