web.archive.org

sine wave: Definition and Much More from Answers.com

️Wed Jul 01 2015

The graphs of the sine and cosine functions are sinusoids of different phases.

The oscillation of an undamped spring-mass system around the equilibrium is a sine wave.

The sine wave or sinusoid is a function that occurs often in mathematics, physics, signal processing, electrical engineering, and many other fields. Its most basic form is:

$y (t) = A \cdot \sin(\omega t + \theta)$

which describes a wavelike function of time (t) with:

peak deviation from center = A (aka amplitude)
angular frequency $\omega\,$ (radians per second)
phase = θ
- When the phase is non-zero, the entire waveform appears to be shifted in time by the amount θ/ω seconds. A negative value represents a delay, and a positive value represents a "head-start".

General form

In general, the function may also have:

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

which looks like this:

$y(t) = A\cdot \sin(\omega t - kx + \theta) + D.\,$

The wave number is related to the angular frequency by:.

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

where λ 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 propagation of an electrical wave through a conducting plane.

Occurrences

This wave pattern occurs often in nature, including 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 said to be "sinusoidal", because:

$\cos(x) = \sin\left(x +\frac{\pi}{2}\right),$

which is also a sine wave with a phase-shift of п/2. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.

Any non-sinusoidal waveforms, such as square waves or even the irregular sound waves made by human speech, can be represented as 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; this may be described as a different timbre.

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 nearly any periodic waveform. The process is named Fourier analysis, which is a useful analytical tool in the study of waves, heat flow, many other scientific fields, and signal processing theory. Also see Fourier series and Fourier transform.

sine wave: Definition and Much More from Answers.com

General form

Occurrences

Fourier series

See also