A two-dimensional time-dependent hybrid Fourier-Chebyshev method of co
llocation is developed and used for the study of tidal effects in accr
etion disks, under the assumptions of a polytropic equation of state a
nd a standard alpha viscosity prescription. Under the influence of the
m = 1 azimuthal component of the tidal potential, viscous oscillation
s in the outer disk excite an m = 1 eccentric instability in the disk.
While the m = 2 azimuthal component of the tidal potential excites a
Papaloizou-Pringle instability in the inner disk (a saturated m = 2 az
imuthal mode), with an elliptic pattern rotating at about a fraction (
approximate to 1/3) of the local Keplerian velocity in the inner disk.
The period of the elliptic mode corresponds well to the periods of th
e short-period oscillations observed in cataclysmic variables. In cold
disks (r Omega/c(s) = M approximate to 40) we also find a critical va
lue of the viscosity parameter (alpha approximate to 0.01), below whic
h shock dissipation dominates and is balanced by the wave amplificatio
n due to the wave action conservation. In this case the double spiral
shock propagates all the way to the inner boundary with a Mach number
M-s approximate to 1.3.