Standing Sound Waves

Why this chapter

Why this chapter exists

A flute, a clarinet, a bansuri and a pipe organ have no strings at all. Each makes its notes with a column of air. Sound runs along the column, reflects at its ends and builds a standing wave, the way a wave does on a string. The ends decide everything. A closed end holds the air still, and an open end lets it swing freely. These two rules explain the notes of every wind instrument and the hollow sound of a clarinet. They also give a way to measure the speed of sound with a tube of water and a tuning fork.

You will learn

By the end of this chapter you will

Explain why a closed end of a pipe is a displacement node and a pressure antinode, and an open end the reverse.
Show that a pipe open at both ends holds whole half-wavelengths, \(L = n\lambda_n/2\), and has every harmonic, \(f_n = nv/2L\).
Show that a pipe closed at one end holds odd quarter-wavelengths, \(L = n\lambda_n/4\), and has only the odd harmonics, \(f_n = nv/4L\).
Explain the end correction, \(e = 0.6r\), and find the speed of sound from a resonance tube, \(v = 2f(L_2 - L_1)\).
Name and count the overtones of a pipe, match an open and a closed pipe in unison, and find which mode answers a tuning fork.
Draw the pressure picture of any mode, with pressure nodes at the displacement antinodes.

Before you begin

From Standing Waves on a String you should know nodes, antinodes and resonance. You should know the harmonics and overtones of a string with fixed or free ends. From Propagation of Sound Waves you need sound as a longitudinal wave, and the displacement and pressure waves, \(90°\) apart, linked by \(\Delta P = -B\,\partial s/\partial x\). You should know that sound travels in air at about \(340\,\text{m/s}\), and use \(v = f\lambda\).

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Lessons

Air Columns and Their Ends

A column of air in a pipe carries standing sound waves, the way a string does. A closed end must be a displacement node and a pressure antinode. An open end must be a pressure node and a displacement antinode.

Resonance in a column of airA closed end is a displacement nodeAn open end is a displacement antinode

13 min0%

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Pipes Open at Both Ends

A pipe open at both ends has a displacement antinode at each end, so it holds a whole number of half-wavelengths, \(L = n(\lambda_n/2)\). Its resonant frequencies \(f_n = n(v/2L)\) include every harmonic. The first three modes, and the flute worked through.

An open pipe holds whole half wavelengthsEvery harmonic sounds in an open pipeThe modes of an open pipe

14 min0%

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Pipes Closed at One End

A pipe closed at one end has a node at the closed end and an antinode at the open end, so it holds an odd number of quarter-wavelengths, \(L = n(\lambda_n/4)\). Only the odd harmonics, \(f_n = n(v/4L)\), are present. The closed organ pipe and the clarinet worked through.

A closed pipe holds odd quarter wavelengthsOnly odd harmonics in a closed pipeThe modes of a closed pipe

14 min0%

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The End Correction and the Resonance Tube

The antinode at an open end lies a little outside the pipe, by the end correction \(e = 0.6r\), so a pipe sounds as if it were longer. The resonance tube finds the speed of sound from two resonance lengths, \(v = 2f(L_2 - L_1)\), in which the end correction cancels.

The antinode lies beyond the open endThe resonance tubeThe speed of sound from two resonance lengths

14 min0%

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Overtones and Pipes in Unison

Naming and counting the overtones of a pipe below a given frequency. An open pipe and a closed pipe with the same fundamental, the closed one half as long. A pipe that resonates with a tuning fork, and the pressure picture of every mode.

Counting the overtones of a pipeAn open and a closed pipe in unisonA pipe resonating with a tuning forkThe pressure picture of each mode

12 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

air column: The air inside a pipe or a tube, between its two ends. It can carry standing sound waves.
boundary condition: A rule that a wave must obey at the edge of its medium. In a pipe, a closed end must be a displacement node and an open end a displacement antinode.
closed pipe: A pipe closed at one end and open at the other, such as a clarinet or a stopped organ pipe. It has only the odd harmonics, \(f_n = nv/4L\) with \(n\) odd.
displacement antinode: A place in a standing sound wave where the air swings to and fro with the largest amplitude. An open end of a pipe is always one, or very nearly.
displacement node: A place in a standing sound wave where the air never moves. A closed end of a pipe is always one.
effective length: The length that sets the resonances of a pipe: \(L + e\) for a closed pipe and \(L + 2e\) for an open pipe.
end correction: The small distance \(e\) by which the displacement antinode lies outside an open end, because the air just beyond the mouth moves too. For a pipe of inner radius \(r\), \(e \approx 0.6r\).
fundamental frequency: The lowest resonant frequency of a pipe, \(f_1\). It is \(v/2L\) for an open pipe and \(v/4L\) for a closed pipe. It is also called the first harmonic.
harmonic: A resonant frequency that is a whole-number multiple of the fundamental. The \(n\)th harmonic is \(f_n = nf_1\).
harmonic series: The full set of resonant frequencies of a system. For an open pipe it is \(f_1, 2f_1, 3f_1, \dots\), and for a closed pipe only \(f_1, 3f_1, 5f_1, \dots\).
open pipe: A pipe open at both ends, such as a flute. Both ends are displacement antinodes, and it has every harmonic, \(f_n = nv/2L\).
overtone: A resonant frequency above the fundamental, counted in order. In an open pipe the \(n\)th harmonic is the \((n - 1)\)th overtone; in a closed pipe the \(m\)th overtone is the \((2m + 1)\)th harmonic.
pressure antinode: A place in a standing sound wave where the pressure varies most. It lies at a displacement node; a closed end is one.
pressure node: A place in a standing sound wave where the pressure stays at its ambient value. It lies at a displacement antinode; an open end is one.
resonance: The large, steady vibration of a system driven at one of its natural frequencies. In an air column, sound reflected at the ends returns in step with the source and builds a large standing wave.
resonance length: A length of the air column in a resonance tube at which it resonates loudly with the fork. The first two are \(L_1\) and \(L_2\), with \(L_2 - L_1 = \lambda/2\).
resonance tube: An upright glass tube partly filled with water, whose level can be changed. The air column above the water, driven by a tuning fork, is used to find the speed of sound.
timbre: The quality of a musical note, set by the mixture of harmonics that sound with the fundamental. It lets the ear tell a flute from a clarinet playing the same pitch.
tuning fork: A two-pronged steel fork that vibrates at one fixed frequency when struck. It is used as a standard of frequency.
wind instrument: A musical instrument whose note is made by a vibrating column of air, such as a flute, a clarinet, a bansuri or an organ pipe.