Why sine wave is the natural wave, why not something else?

In: 78

It depends on what you mean by ‘natural’

Sinusoids have the property that they are proportional to the negative of their second derivatives. This is analogous to how exponentials are proportional to their own first derivatives. They will show up in systems governed by (relatively) simple mathematics as a result.

A sine wave is the shape you get when tracing the change in y per unit of a circle.

This is not the same as cutting a circle in half and making one long wave, as the x axis does not move continuously.

Starting at the center of the circle, you approach the top, the bottom, and return, where the line traces a perfect circle of X radius.

Other waves can have the same wavelength but higher/lower amplitudes. These are explained with different mathematics.

These waves can be combined into one signal, which will add and subtract all the wave components into a new shape, these can be random wobbles or make shapes like triangle waves and square waves.

However, these require a combination of forces, and are difficult for men to replicate, even if they are found in nature, as they are a complex combination of other more simple forces.

It’s the simplest geometrical (or mechanical) situation: something rotating around an axis (a fixed point) will have a sine (or cosine) movement when projected onto a line.

If you leave aside the movement, sine and cosine are unavoidable each time you have a projection: your shadow on the ground, mapmaking, video game rendering, &c.

On the other hand if things change/move, you can mathematically represent nearly any situation by combining various sines and cosines (so-called Fourier decomposition). It just so happens that sines and cosines have awfully pleasant mathematical properties. A piano note for example can easily by analyzed into a superposition of sine (and cosine) vibrations.

That being said, there are other methods/waves, e. g. [wavelets](https://en.wikipedia.org/wiki/Wavelet).

A sine wave is “natural” because it’s the graph of many natural phenomena. E.g. A weight on a spring bouncing up and down. A pendulum. A water wave. A sound (pressure) wave.

These all have the property that the restoring force is linearly proportional to the displacement. Sinusoidal movement is the result.

The 5….uhh, 8 year old explanation is that sine (or sinusoidal functions like cosine) has a number of unique mathematical properties that no other curve has.

The HS explanation is that sine can describe both circles and is given by very simple second order differential equations.

ETA: it also connects the numbers e, pi, 1, and 0.