Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like :
and the equations governing the motion would be
So I've been wondering why can't simple harmonic motion be represented in form of triangular waves.Although the equations above involve angular momentum so I may be contradicting myself but fundamentally the velocity time is sine function :
$$-\sin(x)$$
and the gradient represents the acceleration is non-uniformly increasing and decreasing.
What if instead of that you use
$$-(\arcsin(\sin(x))$$
Which would represent a triangular wave whose gradient would depict that the acceleration is uniformly accelerating and decelerating.So would this represent harmonic motion or is it fundamentally incorrect.
Best Answer
Harmonic motion in physics isn't so much defined by a periodic solution as it is defined by a certain differential equation. The equation for the harmonic oscillator is $$ \ddot{x} + kx = 0, $$ where $k$ is some constant. The general solution to this equation can be written $$ x(t) = A \sin(\sqrt{k}t) + B \cos(\sqrt{k}t), $$ where $A$ and $B$ depend on the initial conditions.
Sines and cosines simply fall out naturally from the basic equation. Triangle waveforms, on the other hand, are not even differentiable at kinks, so they are not so naturally governed by a differential equation.