# Waveform

A waveform is the pattern of oscillating pressure produced by a sound, represented as a graph in which the vertical axis shows displacement or amplitude and the horizontal axis shows time or distance.<sup>[1](https://www3.cs.stonybrook.edu/~lori/classes/MM/sound.htm)</sup> Sound waves are longitudinal: a vibrating source creates alternating regions of high pressure (compressions) and low pressure (rarefactions), and the waveform shows the time evolution of these pressure variations.<sup>[2](https://musicweb.ucsd.edu/~trsmyth/funAudio/Sound_Waveform.html)</sup>

| Key fact | Detail |
|---|---|
| What it represents | The time evolution of pressure variations in a sound, graphed as amplitude (or displacement) against time or distance<sup>[2](https://musicweb.ucsd.edu/~trsmyth/funAudio/Sound_Waveform.html)</sup><sup> • </sup><sup>[1](https://www3.cs.stonybrook.edu/~lori/classes/MM/sound.htm)</sup> |
| Principal quantities | Amplitude, frequency, and phase describe any acoustic wave<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup> |
| Frequency | Cycles per second, in hertz (Hz); f = 1/T, where T is the period; corresponds to perceived pitch<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup> |
| Amplitude | The size of the pressure change, measured in pascals and usually reported as decibels of sound pressure level (dB SPL)<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup> |
| Human hearing range | Normally given as 20 to 20,000 Hz<sup>[5](https://www.ucl.ac.uk/slade/patrickwhite/know_files/audio_fundamentals.pdf)</sup> |
| Speed in air | Around 343 m/s, varying with temperature and elevation<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup> |
| Digital capture | The recording standard is 44,100 samples per second and the broadcasting standard is 48,000; the sampling rate should be at least twice the highest frequency present<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup> |

## The sine wave and its parameters

The simplest periodic vibration is a sine wave. The distance between two peaks is the period (T), and the number of periods repeating per second is the frequency, f = 1/T, which corresponds to the subjective perception of pitch. The distance between the highest peak and lowest valley represents the peak-to-peak amplitude, of which the amplitude (a) is half for a sine wave, corresponding roughly to the perception of loudness.<sup>[12](https://www.open.edu/openlearn/mod/oucontent/view.php?id=3509&section=6.1)</sup> The starting point of the wave measured relative to a given origin is the phase (φ).<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup> Such a vibration can be written as:

\[ w = a \cdot \sin(2\pi f \cdot t + \varphi) \]

In physics notation, a sound can be modeled as a pressure wave by the change in pressure from average pressure:

\[ \Delta P = \Delta P_{\max} \sin(kx \mp \omega t + \varphi) \]

where \( \Delta P_{\max} \) is the maximum pressure change, \( k = 2\pi/\lambda \) is the wave number, \( \omega = 2\pi f \) is the angular frequency, and \( \varphi \) is the initial phase.<sup>[6](https://openstax.org/books/university-physics-volume-1/pages/17-1-sound-waves)</sup>

**Amplitude and loudness.** For a sine wave, amplitude is half the full pressure change between compression and rarefaction, which is the peak-to-peak amplitude. It is measured in pascals, but because perception of acoustic amplitude follows a logarithmic scale, it is more often reported in decibels of sound pressure level (dB SPL); each step of 20 dB corresponds to a ten-fold increase in pressure.<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup> For reference, the human hearing threshold is 0.00002 Pa (0 dB SPL), normal conversation at 1 m is about 0.0063 Pa (50 dB SPL), and a jet engine at 100 m is about 20 Pa (120 dB SPL).<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup>

**Phase.** The phase of a wave measures time relative to the cycles of compression and rarefaction. It is not perceived directly, but it affects how different waves interact with each other.<sup>[3](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)</sup> A single waveform's phase is usually not discernible at audio rates, but it becomes a factor when two or more waves of differing phase are played simultaneously, or when a waveform is used at a very slow rate as a control signal in synthesis, for example to create vibrato.<sup>[7](https://cmtext.indiana.edu/acoustics/chapter1_waves.php)</sup>

## Complex waveforms and Fourier's theorem

Most sound sources do not produce a pure sine wave. Jean Baptiste Joseph Fourier (1768–1830), a French mathematician, showed that any complex vibration can be described as the summation of sine waves, provided the amplitude, frequency, and phase of each component are known; in musical sound this corresponds to the existence of overtones.<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup> Additive synthesis reverses the process, building complex sounds by adding sine waves.<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup>

