  The present study of the two-stream instability in stellar disks with
counter-rotating components of stars and/or gas is stimulated by recently
discovered counter-rotating spiral and S0 galaxies. Strong linear two-stream
instability of tightly-wrapped spiral waves is found for one and two-armed
waves with the pattern angular speed of the unstable waves always intermediate
between the angular speed of the co-rotating matter ($+\Omega$) and that of the
counter-rotating matter ($-\Omega$). The instability arises from the
interaction of positive and negative energy modes in the co- and
counter-rotating components. The unstable waves are in general convective -
they move in radius and radial wavenumber space - with the result that
amplification of the advected wave is more important than the local growth
rate. For a galaxy of co-rotating stars and counter-rotating stars of
mass-fraction $\xi_* < {1\over 2}$, or of counter-rotating gas of mass-fraction
$\xi_g < {1\over 2}$, the largest amplification is usually for the one-armed
leading waves (with respect to the co-rotating stars). For the case of both
counter-rotating stars and gas, the largest amplifications are for $\xi_*+\xi_g
\approx {1\over 2}$, also for one-armed leading waves. The two-armed trailing
waves usually have smaller amplifications. The growth rates and amplifications
all decrease as the velocity spreads of the stars and/or gas increase. It is
suggested that the spiral waves can provide an effective viscosity for the gas
causing its accretion.
