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Sine wave

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Wave shaped like the sine function

"Sinusoid" redirects here; not to be confused withSinusoid (blood vessel).

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Tracing the y component of acircle while going around the circle results in a sine wave (red). Tracing the x component results in acosine wave (blue). Both waves are sinusoids of the same frequency but different phases.

Asine wave,sinusoidal wave, orsinusoid (symbol:∿) is aperiodic wave whosewaveform (shape) is thetrigonometric sine function. Inmechanics, as a linearmotion over time, this issimple harmonic motion; asrotation, it corresponds touniform circular motion. Sine waves occur often inphysics, includingwind waves,sound waves, andlight waves, such asmonochromatic radiation. Inengineering,signal processing, andmathematics,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 samefrequency (but arbitraryphase) arelinearly 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, thesine andcosinecomponents, respectively.

Audio example

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Sine wave

Five seconds of a 220 Hz sine wave. This is the sound wave described by a sine function withf = 220 oscillations per second.

Problems playing this file? Seemedia help.

A sine wave represents a single frequency with noharmonics and is considered anacoustically pure tone. Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to thefundamental causes variation in thetimbre, which is the reason why the samemusical pitch played on different instruments sounds different.

Sinusoid form

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Sine waves of arbitrary phase and amplitude are calledsinusoids and have the general form:^[1] $y(t)=A\sin(\omega t+\varphi )=A\sin(2\pi ft+\varphi )$ where:

$A {\displaystyle A}$ ,amplitude, the peak deviation of the function from zero.
$t {\displaystyle t}$ , thereal independent variable, usually representingtime inseconds.
$\omega$ ,angular frequency, the rate of change of the function argument in units ofradians per second.
$f {\displaystyle f}$ ,ordinary frequency, thenumber of oscillations (cycles) that occur each second of time.
$\varphi$ ,phase, specifies (inradians) where in its cycle the oscillation is att = 0.
- When $\varphi$ is non-zero, the entire waveform appears to be shifted backwards in time by the amount ${\tfrac {\varphi }{\omega }}$ seconds. A negative value represents a delay, and a positive value represents an advance.
- Adding or subtracting $2\pi$ (one cycle) to the phase results in an equivalent wave.

As a function of both position and time

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The displacement of an undampedspring-mass system oscillating around the equilibrium over time is a sine wave.

Sinusoids that exist in both position and time also have:

a spatial variable $x {\displaystyle x}$ that represents theposition on the dimension on which the wave propagates.
awave number (or angular wave number) $k {\displaystyle k}$ , which represents the proportionality between theangular frequency $\omega$ and the linear speed (speed of propagation) $v {\displaystyle v}$ :
- wavenumber is related to the angular frequency by ${\textstyle k{=}{\frac {\omega }{v}}{=}{\frac {2\pi f}{v}}{=}{\frac {2\pi }{\lambda }}}$ where $\lambda$ (lambda) is thewavelength.

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

$y(x,t)=A\sin(kx-\omega t+\varphi )$ , if the wave is moving to the right, or
$y(x,t)=A\sin(kx+\omega t+\varphi )$ , if the wave is moving to the left.

Since sine waves propagate without changing form indistributed linear systems,^{[definition needed]} they are often used to analyzewave propagation.

Standing waves

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Main article:Standing wave

When two waves with the sameamplitude andfrequency traveling in opposite directionssuperpose each other, then astanding wave pattern is created.

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

Multiple spatial dimensions

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The earlier equation gives the displacement $y {\displaystyle y}$ of the wave at a position $x {\displaystyle x}$ at time $t {\displaystyle t}$ 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 travellingplane wave if position $x {\displaystyle x}$ and wavenumber $k {\displaystyle k}$ are interpreted as vectors, and their product as adot 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

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This section is an excerpt fromSinusoidal plane wave.[edit]

Inphysics, asinusoidal plane wave is a special case ofplane wave: afield whose value varies as asinusoidal function of time and of the distance from some fixed plane. It is also called a monochromatic plane wave, with constantfrequency (as inmonochromatic radiation).

Fourier analysis

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Main articles:Fourier series,Fourier transform, andFourier analysis

French mathematicianJoseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, includingsquare waves. TheseFourier series are frequently used insignal processing and the statistical analysis oftime series. TheFourier transform then extended Fourier series to handle general functions, and birthed the field ofFourier analysis.

Differentiation and integration

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Differentiation

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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:

${\begin{aligned}{\frac {d}{dt}}[A\sin(\omega t+\varphi )]&=A\omega \cos(\omega t+\varphi )\\&=A\omega \sin(\omega t+\varphi +{\tfrac {\pi }{2}})\,.\end{aligned}}$

Adifferentiator has azero at the origin of thecomplex frequency plane. Thegain of itsfrequency response increases at a rate of +20 dB perdecade of frequency (forroot-power quantities), the same positive slope as a 1^st orderhigh-pass filter'sstopband, although a differentiator does not have acutoff frequency or a flatpassband. A n^th-order high-pass filter approximately applies the n^th time derivative ofsignals whose frequency band is significantly lower than the filter's cutoff frequency.

Integration

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

${\begin{aligned}\int A\sin(\omega t+\varphi )dt&=-{\frac {A}{\omega }}\cos(\omega t+\varphi )+C\\&=-{\frac {A}{\omega }}\sin(\omega t+\varphi +{\tfrac {\pi }{2}})+C\\&={\frac {A}{\omega }}\sin(\omega t+\varphi -{\tfrac {\pi }{2}})+C\,.\end{aligned}}$

Theconstant of integration $C {\displaystyle C}$ will be zero if thebounds of integration is an integer multiple of the sinusoid's period.

Anintegrator has apole at the origin of the complex frequency plane. The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1^st orderlow-pass filter's stopband, although an integrator does not have a cutoff frequency or a flat passband. A n^th-order low-pass filter approximately performs the n^th time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.