# Dictionary Definition

sinusoidal adj : having a succession of waves or
curves

# User Contributed Dictionary

## English

### Adjective

# Extensive Definition

The sine wave or sinusoid is a function that
occurs often in mathematics, physics, signal
processing, audition,
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".

The sine wave is important in physics because it
retains its waveshape when added to another sine wave of the same
frequency and arbitrary phase. It is the only periodic waveform
that has this property. This property leads to its importance in
Fourier analysis and makes it acoustically unique.

## General form

In general, the function may also have:

- a spatial dimension, x (aka position), with frequency k (also called wavenumber)
- 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 wavenumber is related to the angular
frequency by:.

- k = = =

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.

In two or three spatial dimensions, the same
equation describes a travelling plane wave if position x and
wavenumber k are interpreted as vectors, and their product as a
dot
product. For more complex waves such as the height of a water
wave in a pond after a stone has been dropped in, more complex
equations are needed.

## 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(x + \pi/2), which is also a
sine wave with a phase-shift of π/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.## See also

sinusoidal in Catalan: Sinusoide

sinusoidal in Danish: Sinusbølge

sinusoidal in German: Sinusoid

sinusoidal in Estonian: Sinusoid

sinusoidal in Spanish: Sinusoide

sinusoidal in French: Signal sinusoïdal

sinusoidal in Italian: Sinusoide

sinusoidal in Japanese: 正弦波

sinusoidal in Portuguese: Senóide

sinusoidal in Russian: Синусоида

sinusoidal in Simple English: Sine wave

sinusoidal in Serbian: Синусоида

sinusoidal in Sundanese: Gelombang sinus

sinusoidal in Finnish: Siniaalto

sinusoidal in Swedish: Sinusvåg

sinusoidal in Turkish: Sinüzoid dalga

sinusoidal in Chinese: 正弦曲線