Be Like Water...

The situation is analogous to water drops onto a confine of water,

$\psi_n$ water drops that are to pass through, re-bounce at lower depths and resurfaces.  $\psi_n$ water drops that are ejected tangentially, moves away radially around the point of impact.  And $\psi_d$ move up and down in simple harmonic motion, without lateral displacement at the point of impact.

The expression for $v_{max}$ does allow for water drops that bounce immediately,

$v_{ { max } }=\pm\sqrt{\cfrac { 1 }{ m } \cfrac { \psi _{ n } }{ \psi _{ max } }\left\{ \psi _{ n }-\psi _{ max } \right\}  e^{ \psi _{ max } }\left( { e^{ 2\psi _{ max } }-1 } \right) ^{ 1/2 }}$

where the negative root moves the ejected particle back along the direction of impact.