Aims. In this paper, we study the propagation and stability of nonlinear sound waves in accretion disks. Methods. 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. Results. When the wave frequency is between Ω and 2Ω (where Ω 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. Conclusions. 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.