The sine wave or sinusoid is a function that occurs often in mathematics, signal processing, alternating-current power engineering, and other fields. Its most basic form is:

$y\; =\; A\backslash cdot\; \backslash sin(\backslash omega\; t\; -\; \backslash varphi)$

which describes a wavelike function of time ($t\backslash ,$) with:

• peak deviation from center  = $A\backslash ,$ (aka amplitude)
• angular frequency $\backslash omega\backslash ,$ (radians per second)
• initial phase (t=0) = $-\backslash varphi$
  • $\backslash varphi$ is also referred to as a phase shift. E.g., when the initial phase is negative, the entire waveform is shifted toward future time (i.e. delayed). The amount of delay, in seconds, is $\backslash varphi\; /\; \backslash omega$.

General form

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In general, the function may also have:

• a spatial dimension, $x\backslash ,$ (aka position), with frequency $k\backslash ,$ (aka wave number)
• a non-zero center amplitude, $D\backslash ,$ (aka DC offset)

which looks like this:

$y\; \backslash \; =\; \backslash \; A\backslash cdot\; \backslash sin(kx\; -\; \backslash omega\; t\; -\; \backslash varphi)\; +\; D$

The wave number is related to the angular frequency by:

$k\; =\; \{\; \backslash omega\; \backslash over\; c\; \}\; =\; \{\; 2\; \backslash pi\; f\; \backslash over\; c\; \}\; =\; \{\; 2\; \backslash pi\; \backslash over\; \backslash lambda\; \}$

where $\backslash lambda$ is the wavelength, $f$ is the frequency, and $c$ is the speed of propagation.

This equation gives a sine wave for a single dimension, thus the generalized equation given above gives the amplitude of the wave at a position $x$ at time $t$ along a single line. This could, for example, be considered the value of a wave along a wire.

A two-dimensional example would describe the amplitude of a two-dimensional wave at a position $(x,y)$ at time $t$. This could, for example, be considered the value of a water wave in a pond after a stone has been dropped in. Although this example is really a three dimensional wave it demonstrates the point; a more accurate example would be the propogation of an electrical wave through a conducting plane.

Occurrences

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This wave pattern occurs often in nature, including in ocean waves, sound waves, and light waves. Also, a rough sinusoidal pattern can be seen in plotting average daily temperatures for each day of the year, although the graph may resemble an inverted cosine wave.

Graphing the voltage of an alternating current gives a sine wave pattern. In fact, graphing the voltage of direct current full-wave rectification system gives an absolute value sine wave pattern, where the wave stays on the positive side of the x-axis.

A cosine wave is also said to be sinusoidal, since it has the same shape but is shifted slightly behind the sine wave on the horizontal axis: $\backslash cos\backslash left(x\; -\backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; \backslash sin\{x\}$

Any non-sinusoidal waveforms, such as square waves or even the irregular sound waves made by human speech, are actually a collection of sinusoidal waves of different periods and frequencies blended together. The technique of transforming a complex waveform into its sinusoidal components is called Fourier analysis.

The human ear can recognize single sine waves because sounds with such a waveform sound "clean" or "clear" to humans; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork.

To the human ear, a sound that is made up of more than one sine wave will either sound "noisy" or will have detectable harmonics.

Wave equation

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The wave equation is one that can satisfy:

$\backslash frac\{1\}\{c^2\}\; \backslash frac\{\backslash partial^2\; y\}\{\backslash partial\; t^2\}\; =\; \backslash frac\{\backslash partial^2\; y\}\{\backslash partial\; x^2\}$

To show this is true:

$\backslash frac\; \{\backslash partial\; y\}\{\backslash partial\; t\}\; =\; -\; \backslash omega\; A\; \backslash cos\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \{\backslash partial^2\; y\}\{\backslash partial\; t^2\}\; =\; -\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \{\backslash partial\; y\}\{\backslash partial\; x\}\; =\; -\; k\; A\; \backslash cos\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \{\backslash partial^2\; y\}\{\backslash partial\; x^2\}\; =\; -\; k^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

and inserting the second partials into the wave equation yields:

$\backslash frac\{1\}\{c^2\}\; \backslash left(\; -\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; \backslash right)\; =\; -\; k^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

and removing common terms

$\backslash frac\{1\}\{c^2\}\; \backslash omega^2\; =\; k^2$

and since $k\; =\; \backslash frac\{\backslash omega\}\{c\}$ (from above) they are shown to be equivalent. Thus, $y$ satisfies the wave equation.

Helmholtz equation

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The Helmholtz equation is one that can satisfy:

$\backslash frac\{\backslash partial^2\; y\}\{\backslash partial\; t^2\}\; +\; \backslash omega^2\; y\; =\; 0$

Substituting in the second time partial from above

$-\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; +\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; =\; 0$

which is clearly true.

Fourier Series

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In 1822, Joseph Fourier, a French mathematician, discovered that sinusoidal waves can be used as simple building blocks to 'make up' and describe any periodic waveform. The process is named Fourier series. This is a useful analytical tool in signal processing theory.

See also

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• Wave equation
• Helmholtz equation
• Fourier transform
• Harmonic series (mathematics)
• Harmonic series (music)
• Pure tone
• Pseudo sine wave

Trigonometry | Wave mechanics

Sinusoid | Sinusoide | Signal sinusoïdal | Sinusoide | 正弦波 | Senóide | Синусоида | Синусоида | Siniaalto | Sinusvåg | 正弦曲線