Following is the image of a 3D standing electron wave in circular form. Each of its loop moves up and down (while the adjacent loop is supposed to be $\pi$ radians out of phase) i.e. out of the plane and also into the plane. I think the wave equation should be something of the form $z=\sqrt{r^2-x^2-y^2}\sin(?)$. What should be ? here ? It should be something that is dependent on time but I'm not being able to figure it out.
On
I think you would be better asking this on the physics stack exchange. However, I will propose a possible answer.
Often systems of equations have solutions with terms of the form $\sin(kx-\omega t) $. In this case that would cause us to have
$$ ? = k\theta - \omega t $$
In your diagram there are 4 periods of oscillation per rotation, thus
$$ k \pi/2 = 2\pi $$
$$ k=4 $$
Thus $$ ? = 4\theta - \omega t $$
The diagram doesn't give us any information about the frequency $\omega$, so that's as far as guessing will take us.
Let $\theta$ denote a polar angle in the $(x, y)$-plane. If you mean a standing wave with amplitude $A$, frequency $\omega$ with respect to time $t$, and $n$ waves in one turn, you want a formula of the type $$ z(\theta) = A\sin(n\theta)\sin(2\pi\omega t). $$ (The overall phase of either trig function may be changed harmlessly.)