View source for Sine wave

{{Short description|Wave shaped like the sine function}}
{{redirect-distinguish|Sinusoid|Sinusoid (blood vessel)}}
{{onesource|date=January 2024}}
[[File:Sine and cosine animation.gif|class=skin-invert-image|thumb|upright=1.35|Tracing the y component of a [[circle]] while going around the circle results in a sine wave (red). Tracing the x component results in a [[cosine]] wave (blue). Both waves are sinusoids of the same frequency but different phases.]]

A '''sine wave''', '''sinusoidal wave''', or '''sinusoid''' (symbol: '''∿''') is a [[periodic function|periodic wave]] whose [[waveform]] (shape) is the [[trigonometric function|trigonometric]] [[sine|sine function]]. In [[mechanics]], as a linear [[motion]] over time, this is ''[[simple harmonic motion]]''; as [[rotation]], it corresponds to ''[[uniform circular motion]]''. Sine waves occur often in [[physics]], including [[wind wave]]s, [[sound]] waves, and [[light]] waves, such as [[monochromatic radiation]]. In [[engineering]], [[signal processing]], and [[mathematics]], [[Fourier analysis]] decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

When any two sine waves of the same [[frequency]] (but arbitrary [[phase (waves)|phase]]) are [[linear combination|linearly combined]], the result is another sine wave of the same frequency; this property is unique among periodic waves. Conversely, if some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero and a quarter cycle, the ''sine'' and ''cosine'' [[vector component|components]], respectively.

== Audio example ==
{{Listen
| filename     = 220 Hz sine wave.ogg
| title        = Sine wave
| description  = Five seconds of a 220 Hz sine wave. This is the [[Sound#Waves|sound wave]] described by a sine function with ''f'' = 220 oscillations per second.
}}

A sine wave represents a single frequency with no [[harmonic]]s and is considered an [[Acoustics|acoustically]] [[pure tone]]. Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to the [[Fundamental frequency|fundamental]] causes variation in the [[timbre]], which is the reason why the same [[Pitch (music)|musical pitch]] played on different instruments sounds different.

==Sinusoid form==
Sine waves of arbitrary phase and amplitude are called ''sinusoids'' and have the general form:<ref>{{Cite web |last=Smith |first=Julius Orion |title=Sinusoids |url=https://ccrma.stanford.edu/~jos/st/Sinusoids.html |access-date=2024-01-05 |website=ccrma.stanford.edu}}</ref>
<math display="block">y(t) = A\sin(\omega t + \varphi) = A\sin(2 \pi f t + \varphi)</math>
where:
* ''<math>A</math>'', ''[[amplitude]]'', the peak deviation of the function from zero.
* <math>t</math>, the [[Real number|real]] [[independent variable]], usually representing [[time]] in [[seconds]].
* <math>\omega</math>, ''[[angular frequency]]'', the rate of change of the function argument in units of [[radians per second]].
* ''<math>f</math>'', ''[[ordinary frequency]]'', the ''[[Real number|number]]'' of oscillations ([[Turn (angle)|cycles]]) that occur each second of time.
* <math>\varphi</math>, ''[[phase (waves)|phase]]'', specifies (in [[radian]]s) where in its cycle the oscillation is at ''t'' = 0.
** When <math>\varphi</math> is non-zero, the entire waveform appears to be shifted backwards in time by the amount <math>\tfrac{\varphi}{\omega}</math> seconds. A negative value represents a delay, and a positive value represents an advance.
** Adding or subtracting <math>2\pi</math> (one cycle) to the phase results in an equivalent wave.

== As a function of both position and time ==
[[Image:Animated-mass-spring.gif|right|The displacement of an undamped [[Spring mass system|spring-mass system]] oscillating around the equilibrium over time is a sine wave.|thumb|246x246px]]
Sinusoids that exist in both position and time also have:

* a spatial variable <math>x</math> that represents the ''position'' on the dimension on which the wave propagates. 
* a [[wave number]] (or angular wave number) <math>k</math>, which represents the proportionality between the [[angular frequency]] <math>\omega</math> and the linear speed ([[phase velocity|speed of propagation]]) <math>v</math>:
** wavenumber is related to the angular frequency by <math display="inline"> k {=} \frac{\omega}{v} {=} \frac{2 \pi f}{v} {=} \frac{2 \pi}{\lambda}</math> where <math>\lambda</math> ([[lambda]]) is the [[wavelength]]. 

Depending on their direction of travel, they can take the form:

*<math>y(x, t) = A\sin(kx - \omega t + \varphi)</math>, if the wave is moving to the right, or
*<math>y(x, t) = A\sin(kx + \omega t + \varphi)</math>, if the wave is moving to the left.
Since sine waves propagate without changing form in ''distributed linear systems'',{{Definition needed|date=August 2019}} they are often used to analyze [[wave propagation]].

=== Standing waves ===
{{Main|Standing wave}}
When two waves with the same [[amplitude]] and [[frequency]] traveling in opposite directions [[superposition principle|superpose]] each other, then a [[standing wave]] pattern is created.

On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string's [[resonant]] frequencies are the string's only possible standing waves, which only occur for wavelengths that are twice the string's length (corresponding to the [[fundamental frequency]]) and integer divisions of that (corresponding to higher harmonics).

=== Multiple spatial dimensions ===
The earlier equation gives the displacement <math>y</math> of the wave at a position <math>x</math> at time <math>t</math> 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 <math>x</math> and wavenumber <math>k</math> 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.

==== Sinusoidal plane wave ====
{{excerpt|Sinusoidal plane wave}}

== Fourier analysis ==
{{main|Fourier series|Fourier transform|Fourier analysis}}

French mathematician [[Joseph Fourier]] discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including [[Square wave (waveform)|square wave]]s. These [[Fourier series]] are frequently used in [[signal processing]] and the statistical analysis of [[time series]]. The [[Fourier transform]] then extended Fourier series to handle general functions, and birthed the field of [[Fourier analysis]].

== Differentiation and integration ==
{{See also|Phasor#Differentiation and integration}}

=== Differentiation ===
[[Derivative|Differentiating]] any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:

<math>\begin{align}
\frac{d}{dt} [A\sin(\omega t + \varphi)] &= A \omega \cos(\omega t + \varphi) \\
&= A \omega \sin(\omega t + \varphi + \tfrac{\pi}{2}) \, .
\end{align}</math>

A [[differentiator]] has a [[Zeros and poles|zero]] at the origin of the [[complex frequency]] plane. The [[Gain (electronics)|gain]] of its [[frequency response]] increases at a rate of +20&nbsp;[[Decibel|dB]] per [[Decade (log scale)|decade]] of frequency (for [[root-power]] quantities), the same positive slope as a 1{{Sup|st}} order [[high-pass filter]]'s [[stopband]], although a differentiator does not have a [[cutoff frequency]] or a flat [[passband]]. A n{{Sup|th}}-order high-pass filter approximately applies the n{{Sup|th}} time derivative of [[signals]] whose frequency band is significantly lower than the filter's cutoff frequency.

=== Integration ===
[[Integral|Integrating]] any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle:

<math>\begin{align}
\int A \sin(\omega t + \varphi) dt &= -\frac{A}{\omega} \cos(\omega t + \varphi) + C\\
&= -\frac{A}{\omega} \sin(\omega t + \varphi + \tfrac{\pi}{2}) + C\\
&= \frac{A}{\omega} \sin(\omega t + \varphi - \tfrac{\pi}{2}) + C \, .
\end{align}</math>

The [[constant of integration]] <math>C</math> will be zero if the [[bounds of integration]] is an integer multiple of the sinusoid's period.

An [[integrator]] has a [[Zeros and poles|pole]] at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20&nbsp;dB per decade of frequency (for root-power quantities), the same negative slope as a 1{{Sup|st}} order [[low-pass filter]]'s stopband, although an integrator does not have a cutoff frequency or a flat passband. A n{{Sup|th}}-order low-pass filter approximately performs the n{{Sup|th}} time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.

== See also ==
{{div col }}
* [[Crest (physics)]]
* [[Complex exponential]]
* [[Damped sine wave]]
* [[Euler's formula]]
* [[Fourier transform]]
* [[Harmonic analysis]]
* [[Harmonic series (mathematics)]]
* [[Harmonic series (music)]]
* [[Helmholtz equation]]
* [[Instantaneous phase]]
* [[In-phase and quadrature components]]
* [[Least-squares spectral analysis]]
* [[Oscilloscope]]
* [[Phasor]]
* [[Pure tone]]
* [[Simple harmonic motion]]
* [[Sinusoidal model]]
* [[Wave (physics)]]
* [[Wave equation]]
* [[Tilde#Electronics|∿]] the sine wave symbol (U+223F)
{{div col end}}

== References ==
{{reflist}}

== External links ==
* {{Cite web |date=2021-11-17 |title=Sine Wave |url=https://mathematicalmysteries.org/sine-wave/ |access-date=2022-09-30 |website=Mathematical Mysteries |language=en}}

{{Waveforms}}

[[Category:Trigonometry]]
[[Category:Wave mechanics]]
[[Category:Waves]]
[[Category:Waveforms]]
[[Category:Sound]]
[[Category:Acoustics]]