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• The new scheme is compact and efficient. The three terms: u xx , u yy and u zz are approximated with high order accuracy separately. Operator-splitting techniques such as ADI is not needed here. Therefore, the new method is suitable for parallel computation to further improve efficiency. • Higher order compact boundary conditions approximation for u xx , u yy and u zz are derived. • Richardson extrapolation and Runge–Kutta method have been incorporated to improve the accuracy in time to fourth-order. • An energy method has been used to prove the conditional stability of the new scheme for the case of variable velocity. In this paper we developed a new explicit compact high-order finite difference scheme to solve the 3D acoustic wave equations with spatially variable acoustic velocity. The boundary conditions for the second derivatives of spatial variables have been derived by using the wave equation and the boundary conditions themselves. Theoretical analysis shows that the new scheme has an accuracy order of O (τ 2) + O (h 4), where τ is the time step and h is the grid size. Combined with Richardson extrapolation or Runge–Kutta method, the new method can be improved to 4th-order accuracy in time, which has been implemented and verified in this paper. Four numerical experiments are conducted to validate the efficiency and accuracy of the new scheme. The stability of the new scheme has been proved by an energy method, which shows that the new scheme is conditionally stable with a Currant–Friedrichs–Lewy (CFL) number which is slightly lower than that of the Padé approximation based method. However, the new scheme is much simpler to implement. [ABSTRACT FROM AUTHOR]