The problem of calculating the scattering of a plane sound wave by a hard three-dimensional body is considered. It is assumed that the surface of the body is given by an unstructured triangular mesh, which is important for practical applications. We develop the boundary element method based on the regularized Burton – Miller integral equation with parameter $\alpha$. The use of this equation solves the problem of non-uniqueness of the solution. Despite the fact that this approach has been studied by many authors, some aspects remained unexplored, in particular, the regularization for unstructured meshes and justification for the collocation method for regularized Burton – Miller equation. We provide some answers to these questions. The regularized Burton – Miller equation and its discrete justified version based on the collocation method are proposed. This made it possible to develop a robust numerical method that works for arbitrary wavenumbers. The method involves integration over Voronoi cells and estimation of the surface gradient of the acoustic potential using adjacent vertices. In order to validate and test the numerical method and justify the choice of the parameter $\alpha$ for the case of a sphere, we derive an analytical solution directly from the Burton – Miller equation and the Jackson spherical expansion of the Green's function. The results of the software implementation are presented.