Lately I have been completely absorbed in the mathematics of sine waves, how they work, and how they are intertwined with many important maths principles that we follow today. I was thinking to myself "would it be possible to generate a near-perfect sine wave by hand using some rope". Now, when I say "generate a sine wave by hand", I mean to swing a piece of rope back and forth while the other end is tied to some fixed point in such a way that if you were to freeze the piece of rope in time and measure the amplitude of the wave, you would get 2 (-1 to +1), and get some wavelength ($\ \lambda$ ) that is similar to the value of $\ 2 \pi$. Now this may seem simple – yes. However, when you look into it that way of thinking rapidly falls apart.
Firstly, as we all know, $\ \pi $ is irrational. You cannot possibly write out all digits of $\ \pi $ on paper. This quickly terminates the thinking that the answer is "yes". But the answer isn't no either.
Well, why don't we ask a different question – How close can you get?
If we define our hand as an object with just one parameter – the speed through a defined medium (air) at which it can move (ie: The fastest speed at which a hand can move). I will define this speed as $\ \mu $. By changing $\ \mu $ we can modify how fast the hand can move.
If we take a look at the$\ \sin x$ function, we will see that it repeats its cycle every $\ 2 \pi $ units, and moves up and down between the numbers -1 and 1. So that gives us 2 variables to try and get close to. So if we want our hand to move back and forth at an amplitude of 2, we would need our hand to move up and down every 2 units (so if our units were centimetres (CM), the sine wave would need to reach its highest point in the first second, and its lowest point in the next). But then the sine wave needs to have $\ \lambda = 2 \pi $ (or something close to it). So what would the value of $\ \mu $ be?
Best Answer
I worked really hard to solve this question, and I think this will work.
First of all, because the speed at which you move your hand is non-linear, I cannot represent it as a fixed value. It must be written as a function of $\ \mu (t) $. Here is what I have come up with: $$\ \mu(t) = \frac{\int_{0}^t \sin x dx}{t}$$ Explanation:
We take the antiderivative of $\ \sin x$ from $\ 0 $ to $\ t $ since we want to get the distance travelled by the sine wave up until that point (the area underneath a sine wave will be the total distance travelled by the wave up until that point) and since the value of $\ t$ is also the time that we have been moving our hand, we then divide the distance by it (see: speed) to get the speed of your hand at that point. Voila! We get our speed that our hand must be moving at at that point in time.
If there are any mistakes, please comment on them and I will modify my answer!