Wave in Pipes and Strings

1. Introduction

Musical instruments produce sustained notes by setting up stationary (standing) waves in a bounded medium. In wind instruments (flute, organ pipe, clarinet) the vibrating medium is the air column inside a pipe; in stringed instruments (guitar, sitar, sonometer) it is a stretched string. In both cases a wave and its reflection superpose to form nodes and antinodes, and only certain frequencies (the harmonics) are permitted.

Two useful definitions apply throughout:

• Harmonics: the notes whose frequencies are integral multiples of the fundamental (lowest) frequency. The fundamental itself is the first harmonic.
• Overtones: the notes of frequency higher than the fundamental that an instrument can produce. The first overtone is the next possible frequency above the fundamental.

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2. Organ Pipes

An organ pipe is a hollow tube (of wood or metal) used to produce sound by setting the enclosed air column into vibration. A blast of air at one end creates longitudinal stationary waves. There are two types:

• Closed organ pipe: one end closed, the other open (for example a whistle).
• Open organ pipe: both ends open (for example a flute).

At a closed end the air cannot move, so a node is always formed there. At an open end the air is free to vibrate with maximum amplitude, so an antinode is formed there.

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3. Stationary Waves in a Closed Organ Pipe

Let $L$ be the length of the pipe and $v$ the speed of sound in air. A closed end has a node, the open end an antinode.

3.1 Fundamental mode (first harmonic)

There is one node (closed end) and one antinode (open end). The length holds a quarter wavelength:

$$ \begin{aligned} L &= \frac{\lambda_0}{4} \implies \lambda_0 \ &= 4L \ f_0 &= \frac{v}{\lambda_0} \ &= \frac{v}{4L} \end{aligned} $$

3.2 First overtone (third harmonic)

The next possible mode has two nodes and two antinodes:

$$ \begin{aligned} L &= \frac{3\lambda_1}{4} \implies \lambda_1 \ &= \frac{4L}{3} \ \qquad f_1 &= \frac{v}{\lambda_1} \ &= \frac{3v}{4L} \ &= 3f_0 \end{aligned} $$

3.3 Second overtone (fifth harmonic)

$$ \begin{aligned} L &= \frac{5\lambda_2}{4} \implies \lambda_2 \ &= \frac{4L}{5} \ \qquad f_2 &= \frac{5v}{4L} \ &= 5f_0 \end{aligned} $$