The graphs of the sine and cosine functions are sinusoids of different phases.
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, audition, electrical engineering, and many other fields. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Its most basic form is:

$y (t) = A \cdot \sin(\omega t + \theta)$

which describes a wavelike function of time (t) with:

• peak deviation from center  = A (aka amplitude)
• angular frequency $\omega\,$ (radians per second)
• phase = θ
• When the phase is non-zero, the entire waveform appears to be shifted in time by the amount θ/ω seconds. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 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 DC offset)

which looks like this:

$y(t) = A\cdot \sin(\omega t - kx + \theta) + D.\,$

The wavenumber is related to the angular frequency by:. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters Direct current ( DC) is the unidirectional flow of Electric charge.

$k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }$

where λ is the wavelength, f is the frequency, and c is the speed of propagation. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Frequency is a measure of the number of occurrences of a repeating event per unit Time. The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space

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. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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 waves, sound waves, and light waves. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. Ocean surface waves are Surface waves that occur on the Free surface of the Ocean. Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 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. An alternating current ( AC) is an Electric current whose direction reverses cyclically as opposed to Direct current, whose direction remains constant In fact, graphing the voltage of direct current full-wave rectification system gives an absolute value sine wave pattern, where the wave stays on the positive side of the x-axis. Direct current ( DC) is the unidirectional flow of Electric charge. A rectifier is an electrical device that converts Alternating current (AC to Direct current (DC a process known as rectification. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

A cosine wave is said to be "sinusoidal", because cos(x) = sin(x + π / 2), 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 waves or even the irregular sound waves made by human speech, can be represented as a collection of sinusoidal waves of different periods and frequencies blended together. Non-sinusoidal waveforms are Waveforms that are not pure Sine waves They are usually derived from simple math functions A square wave is a kind of Non-sinusoidal waveform, most typically encountered in Electronics and Signal processing. Speech refers to the processes associated with the production and perception of Sounds used in Spoken language. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks Frequency is a measure of the number of occurrences of a repeating event per unit Time. The technique of transforming a complex waveform into its sinusoidal components is called Fourier analysis. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series

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. The ear is the sense organ that detects Sounds The Vertebrate ear shows a common biology from Fish to Humans with variations Human whistling is the production of Sound by means of a constant stream of air from the mouth Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many A tuning fork is an acoustic Resonator in the form of a two-pronged Fork with the tines formed from a U-shaped bar of elastic

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. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer In Music, timbre (ˈtæm-bər' like timber, or, from Fr timbre tɛ̃bʁ is the quality of a Musical note or sound that distinguishes different

## 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. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist The process is named Fourier analysis, which is a useful analytical tool in the study of waves, heat flow, many other scientific fields, and signal processing theory. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Also see Fourier series and Fourier transform. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and