We consider the domain decomposition method approach to solve the Helmholtz equation. A new double sweep based approach is presented valid for any type of interface boundary conditions and that benefits from the overlap. It makes use of a splitting of the local problems in the subdomain. Despite of the fact that a first order interface boundary conditions is used, the splitting double sweep method demonstrates good stability properties with respect to the number of subdomains and the frequency even for heterogeneous media. Convergence is improved when compared to the double sweep method for all of our test cases: waveguide, open cavity and wedge problems.