Superposing two sound waves of nearly equal frequency, sin(2π f1 t) + sin(2π f2 t), and applying the product-to-sum formula gives 2 cos(2π (f1−f2)/2 · t) sin(2π (f1+f2)/2 · t). The amplitude oscillates at |f1 − f2| Hz, which is perceived as a beat.

Superposition of Waves

Generating complex waveforms by combining sinusoidal waves

Introduction (High School Level)

Introduction

Adding several simple sinusoidal waves together produces more complex waveforms. This "principle of superposition" has wide-ranging applications, from musical chords to the analysis of electrical circuits.

Principle of Superposition

Given two waves $f_1(x)$ and $f_2(x)$, their sum

$$f(x) = f_1(x) + f_2(x)$$

is itself a wave. This is the principle of superposition.

Superposition of Equal Frequencies

Two sinusoidal waves sharing the same angular frequency $\omega$:

$$f_1(x) = A_1\sin(\omega x + \phi_1), \quad f_2(x) = A_2\sin(\omega x + \phi_2)$$

combine to give another sinusoidal wave at the same angular frequency:

$$f_1(x) + f_2(x) = A\sin(\omega x + \phi)$$

where $A$ and $\phi$ are determined from the original amplitudes and phases.

Beats

Superposing two waves whose frequencies are slightly different produces beats:

$$\sin(\omega_1 x) + \sin(\omega_2 x) = 2\cos\left(\dfrac{\omega_1 - \omega_2}{2}x\right)\sin\left(\dfrac{\omega_1 + \omega_2}{2}x\right)$$

Figure 1: Beats. Superposing two waves of nearly equal frequency (Wave 1 and Wave 2), they reinforce where their phases align and cancel where they are out of phase, so the amplitude of the sum varies slowly (dashed line = envelope).

When $\omega_1 \approx \omega_2$, the right-hand side takes the form of:

a wave oscillating at the higher frequency $\dfrac{\omega_1 + \omega_2}{2}$
whose amplitude is modulated at the lower frequency $\dfrac{\omega_1 - \omega_2}{2}$

Musicians use exactly this phenomenon when tuning an instrument: by listening to the beat frequency and adjusting until it disappears, they bring the two pitches into unison.

Example: playing a 440 Hz tone and a 441 Hz tone together produces a beat heard about $|441-440|=1$ time per second. In general, the number of beats per second is approximately the difference of the two frequencies.

Overtones and Timbre

For instruments with a clear pitch, such as a violin or a flute, the sound is well approximated by a superposition of a fundamental frequency (the fundamental) and integer multiples of it (the overtones).

Denoting the fundamental angular frequency by $\omega$:

$$f(x) = A_1\sin(\omega x) + A_2\sin(2\omega x) + A_3\sin(3\omega x) + \cdots$$

$\sin(\omega x)$: fundamental (first harmonic)
$\sin(2\omega x)$: second harmonic (one octave higher)
$\sin(3\omega x)$: third harmonic

The relative amplitudes $A_n$ of the harmonics determine the timbre of each instrument. The same pitch "A" sounds different on a flute versus a violin because the two instruments have different overtone compositions (the amplitude ratios of the harmonics).

Strictly speaking, each overtone has not only an amplitude but also a phase (a shift relative to $\sin$). The ear is rather insensitive to phase (Ohm's acoustic law), so timbre is determined mainly by the amplitude ratios of the overtones, but at low frequencies the auditory nerve synchronises to the temporal fine structure of the waveform (phase locking), so differences in phase can be perceived.

Connection to Fourier analysis

The idea of expressing any periodic function as a sum of sinusoidal waves — a Fourier series — is precisely the mathematical generalisation of this notion of harmonics.

The integer-multiple picture is an idealization. It is only an approximation. The vibrations of plates and membranes, as in cymbals, bells, and drums, have overtones that are not integer multiples (inharmonic overtones), and even on a piano the higher overtones are slightly sharp because of the stiffness of the strings (inharmonicity). Only sounds whose overtones are exact integer multiples are perfectly periodic; other sounds are handled not by Fourier series but by the more general frequency analysis of the Fourier transform.

Synthesizing Waveforms

Example: Approximating a Square Wave

The square wave of period $2\pi$ (a wave that alternates between $+1$ and $-1$) can be approximated by a sum of odd-numbered sine waves:

$$f(x) \approx \dfrac{4}{\pi}\left(\sin x + \dfrac{1}{3}\sin 3x + \dfrac{1}{5}\sin 5x + \cdots\right)$$

$\sin x$ alone: a smooth sinusoidal curve
$\sin x + \dfrac{1}{3}\sin 3x$: a slightly more angular shape
Adding more terms: the approximation converges toward the square wave

Figure 2: Approaching a square wave by superposing sine waves (increasing the number of terms N gets closer to the square wave).

As this shows, superposing a sufficient number of sinusoidal waves can approximate any periodic function. This is the central idea of Fourier series.

Summary

Waves can be added together (principle of superposition).
Superposing waves of nearly equal frequency produces beats.
The timbre of a musical instrument is determined by its harmonic composition.
Even complex periodic waveforms can be constructed from superposed sinusoidal waves.

References

Ohm's acoustic law — Wikipedia
T. Reichenbach, A. J. Hudspeth, "Discrimination of Low-Frequency Tones Employs Temporal Fine Structure", PLoS One, 7(9): e45579, 2012. doi:10.1371/journal.pone.0045579

Frequently Asked Questions

Q1: What is the superposition of waves (principle of superposition)?

A: In a linear system, when multiple waves are present simultaneously, the combined result equals the sum of each wave as if it existed independently. Mathematically this follows from linearity: the wave $f_1(x) + f_2(x)$ is the same as the sum of $f_1$ and $f_2$ taken separately. It applies to musical chords, light interference, and many other phenomena.

Q2: How do beats arise?

A: Superposing two sound waves of nearly equal frequency, $\sin(2\pi f_1 t) + \sin(2\pi f_2 t)$, and applying the product-to-sum formula gives $2\cos\!\left(2\pi \frac{f_1-f_2}{2} t\right) \sin\!\left(2\pi \frac{f_1+f_2}{2} t\right)$. The amplitude oscillates at $|f_1 - f_2|$ Hz, which is perceived as a beat.

Q3: How does Fourier analysis relate to the superposition of waves?

A: Fourier analysis is the theory of representing any periodic function as a Fourier series, i.e. a superposition of sines and cosines. Conversely, understanding what waveforms such a sum of sines can produce makes it possible to analyse and synthesise the frequency content of audio signals and electrical waveforms.