sinusoid

A sinusoid, or sine wave, is a periodic, smoothly varying function that models simple harmonic motion and many physical signals. It is described by sine or cosine curves and is characterized by a constant frequency and amplitude.

In continuous time it can be written as y(t) = A sin(ω t + φ) or y(t) = A cos(ω t + φ), where A is the amplitude, ω is the angular frequency, f = ω/(2π) is the ordinary frequency, and φ is the phase. The period is T = 2π/ω = 1/f. A and φ determine the waveform’s peak height and horizontal shift.

Sinusoids have a single frequency and constant amplitude; they repeat exactly every period and are symmetric about their midline. They are the fundamental building blocks of many signals and phenomena.

They are central in physics and engineering: many signals can be approximated locally by sinusoids, and they describe light and sound waves, electrical alternating current, and oscillatory motion. In Fourier analysis, any periodic signal can be decomposed into a sum of sinusoids of different frequencies, amplitudes, and phases.

A common complex representation uses A e^{i(ω t + φ)}, whose real part gives A cos(ω t + φ). In discrete time, a sinusoid takes the form x[n] = A sin(ω0 n + φ), with ω0 in radians per sample. Applications span signal synthesis, communications, system testing, and the analysis of physical processes. Sinusoids are often regarded as the simplest and most important periodic signals in science and engineering.