Q2Electromagnetics Waves

Definition: a uniform plane wave is an idealized electromagnetic wave whose surfaces of constant phase (and constant amplitude) are infinite planes perpendicular to the direction of propagation; the field has the same magnitude and phase at every point on any such plane, varying only along the direction of propagation (and in time), not across the transverse plane itself. Although no real, finite source can produce a truly infinite plane wave (any real source produces a spherical wave that only locally approximates a plane wave far from the source, as in the far-field discussion elsewhere in this paper), the uniform plane wave is an extremely useful idealization because it satisfies Maxwell's equations exactly and captures the essential local behaviour of any wave far from its source.

Proof that the field is independent of two dimensions: take the propagation direction to be z, and consider fields assumed uniform over the transverse (x,y) plane, i.e., assume as an ansatz that E and H depend only on z and t: E = E(z,t), H = H(z,t), with ∂E/∂x = ∂E/∂y = 0 and similarly for H. Substituting this ansatz into the source-free Maxwell curl equations:

Expanding the curl operator in Cartesian coordinates, every term involving ∂/∂x or ∂/∂y vanishes identically by the assumption of transverse uniformity, leaving only the ∂/∂z terms — the curl equations reduce to a much simpler 1-D system relating ∂Ex/∂z, ∂Ey/∂z to ∂Hy/∂t, ∂Hx/∂t (and similarly for the H-curl equation), plus the requirement (from ∇·E=0, ∇·H=0 applied to a z-only-dependent field) that Ez = Hz = 0 — confirming the wave is purely transverse (TEM), with no z-component of either field. Solving this reduced 1-D system yields the standard travelling-wave solution E(z,t) = E0cos(ωt−βz), depending only on z and t exactly as assumed, self-consistently verifying the ansatz: because the reduced Maxwell equations obtained under the transverse-uniformity assumption admit a valid nontrivial solution of exactly the assumed z,t-only functional form, the assumption is confirmed consistent, demonstrating that a field with no x- or y-dependence anywhere in space is indeed an exact, self-consistent solution of Maxwell's equations — precisely the defining property of a uniform plane wave, independent of the two transverse dimensions x and y.