Sine wave Totally Explained

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: ť $y (t) = A cdot sin(omega t + heta)$

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: ť

y(t) = Acdot sin(omega t - kx + 	heta) + D.,

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

k = 
ight),

which is also a sine wave with a phase-shift of п/2. Because of this "head start", it's 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's 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.

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