Helmholtz's equation for EM wave

Helmholtz's equation is the wave equation for time-harmonic fields, written in phasor form. It describes how E and H vary in space for a sinusoidal wave of angular frequency ω.

Derivation

Consider a homogeneous, linear, isotropic medium with ε, μ, σ and no free charge (ρv = 0). Maxwell's equations are

∇×E = −μ ∂H/∂t        ∇×H = σE + ε ∂E/∂t
∇·E = 0               ∇·H = 0

Take the curl of the first equation and substitute the second:

∇×(∇×E) = −μ ∂(∇×H)/∂t = −μσ ∂E/∂t − με ∂²E/∂t²

Use the identity ∇×(∇×E) = ∇(∇·E) − ∇²E = −∇²E (because ∇·E = 0):

∇²E = μσ ∂E/∂t + με ∂²E/∂t²      (general wave equation)

For fields varying as e^{jωt}, ∂/∂t → jω and ∂²/∂t² → −ω²:

∇²E_s = jωμ(σ + jωε) E_s = γ² E_s

So

∇²E_s − γ²E_s = 0, with γ² = jωμ(σ + jωε)

and in the same way ∇²H_s − γ²H_s = 0. Here γ = α + jβ is the propagation constant.

Special cases

• Lossless dielectric (σ = 0): γ = jβ, γ² = −ω²με, so

  ∇²E_s + k²E_s = 0, k = β = ω√(με)

  This is the usual form of Helmholtz's equation, with k the wave number.

• Free space: k = ω/c = ω√(μ₀ε₀).

• Good conductor: γ ≈ (1 + j)√(πfμσ).

The solutions are travelling waves, e.g. E_s = E₀ e^{−γz} for the +z direction.