In mathematics and signal processing, a sine wave or sinusoid is a waveform whose graph is identical to the generalized sine function:

$y\; \backslash \; =\; \backslash \; A\; \backslash sin(kx\; -\; \backslash omega\; t\; -\; \backslash varphi)\; +\; D$

where $A$ is the amplitude, $k$ is the wave number, $\backslash omega$ is the angular frequency, $\backslash varphi$ is the phase shift, and $D$ is the vertical offset (sometimes called the DC offset) at position $x$ and at time $t$.

The wave number is related to the angular frequency by

$k\; =\; \backslash omega\; \backslash over\; c\; =\; 2\; \backslash pi\; f\; \backslash over\; c\; =\; 2\; \backslash pi\; \backslash over\; \backslash lambda$

where $\backslash lambda$ 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,

A two-dimensional example would describe the amplitude of a two-dimensional wave at a position $(x,y)$ at time $t$.This could,

Common form

While the above form is a very generalized form, one common — and more simple — form is:

$y\; =\; A\; \backslash sin(\backslash omega\; t\; -\; \backslash varphi)$

which describes a wave with frequency $\backslash omega$ (in radians) and phase of $\backslash varphi$. It is a specific case of the above by letting $x=0$ and $D=0$.

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: $\backslash cos\backslash left(x\; -\backslash frac\backslash pi2\backslash right)\; =\; \backslash sinx$

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

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:

$\backslash frac1c^2\; \backslash frac\backslash partial^2\; y\backslash partial\; t^2\; =\; \backslash frac\backslash partial^2\; y\backslash partial\; x^2$

To show this is true:

$\backslash frac\; \backslash partial\; y\backslash partial\; t\; =\; -\; \backslash omega\; A\; \backslash cos\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \backslash partial^2\; y\backslash partial\; t^2\; =\; -\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \backslash partial\; y\backslash partial\; x\; =\; -\; k\; A\; \backslash cos\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

$\backslash frac\; \backslash partial^2\; y\backslash partial\; x^2\; =\; -\; k^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

and inserting the second partials into the wave equation yields:

$\backslash frac1c^2\; \backslash left(\; -\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; \backslash right)\; =\; -\; k^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)$

and removing common terms

$\backslash frac1c^2\; \backslash omega^2\; =\; k^2$

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

Helmholtz equation

The Helmholtz equation is one that can satisfy:

$\backslash frac\backslash partial^2\; y\backslash partial\; t^2\; +\; \backslash omega^2\; y\; =\; 0$

Substituting in the second time partial from above

$-\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; +\; \backslash omega^2\; A\; \backslash sin\; (k\; x\; -\; \backslash omega\; t\; -\; \backslash varphi)\; =\; 0$

which is clearly true.

See also

• Wave equation
• Helmholtz equation
• Fourier transform
• Harmonic series (mathematics)
• Harmonic series (music)
• Pure tone
• Pseudo sine wave Sinusoid

Sinusoide Sinusoide Siniaalto Senóide Синусоида