A new and efficient numerical method to solve nonlinear sound propagation is presented. The frequency-domain Burgers equation, which includes nonlinear steepening and atmospheric absorption, is formulated in the form of the real and imaginary parts of the pressure. The new formulation effectively eliminates possible numerical issues associated with zero amplitude at higher frequencies occurring in a previous frequency-domain algorithm. In addition, to circumvent a high frequency error that can occur in the truncated higher frequencies, a split algorithm, in which the Burgers equation is solved below a cut-off frequency and a recursive analytic expression is used beyond the cut-off frequency, is developed. Finally, the Lanzcos smoothing filter is incorporated to remove the Gibbs phenomenon. The new method is found to successfully eliminate high frequency numerical oscillations and to provide excellent agreement with the exact solution for an initially sinusoidal signal with only a few harmonics. The new method is applied to a broad range of applications with a comparison to other methods to assess the robustness and numerical efficiency of the method. These include sonic boom, broadband supersonic jet engine noise, and helicopter high-speed impulsive noise. It is shown that the new method provides the most accurate and fast predictions compared to the other methods for all the application problems.