Sinusoïdale

Sinusoïdale, or sinusoidal, refers to a curve or waveform that describes a smooth periodic oscillation. The standard forms are f(t) = A sin(ω t + φ) or f(t) = A cos(ω t + φ), where A is the amplitude, ω the angular frequency, and φ the phase. The signal repeats every period T = 2π/ω; the ordinary frequency is f = ω/(2π). Sinusoïdal functions have constant amplitude and symmetrical oscillation around zero.

Sinusoïds can be expressed using complex exponentials via Euler's formula: f(t) = Re{A e^{i(ω t + φ)}}. They are related by sin(x) = cos(x − π/2). A damped sinusoid uses an envelope e^{−α t} (α ≥ 0), yielding forms like e^{−α t} sin(ω_d t).

Because linear time-invariant systems respond to sinusoidal inputs with the same frequency, sinusoïds form a basis for Fourier analysis: any periodic signal can be decomposed into a sum of sinusoids with different amplitudes and phases. In physics and engineering, sinusoidal waves model alternating current, sound waves, and many mechanical vibrations. They also provide idealized representations of optical and radio-frequency signals, where phase and amplitude carry information.