Suppose I have a monochromatic electromagnetic plane wave with the E-field linearly polarized in the x-direction (and the B-field linearly polarized in the y-direction). Then the Poynting vector should be pointing in the z direction with a magnitude equal to the product of the B and E-field magnitudes divided by the magnetic constant. But because they are complex, the magnitudes of the B and E field don't depend on time or position, which doesn't make sense, as the Poynting vector shouldn't be constant over all time. Of course, this can be solved by taking only the real parts of both solutions and multiplying them, but this would require breaking up the complex exponentials into cosines and sines. Is there anyway to do this without having to break up the complex exponentials?