Critical behaviors of the sound attenuation in a spin-1 Ising system with bilinear (J) and biquadratic (K) interactions are investigated within the framework of cluster variation method and Onsager theory of irreversible thermodynamics. The sound wave is assumed to couple mainly to the order parameter fluctuations which decay via order parameter relaxation processes. Two relaxation times are obtained and an expression is found for the sound attenuation coefficient (alpha) in terms of these relaxation times. The temperature behavior of the sound attenuation near the phase transition temperatures (T-c) is analyzed according to various values of Onsager coefficients (gamma(ij)) and sound frequency (omega). For TT-c the data give evidence that there is no relaxational contribution to sound attenuation coming from order parameter fluctuations. On the other hand, a convergence is found in attenuation just below the critical and the tricritical points as (T-c-T), while a jump-discontinuity is observed for the first-order behavior. The frequency variation of the sound attenuation is also investigated and in addition to omega(2)-attenuation dependence observed in the hydrodynamic regime it is observed that in the high frequency region the attenuation is independent of omega and the ratio of two interaction parameters (J/K). (C) 2003 American Institute of Physics.