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Plane wave solutions to the wave equation have the form E = E₀ei(k·r - ωt) where ω = ck. The transverse nature of electromagnetic waves (E and B perpendicular to k and to each other) follows from Maxwell's equations. The dispersion relation ω = ck is exact in vacuum, independent of frequency.
From the electromagnetic wave equation you know that both E and B satisfy ∇²E = (1/c²)∂²E/∂t², a linear PDE with constant coefficients. The most natural solutions are plane waves: fields of the form E = E₀ e^{i(k·r − ωt)}, where k is the wave vector pointing in the direction of propagation and ω is the angular frequency. These solutions are "plane" because at any fixed time, the field is identical everywhere on a plane perpendicular to k — the phase k·r − ωt is constant on such planes, which are the wavefronts. Every other solution — beams, pulses, standing waves — can be built as a superposition of plane waves through Fourier decomposition.
Substituting the plane wave ansatz into the wave equation immediately gives the dispersion relation: k² = ω²/c², or equivalently ω = ck. This tells you that all electromagnetic waves in vacuum travel at the same speed c, regardless of frequency. This non-dispersive character distinguishes vacuum from any material medium — in glass or water, different frequencies travel at different speeds (which is why prisms split white light into colors). The dispersion relation ω = ck is linear in k, which implies the group velocity and phase velocity are both equal to c, and a wave packet of any bandwidth travels without spreading.
The truly remarkable consequence comes from applying Gauss's law ∇·E = 0 to the plane wave. Taking the divergence of E = E₀ e^{i(k·r − ωt)}, the spatial derivative acts on the exponent to pull down a factor of ik, giving ik·E₀ = 0. This means k·E₀ = 0: the electric field is perpendicular to the propagation direction. The wave is transverse. Similarly, Gauss's law for magnetism (∇·B = 0) forces k·B = 0 — the magnetic field is also transverse. Faraday's law then fixes the relationship between E and B: B = (k × E)/ω = k̂ × E/c. The three vectors k, E, and B form a right-handed orthogonal triplet, with |B| = |E|/c. This geometry — not assumed but derived from Maxwell's equations — is the fundamental structure of all electromagnetic radiation.
The plane wave solution is also the foundation for understanding polarization: since E₀ can point in any direction perpendicular to k, there is a two-dimensional space of polarization states. Linearly polarized waves have E₀ pointing along a fixed direction; circularly polarized waves are superpositions of two orthogonal linear polarizations with a 90° phase offset. The energy flux carried by the wave — the Poynting vector S = E × H/μ₀ — points in the k direction with magnitude |S| = |E|²/(μ₀c) = c²ε₀|E|², averaging over a cycle to give the intensity. All of this rich structure follows from the single elegant fact that Maxwell's equations in vacuum admit plane wave solutions with ω = ck.