// engineering › Fourier Waveforms
See how amplitude, frequency and phase shape the pure sine and cosine that every Fourier series is built from.
y(t) = Asin(2pi f t + varphi) text{and} Acos(2pi f t + varphi)
A mind behind this: Joseph Fourier 1768–1830
Almost every periodic signal in the physical world — AC mains, sound, light, radio — can be written as a sum of sines and cosines. They are the 'atoms' of vibration: a sine wave keeps its shape when you differentiate it, pass it through a linear circuit, or send it down a wire, which is exactly why engineers describe everything in terms of them.
Amplitude (A) is how tall the wave is — loudness for sound, voltage for a signal. Frequency (f) is how many cycles happen per second, measured in hertz — pitch for sound, colour for light. Phase (φ) shifts the whole wave left or right in time. Drag each slider here and you can see exactly which knob does what.
cos(x) = sin(x + π/2). They are the same wave a quarter-cycle apart. That 90° relationship is why engineers pair them: any sinusoid of a given frequency can be made from a little sine plus a little cosine, which is the whole basis of the Fourier series.
Tuning a guitar string (a near-pure sine at its fundamental), the 50/60 Hz hum from mains power, the carrier wave of an FM radio station, and the swing of a pendulum all behave like the sinusoids on this page. Audio synthesisers literally start from a sine oscillator and build everything else on top.
Phase decides where in its cycle a wave starts. Two identical speakers slightly out of phase can cancel each other (that is how noise-cancelling headphones work) or reinforce each other. In power systems, the phase difference between voltage and current determines how much usable power you actually get.