Why do we see Sine Waves in nature so often?

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I’m studying engineering, and I see sine waves everywhere I look. why is this waveform so common in nature?

In: Mathematics

Most examples I can think of are due to things trying to get to a stable position.

Swinging pendulum: stable position is at the bottom, but it still has enough energy to overshoot back up to the same height on the opposite side (if undamped).

Spring oscillating: stable position is at δ=0, but overshoot on the way back, same as the pendulum.

Same thing with ripples in water.

A/C current: not exactly sure what induces the oscillation, but I know it goes back and forth between positive and negative, and the smoothest way to do that is a sine wave.

Vibrations through solid objects (probably liquids and gas too, and therefore sound waves) work for the same reason as a spring.

No idea about E&M waves.

Can you give some examples? If you’ve recently been studying sine waves you may be experiencing [Selection bias](https://en.wikipedia.org/wiki/Selection_bias) where you notice things more when you’ve been made aware of them.

Wave phenomenon appear all the time in nature but I’d disagree sinusoids in particular are showing up a lot. What distinguishes sinusoids from other waves is their mathematical convenience, being tied to a simple, linear differential equation and exponentiation with complex numbers. That mathematical convenience means we’re often expressing non-sinusoidal waves as sums of sinusoids or outright approximating a non-sinusoid with a sinusoid, often via simplifying assumptions.

For instance, a pendulum swing isn’t actually a sine wave, it’s just common practice to take the ODE governing a pendulum’s motion, which has no explicit solution in elementary functions, and literally replace it with another ODE that has a sine wave solution, justifying it by saying something about a small angle approximation. To catalog the pendulum’s swing as a natural occurrence of a sine wave is to not critically examine the simplifying steps we took in our analysis.

We see exponential growth and exponential decay all over the place in nature. In the heating and cooling of materials, in the rates of osmosis or chemical reactions, in population growth of bacteria, in finance and economics, etc. etc. etc.

This is “because” exponential functions are the solution to a very common class of differential equation. Specifically any equation which relates the rate of change of a quantity to it’s current size – that is, any equation of the form f'(x) = a*f(x) – has an exponential solution. The exponential function f(x)=e^ax is the *eigenfunction* of first order differentiation.

Sine and cosine are, likewise, the eigenfunctions of second order differentiation, at least for negative eigenvalues. They solve equations of the form f”(x)=a*f(x) for negative values of a and while this sort of differential equation isn’t quite as common as the one above it still shows up frequently. This is “why” you see sine and cosine so often: whenever the rate of the rate of change of a quantity is negatively proportional to the current magnitude of that quantity you get a sine wave.

I’m no maths expert.

Wouldn’t it be just because so many things rotate, revolve, turn, etc. in ways that are described by circles/ellipses? And whenever you have circles, you have sine waves, and lots of delicious pi.

Too simple?