Based on the idea that the modified wavenumbers should be as close to the exact wavenumbers as possible, an optimal tridiagonal fourth-order compact difference scheme and an interpolation scheme on the staggered grid system are proposed in this paper. Although its accuracy is of the 4th order, the optimal scheme enjoys a high resolution and at the same time preserves the characteristics of the group velocity. The numerical calculations show that the maximum resolvable wavenumbers obtained with the optimal compact difference (interpolation) scheme is 0.86π (0.63π). The group velocity can be preserved for the wavenumber less than 0.75π. All these values are better than those obtained from the standard compact schemes of fourth or sixth order. Finally, the optimal scheme, the standard fourth order compact scheme and the sixth order compact schemes are employed to calculate the first derivation and the propagations of small scale waves. The results show that the optimal scheme is superior to other two schemes with respect to the reduction of errors and the preservation of the group velocity.