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.

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 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.

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 main way a sine wave is usually defined is that it’s what you get when you measure the height of a point as it travels around the edge of a circle. However, there is another way to get a sine wave:

A sine wave is the pattern of motion you get when you dangle an object from the end of a spring and it bounces up and down.

The force that the spring exerts on the object is linearly proportional to the distance that the spring has been stretched, (AKA the longer the spring stretches the harder it pulls) and that’s really the only thing going on here. A sine wave naturally emerges whenever you have a system where a parameter’s acceleration is inversely linearly proportional to its distance from some equilibrium value (inversely because the pull is back towards the center). In other words, the only requirement for a sine wave is you have some system that pulls some parameter towards a resting value and pulls harder the further away that parameter is.

That’s really not an uncommon scenario, so it pops up literally everywhere in nature from electromagnetic waves to sound waves to chemical reactions.

It so happens that things that repeat in a pattern do so in simple ways or can be decomposed into the summation of simpler patterns. One of the simplest ways of oscillating (i.e. “waving”) can be represented by some value going up and down at a particular rate of repetition. This simple way is also smooth and not erratic (see sawtooth waves). Physical phenomenon tend to be smooth and continuous. It’s therefore easier to use simple and smooth things *to model* physical phenomenon bc they represent reality well when we use them. They are also amazing to work with mathematically and have properties that make it *a lot* better to deal with algebraically.

So really, we have this mathematical object that we call the sine wave and it is perhaps deemed “natural” because it helps in describing so many things. This choice arose mostly out of natural selection bc it’s so simple and useful when we connect the abstract mathematical world to the physical one. We could equivalently use any other periodic function that has similar properties to do the same thing – it so happens that the sine wave is one of the simplest bases to work off of.

I’ll also add that very few things in nature are a single sine wave. Periodic phenomenon, or anything wave-like, are usually better represented as a summation of them, where each individual sine component can have a separate frequency and/or amplitude.

Most math describes the natural world. A sine wave is the natural response to an object in this universe to a perturbation. If I have a ball hanging from a string and I raise the ball and let it go, we all know, the ball will swing. This is what we observe from every ball and string no matter where we are in this universe. This of course assumes gravity is a constant. The value (y) of the sine function is the velocity of the ball at the position (x). At the top and bottom of the sine wave (maxima and minima) the velocity is zero. If there was no friction from the air or the string, the ball would swing for a very, very long time.

Walter Lewin MIT 802 does a great demo, 802 is a great physics class, it’s worth watching.

That’s not quite true. For example, if you study the KdV equation which describes shallow water wave, the natural wave is the *soliton*. Same goes for light in a fiber optic. These can be related to asymptotically sinusoidal wave satisfying the Schrodinger equation, through the process call inverse scattering.

In general, the natural wave we want to study are solitons. Over time, solitons keep its shape as it moves, while being sturdy and highly resistant against disturbance. This allows us to think of them as concrete object.

When it comes to physical field, it’s common for those field to satisfy a law that is both time and space invariant. After all, we expect the law of physics to be unchanged over time and space, so as long as there are no external forces we expect these fields to be space and time invariant. In that case, complex exponential on imaginary axis has a very specific property: it diagonalizes the translation, in other word, translation only causes a multiplication by a constant factor. If the equation is also linear, which is a common situation when you’re looking at *perturbed* equation (that is, equation describe tiny changes from the equilibrium), then these equation are linear and hence has superposition, so complex exponential on imaginary axis satisfies all the other properties of solitons automatically. This makes them the most natural wave. Sine wave is equivalent to these wave, but in real number.

If the law is not space invariant but time invariant, due to an effect that is localized in space, one can look at the far-field approximation: what do the waves look like far enough, where they behave almost as if there are no external forces, and the above analysis still work approximately.

The difficulty started to come in when you no longer have perturbed equations and have to face non-linear equation. Then you would have to look at solitons that are very different from sine waves.

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