I Apologize

Some posts are still missing, but I found,

$\cfrac{1}{\lambda_{p}}=\cfrac{1}{\lambda_{f}}-\cfrac{1}{\lambda_{i}}$

that follows immediately from,

$E=h.f=h\cfrac{c}{\lambda}$

and

$E_p=E_f-E_i$

that the photon emitted on a energy state transition, $n_i\rightarrow n_f$ is the energy difference between the two states.

I will pick up that line of thought yet...

Still, I am missing $v_{max}=???$.  That post is important because it makes the unlikely combination of ice and sonic boom dangerous.  Very dangerous.