Exercise 6.1.4: Direction finding with three probes

Seventeen further exercises in the spirit of the problems, the home exercises and the worked examples, graded from easy to harder within each group. Exercises 6.1.1–6.1.4 are about uniform plane waves, $k$, $\lambda$, $\eta$ and the triad, Exercises 6.1.5–6.1.6 about power, energy and loss, Exercises 6.1.7–6.1.10 about polarisation, Exercises 6.1.11–6.1.12 about the Doppler effect, and Exercises 6.1.13–6.1.17 mix power, exposure and polarisation. Besides the relations listed at the head of this subsection they use the lossy forms $k=\omega \sqrt{\mu {\epsilon }_c}=\beta -\mathrm{j}\alpha$, ${\epsilon }_c=\epsilon (1-\mathrm{j}\sigma /\omega \epsilon )$, $\eta =\sqrt{\mu /{\epsilon }_c}=|\eta |{\mathrm{e}}^{\mathrm{j}{\theta }_{\eta }}$ and ${\mathcal{P}}_{av}=\frac{|E_0{|}^2}{2|\eta |}{\mathrm{e}}^{-2\alpha a_k\cdot r}\cos {\theta }_{\eta }{a}_k$ (Exercise 6.1.6 only), the average energy density $w_{av}=\frac{1}{4}(\epsilon |E{|}^2+\mu |H{|}^2)$, Faraday's law $\mathcal{E}=-\mathrm{j}\omega \Phi$, and the polarisation match $|p_w\cdot p_a^{\ast }{|}^2$. A field factor ${\mathrm{e}}^{-\alpha z}$ is $8.686\alpha z\mathrm{dB}$.