The Wave-Envelope (WE) Method is a technique that has been applied to acoustic problems to model the unbounded domain surrounding an acoustic source. Here it is applied to electromagnetic wave problems. A key feature of the method is that it uses a change of dependent variable to remove the wave-like qualities of the solution and thereby permits the use of arbitrarily large elements in the exterior domain. This makes it possible to apply to wave problems some of the techniques developed for magnetostatics, such as Transformation Methods. These methods map the very large exterior domain into a smaller region which is more easily meshed. The combined Wave-Envelope Transformation Method is applied to two time-harmonic electromagnetic wave problems: radiation of a single cylindrical wave function; and scattering of an incident plane wave by a perfectly-conducting circular cylinder. In both cases the near-zone electric field is compared to the exact solution, for a range of frequencies.