Abstract Wave envelope elements provide an alternative to boundary element methods for the solution of exterior wave problems. Applications to date have been to axisymmetric geometries and have involved the use of large, but finite, wave envelope meshes compatibly matched to denser, near field meshes of conventional elements. In the present paper the wave envelope idea is extended by the inclusion of an element mapping, taken directly from infinite element methodology, and by modifications to the residual formulation. Taken in conjunction, these two measures permit the outer boundary of the wave envelope region to be moved to infinity while preserving the boundedness of the acoustic mass and stiffness integrals. The present formulation is developed initially with reference to a simple one-dimensional model and is then applied in two and three dimensions. Comparisons between computed results and those obtained from analytic solutions, where applicable, and from boundary element codes demonstrate the effectiveness of the method.