It is generally held that Lorentz transformation is superior to Galilean transformation. However, this paper reveals that the Lorentz’s application of both transformations to derive the electromagnetic wave equation for two inertial reference frames is logically flawed. This logical mistake makes the superiority of Lorentz transformation questionable. The paper also demonstrates that both Lorentz and Galilean transformations can generate standard equations for electromagnetic waves, while both cannot keep the transformed Maxwell’s equations consistent. Since the relativity theory is based on the belief that the Lorentz transformation can keep physics law in the same form for different reference frames, the findings of this paper have important implications: Either the relativity theory is not one hundred percent correct or our understanding of the theory needs updating in light of the new knowledge