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