The premise of the question is faulty. Sine waves never appear in nature. Nor do squares, circles, spheres, logarithms, exponentials, or any other pure mathematical construct. Some things in nature are variously well approximated by these constructs, but you will never find a perfect one anywhere.
Fourier's theorem states that any periodic signal can be constructed with an infinite sum of sine waves. Some quasi-periodic signals in nature will be well approximated with a single frequency component, others will not. In reality there are no sine waves, just occasionally quasi-periodic phenomena. The sine waves are imaginary - a construct to ease our own cognition and calculation.
We could just as well have decided that triangle waves would be the basis of our mathematics - in that case we could represent sinusoidal motion as a fourier series of triangle components. The mathematics would become entirely more complex but the model would nevertheless be perfectly valid. The sine wave is a natural pure component, however, so we use it because it is the most primitive and the simplest and clearest to manipulate.
Roll a lumpy, odd shaped rock down a hill. Is there a sine wave there? Maybe. But there are other things as well. The rise and fall of rivers might be square-ish, but with logarithmic-ish rises and exponential-ish falls. A bolt of lightning you might think of as a dirac delta function on human timescales, but something completely different at 100,000fps. If you look closely enough you can find circles, squares, sine waves, square waves, straight lines, logarithms, exponentials, and all manner of other things in nature. Speakers of Arabic often find "words" written in trees, bushes, vines, etc, much as you might see a rabbit in the clouds.
Are any other things in nature approximated by rabbits? It's a silly question, depending on how you think about it.