# Sine wave

The graphs of the sine and cosine functions are sinusoids of different phases.
The sine wave or sinusoid is a function that occurs often in mathematics, physics, signal processing, electrical engineering, and many other fields. Its most basic form is:

which describes a wavelike function of time (t) with:
• peak deviation from center  = A (aka amplitude)
• angular frequency (radians per second)
• phase = ?
• When the phase is non-zero, the entire waveform appears to be shifted in time by the amount θ/ω seconds. A negative value represents a delay, and a positive value represents a "head-start".

## General form

In general, the function may also have:
• a spatial dimension, x (aka position), with frequency k (also called wave number)
• a non-zero center amplitude, D (also called DC offset)
which looks like this:

The wave number is related to the angular frequency by:.

where λ is the wavelength, f is the frequency, and c is the speed of propagation.

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.

A two-dimensional example would describe the amplitude of a two-dimensional wave at a position (xy) at time t. This could, for example, be considered the value of a water wave in a pond after a stone has been dropped in. Although this example is really a three dimensional wave it demonstrates the point; a more accurate example would be the propagation of an electrical wave through a conducting plane.

## Occurrences

This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves. 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. 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.

A cosine wave is said to be "sinusoidal", because:

which is also a sine wave with a phase-shift of п/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. The technique of transforming a complex waveform into its sinusoidal components is called Fourier analysis.

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.

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.

## 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. 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. Also see Fourier series and Fourier transform.

Sine, square, triangle, and sawtooth waveforms
A sinusoid is a small blood vessel similar to a capillary but with a discontinuous endothelium.

Sinusoids are found in the liver, lymphoid tissue, endocrine organs, and hematopoietic organs such as the bone marrow and the spleen.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
Signal processing is the analysis, interpretation and manipulation of signals. Signals of interest include sound, images, biological signals such as ECG, radar signals, and many others.
Electrical engineering (sometimes referred to as electrical and electronic engineering) is an engineering field that deals with the study and/or application of electricity, electronics and electromagnetism.
angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity.
radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
phase can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time.
Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m−1). Wavenumber is the spatial analogue of frequency.
Direct current (DC or "continuous current") is the constant flow of electric charge. This is typically in a conductor such as a wire, but can also be through semiconductors, insulators, or even through a vacuum as in electron or ion beams.
In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek letter lambda (λ). Examples of wave-like phenonomena are light, water waves, and sound waves.
FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. A sequel, titled Amplitude was released in 2003.
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate.
wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions.
Ocean surface waves are surface waves that occur in the upper layer of the ocean. They usually result from wind or geologic effects and may travel thousands of miles before striking land. They range in size from small ripples to huge tsunamis.
Sound is a disturbance of mechanical energy that propagates through matter as a wave (through fluids as a compression wave, and through solids as both compression and shear waves).
Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
alternating current (AC) is an electrical current whose magnitude and direction vary cyclically, as opposed to direct current, whose direction remains constant. The usual waveform of an AC power circuit is a sine wave, as this results in the most efficient transmission of
Direct current (DC or "continuous current") is the constant flow of electric charge. This is typically in a conductor such as a wire, but can also be through semiconductors, insulators, or even through a vacuum as in electron or ion beams.
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
Non-sinusoidal waveforms are waveforms that are not pure sine waves. They are usually derived from simple math functions. While a pure sine consists of a single frequency, non-sinusoidal waveforms can be described as containing multiple sine waves of different frequencies.
square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels.
Speech communication refers to the processes associated with the production and perception of sounds used in spoken language. A number of academic disciplines study speech and speech sounds, including acoustics, psychology, speech pathology, linguistics, and computer science.
Periodicity is the quality of occurring at regular intervals or periods (in time or space) and can occur in different contexts:
• A clock marks time at periodic intervals.
• A metronome ticks at periodic intervals of time.