Add two sine waves of the same pitch and you always get another
Mix two sine waves that share a frequency, whatever their sizes or timing, and the result is still a clean sine wave at that frequency. No other periodic wave shape behaves this way. That quirk explains why the sine wave, a pure tone with no harmonics, is the basic building block of sound, light and signal processing.
A sine wave traces the shape of the trigonometric sine function. Viewed as motion along a line it is simple harmonic motion, like a weight bobbing on a spring; viewed as rotation it is a point moving steadily around a circle. Three numbers describe one: amplitude, the height of its peaks; frequency, how many cycles pass each second; and phase, where in its cycle it starts. Shifting the phase slides the whole wave earlier or later in time, and a quarter-cycle shift turns a sine into a cosine.
The closure property follows from a trigonometric identity. Expand each wave with the angle-addition formula and everything collapses into one sine part plus one cosine part, which a right triangle then folds back into a single sinusoid with new amplitude and phase. Engineers prize a consequence: sine waves pass through linear time-invariant systems, such as many filters and circuits, emerging with only their size and timing altered. And because they travel through linear media without changing shape, they are the standard tool for analysing how waves propagate.
Music shows the other side. A lone sine wave sounds like a pure tone. Real instruments add higher harmonics on top of the fundamental, and the particular blend is timbre, the reason a violin and a flute playing the same pitch sound different. Fourier analysis runs this in reverse, breaking complicated signals into sums of sine waves with various frequencies, phases and strengths.
When two identical waves travel in opposite directions, they combine into a standing wave that seems to vibrate in place. A plucked string does this with waves bouncing off its fixed ends, so it can sustain only wavelengths twice its length, the fundamental, or whole-number fractions of that, the harmonics. The same mathematics describes wind waves on water, sound in air and single-colour light.
Source: Sine wave
