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.