Sine wave

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The sine wave or sinusoid is a function that occurs often in mathematics, signal processing, alternating-current power engineering, and other fields. Its most basic form is:

[y = A\cdot \sin(\omega t - \varphi)]

which describes a wavelike function of time ([t\,]) with:

peak deviation from center = [A\,] (aka amplitude)
angular frequency [\omega\,] (radians per second)
initial phase (t=0) = [-\varphi]
*[\varphi] is also referred to as a phase shift. E.g., when the initial phase is negative, the entire waveform is shifted toward future time (i.e. delayed). The amount of delay, in seconds, is [\varphi / \omega].

Contents

1 General form
2 Occurrences
3 Wave equation
4 Helmholtz equation
5 Fourier Series
6 See also

General form

In general, the function may also have:

a spatial dimension, [x\,] (aka position), with frequency [k\,] (aka wave number)
a non-zero center amplitude, [D\,] (aka DC offset)

which looks like this:

[ y \ = \ A\cdot \sin(kx - \omega t - \varphi) + D]

The wave number is related to the angular frequency by:

[ k = = = ]

where [\lambda] 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 [(x,y)] 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 propogation of an electrical wave through a conducting plane.

Occurrences

This wave pattern occurs often in nature, including in 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 also said to be sinusoidal, since it has the same shape but is shifted slightly behind the sine wave on the horizontal axis: [\cos\left(x -\frac\right) = \sin]

Any non-sinusoidal waveforms, such as square waves or even the irregular sound waves made by human speech, are actually 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.

[Sine wave sound sample] ([file info])
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* Problems listening to the file? See [Media helpmedia help].

Wave equation

The wave equation is one that can satisfy:

[\frac \frac = \frac]

To show this is true:

[\frac = - \omega A \cos (k x - \omega t - \varphi)]

[\frac = - \omega^2 A \sin (k x - \omega t - \varphi)]

[\frac = - k A \cos (k x - \omega t - \varphi)]

[\frac = - k^2 A \sin (k x - \omega t - \varphi)]

and inserting the second partials into the wave equation yields:

[\frac \left( - \omega^2 A \sin (k x - \omega t - \varphi) \right) = - k^2 A \sin (k x - \omega t - \varphi)]

and removing common terms

[\frac \omega^2 = k^2]

and since [k = \frac] (from above) they are shown to be equivalent. Thus, [y] satisfies the wave equation.

Helmholtz equation

The Helmholtz equation is one that can satisfy:

[\frac + \omega^2 y = 0]

Substituting in the second time partial from above

[- \omega^2 A \sin (k x - \omega t - \varphi) + \omega^2 A \sin (k x - \omega t - \varphi) = 0]

which is clearly true.

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 any periodic waveform. The process is named Fourier series. This is a useful analytical tool in signal processing theory.