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A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
''
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, of which it is the graph. It is a type of
continuous wave A continuous wave or continuous waveform (CW) is an electromagnetic wave of constant amplitude and frequency, typically a sine wave, that for mathematical analysis is considered to be of infinite duration. It may refer to e.g. a laser or partic ...
and also a smooth
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
. It occurs often in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, as well as in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and many other fields.


Formulation

Its most basic form as a function of time (''t'') is: y(t) = A\sin(2 \pi f t + \varphi) = A\sin(\omega t + \varphi) where: * ''A'', ''
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
'', the peak deviation of the function from zero. * ''f'', '' ordinary frequency'', the ''
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
'' of oscillations (cycles) that occur each second of time. * ''ω'' = 2''f'', ''
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
'', the rate of change of the function argument in units of
radians per second The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega). ...
. * \varphi, '' phase'', specifies (in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s) where in its cycle the oscillation is at ''t'' = 0. When \varphi is non-zero, 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. The sine wave is important in physics because it retains its wave shape 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 variable ''x'' that represents the ''position'' on the dimension on which the wave propagates, and a characteristic parameter ''k'' called wave number (or angular wave number), which represents the proportionality between the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
''ω'' and the linear speed ( speed of propagation) ''ν''; * a non-zero center amplitude, ''D'' which is *y(x, t) = A\sin(kx - \omega t + \varphi) + D, if the wave is moving to the right *y(x, t) = A\sin(kx + \omega t + \varphi) + D, if the wave is moving to the left. The wavenumber is related to the angular frequency by: k = \frac = \frac = \frac where ''λ'' (lambda) is the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, ''f'' is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, and ''v'' is the linear speed. This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement 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 In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
if position ''x'' and wavenumber ''k'' are interpreted as vectors, and their product as a
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. 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.


Cosine

The term sinusoid describes any wave with characteristics of a sine wave. Thus, a cosine wave is also said to be ''sinusoidal'', because \cos(x) = \sin(x + \pi/2), which is also a sine wave with a phase-shift of /2
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s. Because of this head start, it is often said that the cosine function ''leads'' the sine function or the sine ''lags'' the cosine. The term ''sinusoidal'' thereby collectively refers to both sine waves and cosine waves with any phase offset.


Occurrence

This
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
pattern occurs often in nature, including wind waves,
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
waves, and
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
waves. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
with no
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
s. To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics; addition of different sine waves results in a different waveform and thus changes the timbre of the sound. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical note (the same frequency) played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves (which are periodic), then the sound will be perceived to be noisy, as
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
is characterized as being aperiodic or having a non-repetitive pattern.


Fourier series

In 1822, French mathematician Joseph Fourier 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 Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and the statistical analysis of
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
.


Traveling and standing waves

Since sine waves propagate without changing form in ''distributed linear systems'', they are often used to analyze
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative ...
. Sine waves traveling in two directions in space can be represented as u(t, x) = A \sin(kx - \omega t + \varphi) When two waves having the same
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, and traveling in opposite directions, superpose each other, then a
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed endpoints of the string. Therefore, standing waves occur only at certain frequencies, which are referred to as resonant frequencies and are composed of a fundamental frequency and its higher
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
s. The resonant frequencies of a string are proportional to: the length between the fixed ends; the tension of the string; and inversely proportional to the mass
per unit length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m&minus ...
of the string.


See also

*
Crest (physics) A crest point on a wave is the maximum value of upward displacement within a cycle. A crest is a point on a surface wave where the displacement of the medium is at a maximum. A trough is the opposite of a crest, so the minimum or lowest point in a ...
*
Damped sine wave Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
*
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
*
Harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
* Harmonic series (mathematics) *
Harmonic series (music) A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Pitched musical instruments are often based on an acoustic resonator ...
*
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalu ...
* Instantaneous phase * Least-squares spectral analysis * Oscilloscope *
Phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
* Pure tone * Simple harmonic motion * Sinusoidal model * Wave (physics) *
Wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
* the sine wave symbol (U+223F)


References


Further reading

* {{Waveforms Trigonometry Wave mechanics Waves Waveforms Sound Acoustics