Q2Electromagnetics Waves

Wave equation in a conducting medium: starting from Maxwell's curl equations in a linear, homogeneous, conducting medium (conductivity σ, permittivity ε, permeability μ) with time-harmonic fields ~e^{jωt}:

Taking the curl of the first equation and substituting the second gives the homogeneous vector wave equation:

For a uniform plane wave propagating in +z, E = E0 e^{−γz}, and the complex propagation constant γ = α + jβ has real part α (attenuation constant, Np/m) and imaginary part β (phase constant, rad/m), given exactly by:

This is obtained by writing γ² = jωμσ − ω²με = −ω²με(1 − jσ/ωε), taking the complex square root (γ = α+jβ) and separately equating real and imaginary parts of γ² = (α+jβ)² = (α²−β²) + j2αβ to the real and imaginary parts of −ω²με + jωμσ, then solving the resulting simultaneous equations α²−β² = −ω²με and 2αβ = ωμσ, which yields the expressions above.

Phase velocity: by definition vp = ω/β, so directly substituting the β expression above:

Because β does not vary linearly with ω (the bracketed factor itself depends on ω through the loss tangent σ/ωε), vp is frequency-dependent in a lossy medium — the medium is dispersive, unlike a truly lossless dielectric (σ=0) where β = ω√(με) reduces vp to the constant value 1/√(με), independent of frequency.

Group velocity: defined as vg = dω/dβ, the velocity at which a narrowband modulated signal's envelope (and hence information/energy) actually travels. Because β(ω) is a nonlinear function of ω in a lossy/dispersive medium, vg = (dβ/dω)⁻¹ generally differs numerically from vp = ω/β at the same frequency — this vp ≠ vg condition is precisely the definition of a dispersive medium, in which different frequency components of a wave packet travel at slightly different phase speeds, causing the packet to spread (disperse) as it propagates. In the two limiting cases the general formulas simplify usefully: for a good dielectric (σ/ωε ≪ 1, low-loss limit) β ≈ ω√(με)[1 + (1/8)(σ/ωε)²] so vp ≈ vg ≈ 1/√(με) to good approximation (nearly non-dispersive); for a good conductor (σ/ωε ≫ 1, high-loss limit) α ≈ β ≈ √(ωμσ/2), giving vp = ω/β = √(2ω/μσ), which itself depends on √ω — a strongly dispersive result characteristic of the skin-effect regime inside good conductors, where vg likewise differs substantially from vp.

Physical significance of α: unlike a lossless dielectric, where a wave propagates indefinitely without loss, the nonzero attenuation constant α in a conducting medium means the wave amplitude decays exponentially with distance as e^{−αz}, with the characteristic penetration (skin) depth δ = 1/α marking the distance over which the field falls to 1/e of its surface value. In the good-conductor limit this reduces to the familiar skin-depth formula δ = √(2/ωμσ) = 1/√(πfμσ), directly linking this general conducting-medium wave analysis to the skin-effect phenomenon examined quantitatively elsewhere in this paper — both are simply different manifestations of the same complex propagation constant γ = α + jβ evaluated in the high-conductivity limit.

Summary of the two velocity concepts: phase velocity vp describes how fast a point of constant phase (a wave crest) appears to move, while group velocity vg describes how fast the envelope of a modulated wave packet — and therefore the energy and information it carries — actually propagates. In any dispersive medium, including a lossy conductor, these two velocities differ, and only vg is physically constrained to remain at or below the free-space speed of light c; vp may exceed c (as it also does inside a waveguide) without violating causality, precisely because a pure, infinite-duration sinusoid carries no information, and only the group velocity of an actual modulated signal is subject to the relativistic speed limit.