When we are introduced to waves in school, we are often presented with a picture of a sinusoid (or a cosinusoid).
Sinusoids can represent the way many physics phenomena behave, still....
Why are sinusoids so common?
When I first started to be more interested in waves, I was confused.
According to me a triangular wave, or a square wave seemed to be much more "intuitive". At least "easier". Why aren't we choosing them?
Mathematic gives us a little clue, but it doesn't show the full picture.
In fact, they form an orthonormal basis of the vector space of functions on $[0,2\pi]$. (More or less, every function in an interval can be written as a linear combination of them).
Moreover, sines and cosines are eigenfunctions (eigenvectors) of the second-derivative operator (eigenvalue $=-1$).
So sines and cosines are a solution of the [ $x''=-kx$ ] differential equation that is produced by a simple harmonic oscillator, which can be basically anything that obeys $F=-kx$.
But how do we glue all these pieces together to understand why are sinusoids such a significant phenomenon of our world?
Intuitively, I can guess that given the "apparent" analog nature of our world, physic phenomena presenting a back and forth pattern must go through a smooth transition as they reverse their cycle, and this may be why sinusoids are perfect in embodying this job. Though, my road to get a full picture seems still very long.
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2$\begingroup$ The projection of a regular circular motion along a line in its plane is a sinusoid. So one can expect sinusoids to be pretty much as common as circles, which, simply being the set of points at a given distance of another point, are rather common. For example, when one talks about quantum oscillations due to the incrementing of the phase factor, it is in fact a circular motion in the complex plane. $\endgroup$– Stéphane RollandinCommented Jul 14, 2019 at 16:42
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/159021/2451 , physics.stackexchange.com/q/201847/2451 , physics.stackexchange.com/q/414381/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Jul 14, 2019 at 18:38
2 Answers
You have almost the entire answer in your question. But first: I hope you can get past your intuition that triangle waves are more natural. That is an example of one of the biggest stumbling blocks to progress in physics education: the preconceived notion.
There are three important things that contribute.
$\, F=ma\quad\ \ $ — Newton's Second Law.
$\, F = -kx\quad$ — Hooke's Law
Those you have. The missing piece:
- All physical systems obey Hooke's Law if the disturbance is small enough, and in the vast majority of case, the disturbance is small enough.
[Note to pedants: please don't remind me of the many exceptions to 3.) ]
Putting that all together: there are lots of sinusoids in nature.
A spring or a pendulum obeys Hooke's Law unless the spring becomes distorted, or the pendulum swings too high. The chemical bonds between atoms in a solid very nearly obey Hooke's Law, but not exactly, and the deviation from Hooke's Law does have consequences. Electrons in an atom or solid obey Hooke's law when disturbed by an electric field. As a consequence, nearly all of the light reflected from a solid has the same frequency as the incoming light. Nearly all. A tiny fraction, unobservable without sensitive equipment, has twice the frequency due to deviations from Hooke's Law. And so on.
Well, sinusoids per se are not that common in nature at all. Even a tiny bit of nonlinearity essentialy corrupts the pure sine behavior of the idealized oscillator (see the van der Pol and the Duffing oscillators for some popular weakly nonlinear extensions). Based on what you have already stated, maybe a bit better assertion would be that the sinusoids are very common tool we use to simplify our way of describing the nature.
The square or sawtooth wave are more straightforward to draw with a pencil but that's it. Since both of them have discontinuous derivatives, they are in fact very unnatural.
Still to the points you have made, the mentioned features lead to one of the most powerful weapons of the mathematical physics and signal processing: the Fourier transform. It is safe to state that there are almost whole branches of applied physics that entirely rely on this process. It's obviously toxic, but I have met some engineers effectively incapable of thinking in the time domain any more. :-)
I think that the strongest link between the concept of sinusoids and "being natural" (however vague the latter feature is) is their connections to the second order differential equations solutions. These are the cornerstones of classical dynamics (the Newton's 2nd law or the Lagrange equations) and the complete set of Maxwell equations as well.
However, to conclude, I would like to stress once more the point that traditionally we are used to the sinusoids. Hence, it is pretty common to express the solutions employing them even if it would not be directly necessary.
