Sine wave

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

Asine wave,sinusoidal wave,orsinusoid(symbol:∿) is aperiodicwavewhosewaveform(shape) is thetrigonometricsine function.Inmechanics,as a linearmotionover time, this issimple harmonic motion;asrotation,it corresponds touniform circular motion.Sine waves occur often inphysics,includingwind waves,soundwaves, andlightwaves, such asmonochromatic radiation.Inengineering,signal processing,andmathematics,Fourier analysisdecomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

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

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, thesineandcosinecomponents,respectively.

Audio example

A sine wave represents a singlefrequencywith noharmonicsand is considered anacousticallypure tone.Adding sine waves of different frequencies results in a different waveform. Presence of higher harmonics in addition to thefundamentalcauses variation in thetimbre,which is the reason why the samemusical pitchplayed on different instruments sounds different.

Sinusoid form

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

• $A$,amplitude,the peak deviation of the function from zero.
• $t$,therealindependent variable,usually representingtimeinseconds.
• $\omega$,angular frequency,the rate of change of the function argument in units ofradians per second.
• $f$,ordinary frequency,thenumberof 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

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

Sinusoids that exist in both position and time also have:

• a spatial variable$x$that represents thepositionon the dimension on which the wave propagates.
• awave number(or angular wave number)$k$,which represents the proportionality between theangular frequency$\omega$and the linear speed (speed of propagation)$v$:
  • 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,they are often used to analyzewave propagation.

Standing waves

Main article:Standing wave

When two waves with the sameamplitudeandfrequencytraveling in opposite directionssuperposeeach other, then astanding wavepattern is created.

On a plucked string, the superimposing waves are the waves reflected from the fixed endpoints of the string. The string'sresonantfrequencies 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

The earlier equation gives the displacement$y$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 travellingplane waveif position$x$and wavenumber$k$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

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

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

Fourier analysis

Main articles:Fourier series,Fourier transform,andFourier analysis

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

Differentiation and integration

See also:Phasor § Differentiation and integration

Differentiation

Differentiatingany sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:

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

Adifferentiatorhas azeroat the origin of thecomplex frequencyplane. Thegainof itsfrequency responseincreases at a rate of +20dBperdecadeof frequency (forroot-powerquantities), the same positive slope as a 1^storderhigh-pass filter'sstopband,although a differentiator doesn't have acutoff frequencyor a flatpassband.A n^th-order high-pass filter approximately applies the n^thtime derivative ofsignalswhose frequency band is significantly lower than the filter's cutoff frequency.

Integration

Integratingany 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$C$will be zero if thebounds of integrationis an integer multiple of the sinusoid's period.

Anintegratorhas apoleat 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^storderlow-pass filter's stopband, although an integrator doesn't have a cutoff frequency or a flat passband. A n^th-order low-pass filter approximately performs the n^thtime integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.

See also

• 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
• ∿the sine wave symbol (U+223F)

External links

• "Sine Wave".Mathematical Mysteries.2021-11-17.Retrieved2022-09-30.