  We theoretically investigate the evolution of the peak height of an energy
resolved electronic wave-packets ballistically propagating along integer
quantum Hall edge channels at filling factor $\nu=2$. This is ultimately
related to the elastic scattering amplitude for the fermionic excitations
evaluated at different injection energy. We investigate this quantity assuming
a short range capacitive coupling between the edges. Moreover, we also take
into account phenomenologically the possibility of energy dissipation towards
additional degrees of freedom both linear and quadratic in the injection
energy. Comparing with recent experimental data, we rule out the
non-dissipative case as well a quadratic dependence of the dissipation,
indicating a linear energy loss rate as the best candidate to describe the
behavior of the quasi-particle peak at short enough propagation lengths.
