  In this paper, we study the propagation and stability of nonlinear sound
waves in accretion disks. Using the shearing box approximation, we derive the
form of these waves using a semi-analytic approach and go on to study their
stability. The results are compared to those of numerical simulations performed
using finite difference approaches such as employed by ZEUS as well as Godunov
methods. When the wave frequency is between Omega and two Omega (where Omega is
the disk orbital angular velocity), it can couple resonantly with a pair of
linear inertial waves and thus undergo a parametric instability. Neglecting the
disk vertical stratification, we derive an expression for the growth rate when
the amplitude of the background wave is small. Good agreement is found with the
results of numerical simulations performed both with finite difference and
Godunov codes. During the nonlinear phase of the instability, the flow remains
well organised if the amplitude of the background wave is small. However,
strongly nonlinear waves break down into turbulence. In both cases, the
background wave is damped and the disk eventually returns to a stationary
state. Finally, we demonstrate that the instability also develops when density
stratification is taken into account and so is robust. This destabilisation of
freely propagating nonlinear sound waves may be important for understanding the
complicated behaviour of density waves in disks that are unstable through the
effects of self-gravity or magnetic fields and is likely to affect the
propagation of waves that are tidally excited by objects such as a protoplanet
or companion perturbing a protoplanetary disk. The nonlinear wave solutions
described here as well as their stability properties were also found to be
useful for testing and comparing the performance of different numerical codes.
