This is a continuation of my question from Physics, Is there an equation representing the movement of a longitudinal wave?. It is referring to this animation.
I don't know much about notations on longitudinal waves, but here is my modification of the parametric equation: $$\begin{cases}x(\chi, t) &= f\cos\left(t - c\chi\right)+\frac{1}{\pi}g\chi \\ y(t) &= d \sin(h + 2b\chi)\end{cases}, \qquad 0 \leq \chi \leq \pi$$ where $f$ is the damping, $c$ is the stiffness, $g$ is the length of the spring, $d$ is the amplitude, $h$ is the shift, and $2b$ is the 'frequency'. I am not sure if I can call $b$ as half-cycle. The possible values of each variable are as follows: $$0 \leq f \leq \frac{cg}{100\pi} \\ 0 \leq c \leq 10 \\ g > 0 \\ d > 0 \\ -\frac{\pi}{2} \leq h \leq \frac{\pi}{2} \\ b > 0$$
Is there a way to transform this into a helix? This is because, what I think the equation right now, is an orthographic projection. I have no idea how to this, so I can't present any attempt. This is just curiosity, and the equation was obtained after mixing combinations of various functions for parametric equations.
A helix can be obtained by introducing a third coordinate, say $z$, and set $z(\chi) = d \cos(h + 2 b\chi)$. Then $(x(\chi,t),y(\chi),z(\chi))$ is a parametric representation of the oscillating helix at time $t$. Animating this set of equations yields something like this: