  A hydrodynamical description of coherent instabilities that take place in the
longitudinal dynamics of a charged-particle coasting beam in a high-energy
accelerating machine is presented. This is done in the framework of the
Madelung fluid picture provided by the Thermal Wave Model. The well known
coherent instability charts in the complex plane of the longitudinal coupling
impedance for monochromatic beams are recovered. The results are also
interpreted in terms of the deterministic approach to modulational instability
analysis usually given for monochromatic large amplitude wave train propagation
governed by the nonlinear Schr\"odinger equation. The instability analysis is
then extended to a non-monochromatic coasting beam with a given thermal
equilibrium distribution, thought as a statistical ensemble of monochromatic
incoherent coasting beams ("white" beam). In this hydrodynamical framework, the
phenomenon of Landau damping is predicted without using any kinetic equation
governing the phase space evolution of the system.

Accelerator Physics;Plasma Physics