One reason that sine waves appear in nature is that in many physical systems, we can express a general wave as a superposition of sines and cosines of all frequencies, but different frequencies travel at different speeds. This means that even for a messy source of waves, initially looking nothing like a sine wave, observers far away will, at any given time, only see one frequency, and this will look just like a pure sine wave.
We can justify superposing waves and treating different frequencies independently (a linear approximation), when the waves appear as small fluctuations around some equilibrium situation. For example, a flat ocean is in equilibrium, and small disturbances to the surface propagate as waves as gravity acts to restore equilibrium. Mathematically, the sine and cosine waves appear as the natural building blocks of this linearised approximation because they have simple behaviour under differentiation: the slope of a sine is a cosine, and vice versa.
The specific model of the physics then tells us, for any given wavelength (or wavenumber $k$), the frequency $\omega(k)$ (depending on $k$) at which they oscillate/propagate. This is the dispersion relation for the system at hand. For example:
$$\omega \left(k\right) \begin{cases}= \sqrt{g k} & \text{for waves in deep water} \\= c k & \text{for light or sound} \\\propto k^{3/2} & \text{for capillary waves} \end{cases} $$
The last example here happens for small water waves (size around millimetres or smaller) where the main restoring force is surface tension, rather than gravity, as in the first on the list.
The speed at which a given frequency propagates is given by the group velocity $c_g=\frac{d\omega}{dk}$, which for the examples given above is proportional to $\frac{1}{\sqrt{k}}$, $k^0$ (constant), and $\sqrt{k}$ respectively. This shows three qualitatively different behaviours: longer wavelengths (smaller $k$) go faster, all wavelength travel the same speed, or shorter wavelengths go faster.
Any surfer will be able to tell you that when there's a storm across the ocean, and new swell comes in to the beach, the first waves that arrive are always the largest period, longest wavelength, and then as the hours and days pass they get progressively shorter period! The reason that there is a well-defined period, and they look like sine waves (at least until they get too close to the beach and the seabed starts interfering), is precisely this dispersion phenomenon. If you make small waves on a pond/bath with your finger, you may be able to see the opposite phenomenon where the shortest wavelengths spread out fastest.
Sound does not have this dispersion, since all wavelengths move at the same speed. If you play a short burst of white noise on a speaker, with no well-defined frequency, you will hear white noise, even far away, because the different component frequencies will not separate out. The sound waves will then never look anything like sine waves!
In any case, if you make a splash, the surface of the water will be messy at first, but the waves that spread out will look like nice clean sine waves with well-defined peaks and troughs, because all the components of the original mish-mash of wavelengths travel at different speeds and separate out from one another.