Sine wave
From Free net encyclopedia
The sine wave or sinusoid is a function that occurs often in mathematics, signal processing, and other fields. Its most basic form is:
- <math>y = A\cdot \sin(\omega t - \varphi)</math>
which describes a wavelike function of time (<math>t\,</math>) with:
- peak deviation from center = <math>A\,</math> (aka amplitude)
- angular frequency <math>\omega\,</math> (radians per second)
- initial phase (t=0) = <math>-\varphi</math>
- <math>\varphi</math> 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 <math>\varphi / \omega</math>.
Contents |
General form
In general, the function may also have:
- a spatial dimension, <math>x\,</math> (aka position), with frequency <math>k\,</math> (aka wave number)
- a non-zero center amplitude, <math>D\,</math> (aka DC offset)
which looks like this:
- <math> y \ = \ A\cdot \sin(kx - \omega t - \varphi) + D</math>
The wave number is related to the angular frequency by:
- <math> k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }</math>
where <math>\lambda</math> is the wavelength, <math>f</math> is the frequency, and <math>c</math> 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 <math>x</math> at time <math>t</math> 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 <math>(x,y)</math> at time <math>t</math>. 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: <math>\cos\left(x -\frac{\pi}{2}\right) = \sin{x}</math>
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.
Wave equation
The wave equation is one that can satisfy:
- <math>\frac{1}{c^2} \frac{\partial^2 y}{\partial t^2} = \frac{\partial^2 y}{\partial x^2}</math>
To show this is true:
- <math>\frac {\partial y}{\partial t} = - \omega A \cos (k x - \omega t - \varphi)</math>
- <math>\frac {\partial^2 y}{\partial t^2} = - \omega^2 A \sin (k x - \omega t - \varphi)</math>
- <math>\frac {\partial y}{\partial x} = - k A \cos (k x - \omega t - \varphi)</math>
- <math>\frac {\partial^2 y}{\partial x^2} = - k^2 A \sin (k x - \omega t - \varphi)</math>
and inserting the second partials into the wave equation yields:
- <math>\frac{1}{c^2} \left( - \omega^2 A \sin (k x - \omega t - \varphi) \right) = - k^2 A \sin (k x - \omega t - \varphi)</math>
and removing common terms
- <math>\frac{1}{c^2} \omega^2 = k^2</math>
and since <math>k = \frac{\omega}{c}</math> (from above) they are shown to be equivalent. Thus, <math>y</math> satisfies the wave equation.
Helmholtz equation
The Helmholtz equation is one that can satisfy:
- <math>\frac{\partial^2 y}{\partial t^2} + \omega^2 y = 0</math>
Substituting in the second time partial from above
- <math>- \omega^2 A \sin (k x - \omega t - \varphi) + \omega^2 A \sin (k x - \omega t - \varphi) = 0</math>
which is clearly true.
See also
- Wave equation
- Helmholtz equation
- Fourier transform
- Harmonic series (mathematics)
- Harmonic series (music)
- Pure tone
- Pseudo sine wavede:Sinusoid
es:Sinusoide it:Sinusoide fi:Siniaalto pt:Senóide ru:Синусоида sr:Синусоида