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A **sine wave** or **sinusoid** is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (*t*) is:

In mathematics, a **curve** is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that it may be curved.

**Oscillation** is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The term *vibration* is precisely used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current.

A **continuous wave** or **continuous waveform** (**CW**) is an electromagnetic wave of constant amplitude and frequency, almost always a sine wave, that for mathematical analysis is considered to be of infinite duration. Continuous wave is also the name given to an early method of radio transmission, in which a sinusoidal carrier wave is switched on and off. Information is carried in the varying duration of the on and off periods of the signal, for example by Morse code in early radio. In early wireless telegraphy radio transmission, CW waves were also known as "undamped waves", to distinguish this method from damped wave signals produced by earlier *spark gap* type transmitters.

where:

*A*,*amplitude*, the peak deviation of the function from zero.*f*,*ordinary frequency*, the*number*of oscillations (cycles) that occur each second of time.*ω*= 2π*f*,*angular frequency*, the rate of change of the function argument in units of radians per second- ,
*phase*, specifies (in radians) where in its cycle the oscillation is at*t*= 0.When is non-zero, the entire waveform appears to be shifted in time by the amount /*ω*seconds. A negative value represents a delay, and a positive value represents an advance.

The **amplitude** of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

**Frequency** is the number of occurrences of a repeating event per unit of time. It is also referred to as **temporal frequency**, which emphasizes the contrast to spatial frequency and angular frequency. The

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

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.

In mathematics, **Fourier analysis** is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

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*ω*and the linear speed (speed of propagation)*ν*; - a non-zero center amplitude,
*D*

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 travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

which is

- , if the wave is moving to the right
- , if the wave is moving to the left.

The wavenumber is related to the angular frequency by**:**.

where λ (lambda) is the wavelength, *f* is the frequency, and *v* is the linear speed.

In physics, the **wavelength** is the **spatial period** of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter *lambda* (λ). The term *wavelength* is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

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 if position *x* and wavenumber *k* are interpreted as vectors, and their product as a dot 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.

In physics, a **plane wave** is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.

In mathematics, the **dot product** or **scalar product** is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called "the" **inner product** of Euclidean space even though it is not the only inner product that can be defined on Euclidean space; see also inner product space.

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

A cosine wave is said to be *sinusoidal*, because which is also a sine wave with a phase-shift of π/2 radians. Because of this head start, it is often said that the cosine function *leads* the sine function or the sine *lags* the cosine.

The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics.

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 is characterized as being aperiodic or having a non-repetitive pattern.

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 and the statistical analysis of time series.

Since sine waves propagate without changing form in *distributed linear systems*,^{[ definition needed ]} they are often used to analyze wave propagation. Sine waves traveling in two directions in space can be represented as

When two waves having the same amplitude and frequency, and traveling in opposite directions, superpose each other, then a standing wave pattern is created. Note that, on a plucked string, the interfering waves are the waves reflected from the fixed end points 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 harmonics. 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 of the string.

- "Sinusoid".
*Encyclopedia of Mathematics*. Springer. Retrieved December 8, 2013.

The **fundamental frequency**, often referred to simply as the **fundamental**, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as ** f_{0}**, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as

Since the fundamental is the lowest frequency and is also perceived as the loudest, the ear identifies it as the specific pitch of the musical tone [harmonic spectrum]....The individual partials are not heard separately but are blended together by the ear into a single tone.

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In physics and mathematics, the **phase** of a periodic function of some real variable is the relative value of that variable within the span of each full period.

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

In electronics, acoustics, and related fields, the **waveform** of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.

A **chirp** is a signal in which the frequency increases (*up-chirp*) or decreases (*down-chirp*) with time. In some sources, the term *chirp* is used interchangeably with **sweep signal**. It is commonly used in sonar, radar, and laser, but has other applications, such as in spread-spectrum communications.

In physics, **angular frequency***ω* is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

**Fourier optics** is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or *superposition*, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

A **square wave** is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. Although not realizable in physical systems, the transition between minimum and maximum is instantaneous for an ideal square wave.

A **normal mode** of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. When relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones".

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

The concept of **negative and positive frequency** can be as simple as a wheel rotating one way or the other way: a *signed value* of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions per second (hertz) or radian/second.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

**Vibration** is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin *vibrationem*. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

A **damped sine wave** is a sinusoidal function whose amplitude approaches zero as time increases.

In physics, **sinusoidal****plane wave** is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane.

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