What is Dispersion relation; phase velocity? — Physics

Definition 8.1 University Physics — Year 2 · Chapter 8 — Dispersion and Wave Packets

A linear, homogeneous, time-invariant medium admits the plane monochromatic waves $\underline s = A\,\eu^{\iu(\omega t - kx)}$ (complex notation; the physical signal is the real part) provided $\omega$ and $k$ satisfy the dispersion relation of the medium, $\mathcal D(\omega, k) = 0$ , solved as $k(\omega)$ or $\omega(k)$ . For real $k$ the wave travels without deforming at the phase velocity

v_\varphi = \frac\omega k .

The medium is non-dispersive when $v_\varphi$ does not depend on $\omega$ (then $\omega = ck$ and every signal propagates undeformed — the d’Alembert equation) and dispersive otherwise.

Left: three dispersion relations — a straight line for a non-dispersive medium, the Klein–Gordon hyperbola with its cut-off, the parabola of deep-water waves. Right: for the Klein–Gordon relation the phase velocity exceeds c and the group velocity stays below it, with v_ v_g = c2. — Left: three dispersion relations — a straight line for a non-dispersive medium, the Klein–Gordon hyperbola with its cut-off, the parabola of deep-water waves. Right: for the Klein–Gordon relation the phase velocity exceeds $c$ and the group velocity stays below it, with $v_\varphi v_g = c^2$ .

Examples

Example 8.2 (Dispersive and non-dispersive)

(i) A string on an elastic bed (restoring force $-Ky$ per unit length, or a chain of pendulums), $\mu\partial_t^2y = T\partial_x^2y - Ky$ : $\omega^2 = \omega_c^2 + c^2k^2$ with $\omega_c = \sqrt{K/\mu}$ — the Klein–Gordon relation, the same as a plasma’s and a waveguide’s (Chapters 14 and 16): $v_\varphi = c/\sqrt{1 - \omega_c^2/\omega^2} > c$ , and no real $k$ below the cut-off $\omega_c$ . (ii) Deep-water gravity waves: $\omega^2 = gk$ (admitted), $v_\varphi = \sqrt{g/k} = \sqrt{g\lambda/2\pi}$ : long waves are faster — $12.5\,\mathrm{m}/\mathrm{s}$ for $100\,\mathrm{m}$ , $40\,\mathrm{m}/\mathrm{s}$ for $1\,\mathrm{km}$ . (iii) The chain of atoms, $\omega = 2\omega_0|\sin(ka/2)|$ (Chapter 6). (iv) Light in glass, $k = n(\omega)\omega/c$ , which is why a prism spreads a spectrum.

Example 8.7 (Klein–Gordon: faster than light, and not)

For $\omega^2 = \omega_c^2 + c^2k^2$ : $v_\varphi = \omega/k$ and $v_g = c^2k/\omega$ , so $v_\varphi v_g = c^2$ : the phase velocity exceeds $c$ (in the ionosphere, a $10\,\mathrm{MHz}$ wave’s crests move faster than light) but the group velocity, which carries the signal, stays below it. A phase velocity transports no energy and no information — the crests are like the spot of a lighthouse beam sweeping a distant cloud.

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