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The sine wave or '''sinusoid''' is a function that occurs often in Mathematics , Signal Processing , and other fields. Its most basic form is''':'''

:y = A\cdot \sin(\omega t - arphi)

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) = - arphi

  • --- arphi 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 arphi / \omega.




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 - arphi) + D

The wave number is related to the angular frequency by:

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

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.

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 - rac{\pi}{2} ight) = \sin{x}

Any Non-sinusoidal waveforms, such as Square Wave s or even the irregular sound waves made by human Speech , are actually a collection of sinusoidal waves of different Period s 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 .


WAVE EQUATION

The Wave Equation is one that can satisfy:

: rac{1}{c^2} rac{\partial^2 y}{\partial t^2} = rac{\partial^2 y}{\partial x^2}

To show this is true:

: rac {\partial y}{\partial t} = - \omega A \cos (k x - \omega t - arphi)
: rac {\partial^2 y}{\partial t^2} = - \omega^2 A \sin (k x - \omega t - arphi)
: rac {\partial y}{\partial x} = - k A \cos (k x - \omega t - arphi)
: rac {\partial^2 y}{\partial x^2} = - k^2 A \sin (k x - \omega t - arphi)

and inserting the second partials into the wave equation yields:

: rac{1}{c^2} \left( - \omega^2 A \sin (k x - \omega t - arphi) ight) = - k^2 A \sin (k x - \omega t - arphi)

and removing common terms

: rac{1}{c^2} \omega^2 = k^2

and since k = rac{\omega}{c} (from above) they are shown to be equivalent.
Thus, y satisfies the wave equation.


HELMHOLTZ EQUATION

The Helmholtz Equation is one that can satisfy:

: rac{\partial^2 y}{\partial t^2} + \omega^2 y = 0

Substituting in the second time partial from above

:- \omega^2 A \sin (k x - \omega t - arphi) + \omega^2 A \sin (k x - \omega t - arphi) = 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.


SEE ALSO

, Square , Triangle , and Sawtooth waveforms]]