Why are sine waves so fundamental?

Oh man! You landed right in my backyard with this one. Heck, even my
company logo is constructed by summing 2 sine waves.

Why are they so fundamental? Because it takes the least amount of energy/power to move a time varying signal along a path using sinusoidal motion. Why are bubbles spherical? Because it takes the least amount of surface area to enclose the most volume. Why are lines straight? Because the shortest distance between two points is a straight line. Well if you want to move something from side to side repeatedly, a sinusoidal pattern with a sinusoidal velocity and acceleration will take the least amount of energy to do it.

Also by using Fourier analyses you can represent complicated motion by summing a few simple sine waves. In fact all repeating patterns can be represented with sine waves. Once you have the representation you can filter out unwanted signals and amplify the ones you want.
As we speak you are being bombarded with thousands, millions, billions of electromagnetic sine waves. Yet you can turn on your radio, select one frequency and listen to NPR’s Terry Gross on Fresh Air. Magic. Here… I will send out an electromagnetic signal in the UHF region 433.800 MHz to everyone on the planet. It will only take 80 ms to reach everyone on Earth. Hold your breath. .... There. Did you feel it? You’re welcome.

I appreciate that answer, but it still seems strange how nature chooses sine waves. I can see, for example, why bubbles are spherical. Having uniform tension will bring about a spherical shape.

If I pluck a guitar string, it will vibrate at a given frequency plus harmonics. How does it know to do that? What is there about the string that suppresses other frequencies?

It is the same sort of thing as the bubble. The energy required to make the string move back and forth is lowest if you let it speed up and slow down and move like a sine wave. If you try any other pattern you will have to work to slow it down or speed it up. A sine wave naturally slows down to zero at the extreme positions and has the fastest velocity at the midpoint.
A string can move at other harmonics but they take more work. It is the fundamental that has the largest amplitude for the least amount of energy.

Think about a pendulum. It moves in a sine wave pattern. What pattern would you want to to have? A square wave? a triangle wave? Sure you can do it, but you will have to inject extra energy or rapidly extract it to make those shapes.

(In case any stickler brings it up, of course I am ignoring friction and infinite singularity points at the edge of a perfect square wave. I am also using the terms work, energy, and power loosely. We are just talking rough concepts here.)

Thanks, I am going to have to think that over. I found this online book that describes music from a mathematical point of view. I will be looking through it to see what additional information I can get.

You are looking for an intuitive answer so I am trying to answer you in a manner that you can feel. Let’s look at the some of the points along the curve and see if it they make sense.
Imagine a swing. At the extremes (all the way forward and all the way backward) you want your velocity to be zero. That is obvious since you are changing direction. But, you also want to be slowing down smoothly until you reach the end or else that would be equivalent to bumping into a brick wall and that would take extra energy. That means the slope at the ends must be flat. Once you reach an extreme and start swinging the other way, you want the speed to increase gradually until you are right at the mid point (the bottom of the swing). Then you want it to gradually reduce the velocity. Look at the sine curve near the zero point. It has a slope of 1. Compare it to a “curve” made up of alternating semicircles. They kind of look like a sine curve. But, look at how they intersect the X axis. The y=zero points have a slope of infinity. That means at those spots you must move infinitely fast and then you have to slow down infinitely fast. That is not often seen in nature. That takes a lot of energy.

Hey, here is another burst of UHF for you. 433.800 MHz. 100ms. Now. You just got a few hundredths of a picowatt from me. I hope you enjoyed it.
This is Great Question.

Thanks again. I can see that the sine function has the desired characteristics. Now I would like to find something more analytic that shows that the sine function is unique in these characteristics and also shows how other possibilities are selected against.

Nature is the confluence of efficiency and aesthetics. The ultimate design model. Form meets function and thus, beauty to the eye, the ear, all the senses. Mathematically orgasmic for about 9% of the population. The rest of us just, “Ooh, that’s nice”; with a long drawn out sigh of contentment.