Sine wave
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{{redirect|Sinusoid|the blood vessel|Sinusoid (blood vessel)}}
[[Image:Sine_Cosine_Graph.png|thumb|350px|right|The graphs of the sine and cosine functions are sinusoids of different phases.]]
[[Image:Simple harmonic oscillator.gif|right|frame|The oscillation of an undamped spring-mass system around the equilibrium is a sine wave.]]
The '''sine wave''' or '''sinusoid''' is a function that occurs often in [[mathematics]], [[physics]], [[signal processing]], [[hearing (sense)|audition]], [[electrical engineering]], and many other fields. Its most basic form is''':'''
:<math>y (t) = A \cdot \sin(\omega t + \theta)</math>
which describes a wavelike function of time (''t'') with''':'''
*peak deviation from center = ''A'' (aka ''[[amplitude]]'')
*[[angular frequency]] <math>\omega\,</math> ([[radian]]s per second)
*[[Phase (waves)|phase]] = ''θ''
**When the phase is non-zero, the entire waveform appears to be shifted in time by the amount ''θ''/''ω'' seconds. A negative value represents a delay, and a positive value represents a "head-start".
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. 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 frequency ''k'' (also called ''[[wavenumber]]'')
*a non-zero center amplitude, ''D'' (also called ''[[direct current|DC]] offset'')
which looks like this''':'''
:<math> y(t) = A\cdot \sin(\omega t - kx + \theta) + D.\,</math>
The wavenumber is related to the angular frequency by''':'''.
:<math> k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }</math>
where λ is the [[wavelength]], ''f'' is the [[frequency]], and ''c'' is the [[phase velocity|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 surface wave|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 [[rectifier|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 said to be "sinusoidal", because '''<math>\cos(x) = \sin(x + \pi/2),</math>'''
which is also a sine wave with a phase-shift 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.
Any [[non-sinusoidal waveforms]], such as [[square wave]]s or even the irregular sound waves made by human [[Speech communication|speech]], can be represented as a collection of sinusoidal waves of different [[periodicity|period]]s and [[frequency|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]]; 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 'make up' and describe nearly any periodic waveform. The process is named [[Fourier analysis]]. 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]]. It as found applications in many other scientific fields, including probability (in particular, the proof of the [[central limit theorem]] relies upon Fourier analysis), the [[geometry of numbers]], the [[isoperimetric problem]], [[Heisenberg's inequality]], recurrence of [[random walk]]s, and proofs of [[quadratic reciprocity]]. Also see [[Fourier series]] and [[Fourier transform]].
== See also ==
[[Image:Waveforms.svg|thumb|400px|[[sine wave|Sine]], [[square wave|square]], [[triangle wave|triangle]], and [[sawtooth wave|sawtooth]] waveforms]]
* [[Sinusoidal model]]
* [[Simple harmonic motion]]
* [[Wave equation]]
* [[Helmholtz equation]]
* [[Fourier transform]]
* [[Harmonic series (mathematics)]]
* [[Harmonic series (music)]]
* [[Pure tone]]
* [[Pseudo sine wave]]
* [[Instantaneous phase]]
[[Category:Trigonometry]]
[[Category:Wave mechanics]]
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