Verify that E = E₀sin(kz − ωt)x̂ satisfies the wave equation ∇²E = μ₀ε₀∂²E/∂t², and find the relation between ω and k. Then compute the wavelength of a 100 MHz radio wave. Step 1: Only x-component is nonzero: E_x = E₀sin(kz − ωt). ∂²E_x/∂z² = −k²E₀sin(kz − ωt) = −k²E_x. Time derivatives: ∂²E_x/∂t² = −ω²E_x. Step 2: Wave equation needs −k²E_x = μ₀ε₀(−ω²E_x), so k² = μ₀ε₀ω², i.e. ω/k = 1/√(μ₀ε₀) = c. The wave must satisfy ω = ck — the dispersion relation of light in vacuum. Step 3: f = 100 MHz = 10⁸ Hz, ω = 2πf. λ = c/f = 3.00×10⁸/10⁸ = 3.0 m. So k = 2π/λ ≈ 2.09 rad/m. Any (ω, k) pair with ω/k = c is a valid light wave — Maxwell's equations don't pick a frequency; the source does.