  Efficient collective response to external perturbations is one of the most
striking abilities of a biological system. Signal propagation through the group
is an important condition for the imple- mentation of such a response.
Information transfer has been experimentally observed in the turning mechanism
of birds flocks. In this context it is well-known also the existence of density
waves: birds under predation, attempting to escape, give rise to self-organized
density waves that propagates linearly on the flock. Most aspects of this
phenomenon are still not fully captured by theoretical models. In this work we
present a new model for the propagation of the speed (the modulus of the
velocity) fluctuations inside a flock, which is the simplest way to reproduce
the observed density waves. We have studied the full solution of the model in d
= 1 and we found that there is a line in the parameter space along which the
system relaxes in the fastest way with no oscillation after a signal has
passed. This is the critical damping condition. By analyzing the parameters
plane we show that critical damping represents an attractor for a steepest
descent dynamics of the return time of the system. Finally we propose a method
to test the validity of the model through through future experiments.
