  In a recent paper by Lucas and Das Sarma [Physical Review B 97, 115449
(2018)], a solvable model of collective modes in 2D metals was considered in
the hydrodynamic regime. In the current work, we generalize the hydrodynamic
theory to 3D metals for which the calculation of sound modes in a
strongly-coupled quantum Coulomb plasma can be made explicit. The specific
theoretical question of interest is how the usual linearly dispersing
hydrodynamic sound mode relates to the well-known gapped 3D plasmon collective
mode in the presence of long-range Coulomb interaction. We show analytically
that both the zero sound in the collisionless regime and the first sound in the
hydrodynamic region become massive in 3D, acquiring a finite gap because of the
long-range Coulomb interaction, while their damping rates become quadratic in
momentum. We also discuss other types of long-range potential, where the
dispersion of sound modes is modified accordingly. The general result is that
the leading order hydrodynamic sound mode is always given by the leading-order
plasmon frequency in the presence of long-range Coulomb interaction, but the
next-to-leading-order dispersion corrections differ in hydrodynamic and
colllisionless regimes.
