# Sine wave

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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:

## Contents

${\displaystyle y(t)=A\sin(2\pi ft+\varphi )=A\sin(\omega t+\varphi )}$

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 = 2πf, angular frequency , the rate of change of the function argument in units of radians per second
• ${\displaystyle \varphi }$, phase , specifies (in radians) where in its cycle the oscillation is at t = 0.
When ${\displaystyle \varphi }$ is non-zero, the entire waveform appears to be shifted in time by the amount ${\displaystyle \varphi }$/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 ω and the linear speed (speed of propagation) ν;
• a non-zero center amplitude, D

which is

${\displaystyle y(x,t)=A\sin(kx-\omega t+\varphi )+D\,}$, if the wave is moving to the right
${\displaystyle y(x,t)=A\sin(kx+\omega t+\varphi )+D\,}$, if the wave is moving to the left.

The wavenumber is related to the angular frequency by:.

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

where λ (lambda) is the wavelength, f is the frequency, 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 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.

## Occurrences

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

A cosine wave is said to be sinusoidal, because ${\displaystyle \cos(x)=\sin(x+\pi /2),}$ 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.

## 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 and the statistical analysis of time series.

## Traveling and standing waves

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

${\displaystyle u(t,x)=A\sin(kx-\omega t+\varphi )}$

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.

## Related Research Articles

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 f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.

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, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms.

In physics and mathematics, the phase of a periodic function of some real variable is an angle representing the number of periods spanned by that variable. It is denoted and expressed in such a scale that it varies by one full turn as the variable goes through each period. It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or as the variable completes a 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 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, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. 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.

In physics, mathematics, and related fields, a wave is a disturbance of one or more fields such that the field values oscillate repeatedly about a stable equilibrium (resting) value. If the relative amplitude of oscillation at different points in the field remains constant, the wave is said to be a standing wave. If the relative amplitude at different points in the field changes, the wave is said to be a traveling wave. Waves can only exist in fields when there is a force that tends to restore the field to equilibrium.

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.

The Fourier transform (FT) decomposes a function into its constituent frequencies. A special case is the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.

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 frequencyω 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. Angular frequency is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity.

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".

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.

In physics and engineering, a phasor, 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 a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and sinor or even complexor.

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:

The Mathieu equation is a linear second-order differential equation with periodic coefficients. The French mathematician, E. Léonard Mathieu, first introduced this family of differential equations, nowadays termed Mathieu equations, in his “Memoir on vibrations of an elliptic membrane” in 1868. "Mathieu functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude distortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration in a medium with modulated density"

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, a sinusoidalplane 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.