The waveform and the spectrum are the two principal representations of a sound. The signal is the temporal representation, with time on the horizontal axis and amplitude or intensity on the vertical axis; applying a [Fourier transform](https://www.edgepedia.org/fourier-transform) to the signal yields the spectrum, a frequency representation with frequency in hertz on the x-axis and amplitude in decibels on the y-axis.<sup>[8](https://timbreandorchestration.org/writings/timbre-lingo/2019/3/12/signal-vs-spectrum)</sup> The shape of a waveform depends on the phases as well as the amplitudes of its components, so two sounds with similar spectra can have different waveforms.<sup>[9](https://era.ed.ac.uk/server/api/core/bitstreams/f6d35fa0-84ba-40be-a800-ed89d525c956/content)</sup>

## Waveforms in practice

**Measurement.** An oscilloscope plots amplitude against time and is the standard instrument for viewing a waveform directly.<sup>[5](https://www.ucl.ac.uk/slade/patrickwhite/know_files/audio_fundamentals.pdf)</sup> A cycle can be measured from any point on the wave that returns to its origin, not only from zero crossings; frequency can be expressed in seconds per cycle (period) or cycles per second, and one hertz equals one cycle per second.<sup>[7](https://cmtext.indiana.edu/acoustics/chapter1_waves.php)</sup> The hertz became the official international (SI) unit for cycles per second in 1960, named for Heinrich Rudolf Hertz, the 19th-century German physicist who proved the existence of electromagnetic waves.<sup>[7](https://cmtext.indiana.edu/acoustics/chapter1_waves.php)</sup>

**Digital audio.** A waveform's variations of electric current can be measured from time to time by an analog-to-digital converter. To capture all frequencies present, the number of samples per second should be at least twice the highest frequency; the recording standard is 44,100 samples per second and the broadcasting standard is 48,000.<sup>[4](https://computermusic.web.illinois.edu/Notes/acoustics1.html)</sup>

**Timbre.** The waveform view is limited in what it can say about timbre, the quality that distinguishes two sounds of the same pitch and loudness; timbre is more closely related to the sound's spectrum.<sup>[10](https://myweb.liu.edu/jmeschi/f22/mus14a/notes/Chapter%202%20~%20Sound%20Properties%20and%20the%20Waveform%20View.pdf)</sup> Recordings of the same note played on different instruments show differences in the starting transient and the spectrum, both important to timbre.<sup>[11](https://www.animations.physics.unsw.edu.au/labs/sound/sound_lab.html)</sup>

## References

1. [Continuous Media: Audio, Stony Brook University](https://www3.cs.stonybrook.edu/~lori/classes/MM/sound.htm)
2. [Sound Waveform, Tamara Smyth, UCSD](https://musicweb.ucsd.edu/~trsmyth/funAudio/Sound_Waveform.html)
3. [7.1 Acoustic Cues and Signals, OpenStax Introduction to Behavioral Neuroscience](https://openstax.org/books/introduction-behavioral-neuroscience/pages/7-1-acoustic-cues-and-signals)
4. [Acoustics notes, Computer Music, University of Illinois](https://computermusic.web.illinois.edu/Notes/acoustics1.html)
5. [Visualising Fundamentals, UCL Slade](https://www.ucl.ac.uk/slade/patrickwhite/know_files/audio_fundamentals.pdf)
6. [17.1 Sound Waves, OpenStax University Physics Volume 1](https://openstax.org/books/university-physics-volume-1/pages/17-1-sound-waves)
7. [Acoustics Chapter One: Sound Waves, Indiana University Jacobs School of Music](https://cmtext.indiana.edu/acoustics/chapter1_waves.php)
8. [Signal vs. Spectrum, Timbre and Orchestration Resource](https://timbreandorchestration.org/writings/timbre-lingo/2019/3/12/signal-vs-spectrum)
9. [Relationship between the physical and perceptual properties of sound, University of Edinburgh](https://era.ed.ac.uk/server/api/core/bitstreams/f6d35fa0-84ba-40be-a800-ed89d525c956/content)
10. [Introduction to Music Technology, Chapter 2: Sound Properties and the Waveform View](https://myweb.liu.edu/jmeschi/f22/mus14a/notes/Chapter%202%20~%20Sound%20Properties%20and%20the%20Waveform%20View.pdf)
11. [Sound Laboratory, UNSW Physics](https://www.animations.physics.unsw.edu.au/labs/sound/sound_lab.html)
12. [open.edu](https://www.open.edu/openlearn/mod/oucontent/view.php?id=3509&section=6.1)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Wave phenomena and acoustics › Acoustics › Physical acoustics › Acoustic measurement and characterization*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license