What is Complex wavenumber: propagation and attenuation? — Physics

Definition 8.3 University Physics — Year 2 · Chapter 8 — Dispersion and Wave Packets

When the dispersion relation gives, for real $\omega$ , a complex $\underline k = k' - \iu k''$ , the wave is

\underline s = A\,\eu^{-k''x}\,\eu^{\iu(\omega t - k'x)} :

it propagates at $v_\varphi = \omega/k'$ and its amplitude decays as $\eu^{-x/\delta}$ with the attenuation length $\delta = 1/k''$ (it must decay in its direction of propagation: $k'$ and $k''$ of the same sign). If $\underline k$ is purely imaginary, $\underline k = -\iu\kappa$ , the wave is evanescent: $A\,\eu^{-\kappa x}\eu^{\iu\omega t}$ , a standing oscillation whose amplitude dies over the distance $1/\kappa$ without propagating anything — the case of the Klein–Gordon medium below its cut-off, $\kappa = \sqrt{\omega_c^2 - \omega^2}/c$ , and of a metal at optical frequencies.

Examples

Example 8.4 (A damped string)

A string in a viscous fluid, $\mu\partial_t^2y + \alpha\partial_ty = T\partial_x^2y$ : $\underline k^2 = (\omega^2 - \iu\alpha\omega/\mu)/c^2$ . For weak damping ( $\alpha \ll \mu \omega$ ), $\underline k \approx \dfrac\omega c\Bigl(1 - \iu\dfrac{\alpha}{2\mu\omega}\Bigr)$ : the wave keeps its speed and loses amplitude over $\delta = 2\mu c/\alpha = 2Z/\alpha$ , independent of frequency — a dissipative, non-dispersive loss, $8.7\,\mathrm{dB}$ per length $\delta$ .

Example 8.11 (Three cables)

The 1858 transatlantic cable: a single copper wire in gutta-percha, $R \approx 3\,\Omega/\mathrm{km}$ , $\Gamma \approx 0.3\,\text{µ}\mathrm{F}/\mathrm{km}$ , $L = 3000\,\mathrm{km}$ : $R\Gamma = 9 \times 10^{-10}\,\mathrm{s}/\mathrm{m}^{2}$ and $R\Gamma L^2 \approx 8\,\mathrm{s}$ per dot: a word a minute, and the operators’ attempts to force the signal with $2\,\mathrm{kV}$ destroyed the insulation within weeks. A telephone line loaded with coils every $2\,\mathrm{km}$ to satisfy Heaviside’s condition (1900): clear speech over hundreds of kilometres. A modern $50\,\Omega$ coaxial cable: $c = 2 \times 10^{8}\,\mathrm{m}/\mathrm{s}$ , $\Lambda = 0.25\,\text{µ}\mathrm{H}/\mathrm{m}$ , $\Gamma = 100\,\mathrm{pF}/\mathrm{m}$ , and an attenuation, due to the skin effect in the conductors (Chapter 14), growing as $\sqrt f$ : a few decibels per hundred metres at $100\,\mathrm{MHz}$ .

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Examples

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