Electromagnetism / Electrodynamics I · Elementary electromagnetic waves
The vector wave equation alone does not enforce transversality
Problem
A homogeneous source-free medium has positive constants $\epsilon,\mu$ and $v=(\mu\epsilon)^{-1/2}$. Consider the vector wave equation $$\nabla^2\mathbf E-\mu\epsilon\,\partial_t^2\mathbf E=\mathbf0.$$ Let $k=\omega/v$ and propose $$\mathbf E_L=E_0\cos(kz-\omega t)\hat{\mathbf z}.$$ (a) Verify that $\mathbf E_L$ solves the vector wave equation. (b) Compute its divergence and explain why it is not a source-free Maxwell plane wave despite passing that test. (c) Replace $\hat{\mathbf z}$ by $\hat{\mathbf x}$, construct the corresponding B field, and verify the two curl equations. (d) State precisely which Maxwell constraint the wave equation lost.
Hint
Both second derivatives of the proposed field reproduce it with negative scalar factors.
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