KAHIBARO
Table of Contents
Repeating standing-wave patterns
Harmonics are the allowed standing-wave patterns of a system. They appear when a wave is confined to a medium such as a string or an air column, and only certain wavelengths can fit the boundary conditions. Each allowed pattern has its own frequency, called a harmonic frequency.
The idea of harmonics is simple. A standing wave must fit neatly into the length of the system. Because of this, the system cannot vibrate with just any wavelength. It can vibrate only with special wavelengths, and these produce a set of special frequencies.
The lowest allowed frequency is called the fundamental frequency, or first harmonic. Higher allowed frequencies are called overtones or higher harmonics, depending on context.
Harmonics on a string fixed at both ends
For a string fixed at both ends, the ends must always be nodes. This means the standing wave must contain a whole number of half-wavelengths inside the string length $L$.
So the allowed wavelengths are
$$
L = n\frac{\lambda_n}{2}
$$
where $n = 1, 2, 3, \dots$
Solving for wavelength gives
$$
\lambda_n = \frac{2L}{n}
$$
Since wave speed is related to frequency by $v = f\lambda$, the allowed frequencies are
$$
f_n = \frac{v}{\lambda_n} = \frac{nv}{2L}
$$
Here, $f_1 = \dfrac{v}{2L}$ is the fundamental frequency, and the higher harmonics are integer multiples of it:
$$
f_n = n f_1
$$
For a string fixed at both ends,
$$
\lambda_n = \frac{2L}{n}, \qquad f_n = \frac{nv}{2L} = n f_1
$$
Only these frequencies produce standing waves.
Harmonics in air columns
Harmonics also occur in pipes, but the allowed patterns depend on whether the ends are open or closed.
In a pipe open at both ends, both ends are antinodes. The result is mathematically similar to a string fixed at both ends:
$$
\lambda_n = \frac{2L}{n}, \qquad f_n = \frac{nv}{2L}
$$
So all integer harmonics are present, $n = 1, 2, 3, \dots$
In a pipe closed at one end and open at the other, the closed end is a node and the open end is an antinode. Only odd-numbered harmonics fit:
$$
L = \frac{(2n-1)\lambda_n}{4}, \qquad n = 1, 2, 3, \dots
$$
which gives
$$
\lambda_n = \frac{4L}{2n-1}
$$
and
$$
f_n = \frac{(2n-1)v}{4L}
$$
So the frequencies are
$$
f_1, \, 3f_1, \, 5f_1, \dots
$$
Only odd harmonics appear.
For a pipe open at both ends,
$$
f_n = \frac{nv}{2L}
$$
For a pipe closed at one end,
$$
f_n = \frac{(2n-1)v}{4L}
$$
A closed pipe has only odd harmonics.
Harmonic number and overtone number
It is important not to confuse harmonics with overtones. The first harmonic is the fundamental frequency. The second harmonic is twice the fundamental. The third harmonic is three times the fundamental.
An overtone counts frequencies above the fundamental. So the first overtone is the next allowed mode above the fundamental.
For systems that contain every harmonic, such as a fixed string or an open pipe, the relation is simple:
| Harmonic | Frequency | Overtone |
|---|---|---|
| 1st harmonic | $f_1$ | none |
| 2nd harmonic | $2f_1$ | 1st overtone |
| 3rd harmonic | $3f_1$ | 2nd overtone |
For a closed pipe, the first overtone is actually the third harmonic, because the second harmonic does not exist in that system.
| Allowed mode in closed pipe | Frequency | Harmonic name | Overtone name |
|---|---|---|---|
| lowest mode | $f_1$ | 1st harmonic | none |
| next mode | $3f_1$ | 3rd harmonic | 1st overtone |
| next mode | $5f_1$ | 5th harmonic | 2nd overtone |
Why harmonics matter
Harmonics determine the different pitches a vibrating system can produce. More importantly, they shape the sound quality, or timbre, of musical instruments. Two instruments can play the same fundamental frequency but still sound different because the strengths of their higher harmonics are different.
Harmonics also appear far beyond music. Any physical system that supports standing waves can have normal modes with discrete frequencies. Harmonics are the simplest examples of these allowed modes.
A simple example
Consider a string of length $L = 0.60\ \text{m}$ with wave speed $v = 120\ \text{m/s}$.
The fundamental frequency is
$$
f_1 = \frac{v}{2L} = \frac{120}{2(0.60)} = 100\ \text{Hz}
$$
So the first few harmonics are
$$
f_1 = 100\ \text{Hz}, \qquad f_2 = 200\ \text{Hz}, \qquad f_3 = 300\ \text{Hz}
$$
Each one corresponds to a different standing-wave pattern on the same string.
Harmonics are discrete allowed frequencies set by boundary conditions.
For many systems,
$$
f_n = n f_1
$$
But not every system allows every integer value of $n$. Always check the boundary conditions first.