Sine waves can be represented mathematically and it can be shown that any wave can be constructed from an appropriate combination of sine waves (Fourier synthesis)

Any single frequency traveling wave will take the form of a sine wave.
This transverse wave is typical of that caused by a small pebble dropped into a still pool.
The position of an object vibrating in simple harmonic motion will trace out a sine wave as a function of time. (Or if a mass on a spring is carried at constant speed across a room, it will trace out a sine wave.)
