The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:
 $y(t)\; =\; A\; \backslash cdot\; \backslash sin(2\; \backslash pi\; f\; t\; +\; \backslash phi)\; =\; A\; \backslash cdot\; \backslash sin(\backslash omega\; t\; +\; \backslash phi)$
where:
 A, the amplitude, is the peak deviation of the function from zero.
 f, the ordinary frequency, is the number of oscillations (cycles) that occur each second of time.
 ω = 2πf, the angular frequency, is the rate of change of the function argument in units of radians per second
 φ, the phase, specifies (in radians) where in its cycle the oscillation is at t = 0.
 When φ is nonzero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.

Sine wave
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5 seconds of a 220 Hz sine wave

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The sine wave is important in physics because it retains its waveshape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
General form
In general, the function may also have:
 a spatial dimension, x (aka position), with wavenumber k
 a nonzero center amplitude, D
which is
 $y(x,t)\; =\; A\backslash cdot\; \backslash sin(\backslash omega\; t\; \; kx\; +\; \backslash phi\; )\; +\; D.\backslash ,$
The wavenumber 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 λ 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.
In two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, and their product as a dot product.
For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed.
Occurrences
This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves.
A cosine wave is said to be "sinusoidal", because $\backslash cos(x)\; =\; \backslash sin(x\; +\; \backslash pi/2),$
which is also a sine wave with a phaseshift of π/2. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.
The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics; 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; this may be described as a different timbre.
Fourier series
In 1822, Joseph Fourier, a French mathematician, discovered that sinusoidal waves can be used as simple building blocks to describe and approximate any periodic waveform including square waves. Fourier used it as an analytical tool in the study of waves and heat flow. It is frequently used in signal processing and the statistical analysis of time series.
Traveling and standing waves
Since sine waves propagate without changing form in distributed linear systems, they are often used to analyze wave propagation. Sine waves traveling in two directions can be represented as
 $y(t)\; =\; A\; \backslash sin(\backslash omega\; t\; \; kx)$ and $y(t)=\; A\; \backslash sin(\backslash omega\; t\; +\; kx).$
When two waves having the same amplitude and frequency, and traveling in opposite directions, superpose each other, then a standing wave pattern is created.
See also
References
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