6.1.4 Propagation and attenuation constants

The general form of a uniform plane wave is $E(r)=E_0{\mathrm{e}}^{-\mathrm{j}k\cdot r}$ with $k=k_x{a}_x+k_y{a}_y+k_z{a}_z$. In a conducting medium the current $J=\sigma E$ joins the displacement current, and the wave number $k$ becomes

with the complex permittivity ${\epsilon }_c$ of Section 5.6. The wave number is now complex and is composed of the attenuation and the phase constant as in Eq. (6.1.6). Many books, and Section 6.2, use instead the propagation constant

The two descriptions are the same: $\gamma =\mathrm{j}(\beta -\mathrm{j}\alpha )=\alpha +\mathrm{j}\beta$. Both $\alpha$ and $\beta$ are positive for a wave that travels and decays towards $+z$.

With loss the intrinsic impedance is complex too,

and since $H=\frac{1}{\eta }{a}_k\times E$, the phase of $H$ is that of $E$ minus ${\theta }_{\eta }$: the magnetic field lags the electric field by ${\theta }_{\eta }$. For a medium with $\sigma \ge 0$ and real $\mu$ the angle stays in the range $0\le {\theta }_{\eta }<45^{\circ }$, the upper limit being approached by a very good conductor. Figure 6.1.7 shows both fields for ${\theta }_{\eta }=30^{\circ }$.

The explicit $\alpha (\omega )$ and $\beta (\omega )$ of dielectrics, conductors and plasmas are worked out in Section 6.2. In the rest of this section the medium is lossless unless stated otherwise: $\alpha =0$, $\beta =k$, and $\eta$ is real. Exercise 6.1.6 is the one exception, a concrete wall whose ${\epsilon }_r$ and $\sigma$ are found from a measured $\alpha$ and $\beta$.