Q3Electromagnetics Waves
Question
Q.3. Define Phase Velocity.
Answer
Phase velocity is the speed at which a point of constant phase (e.g., a wave crest) travels along the direction of propagation, vp = ω/β; in a waveguide it exceeds the free-space speed of light c.
Phase velocity vp is the rate at which a surface of constant phase of a sinusoidal wave advances through a medium. For a wave of the form cos(ωt − βz), a point of constant phase satisfies ωt − βz = constant, so differentiating gives dz/dt = ω/β, i.e.:
where ω is the angular frequency and β is the phase constant (rad/m). In an unbounded lossless dielectric, vp = 1/√(με), independent of frequency (non-dispersive). Inside a waveguide, however, β = √(k² − kc²) where kc is the cutoff wavenumber of the mode, so vp = ω/β always exceeds c = ω/k; this does not violate relativity because phase velocity carries no information — the group velocity vg = dω/dβ, which does carry information/energy, always satisfies vg ≤ c inside a waveguide, and in fact vp·vg = c² exactly for a lossless waveguide mode.