Home exercise 5.6.6: An oblique wave: \vec E from \vec H

$\nabla \varphi =3a_x+4a_z$, $\beta =|\nabla \varphi |=5rad/m$ and $a_k=0.6a_x+0.8a_z$. $v_p=c/\sqrt{4}=1.5\times 10^{8}m/s$, so

For $H=H_y(x,z)a_y$,

$E\cdot a_k=2.4\pi (4\times 0.6-3\times 0.8)=0$, and $|E|=2.4\pi \times 5=12\pi$, so $|E|/|H|=12\pi /0.2=60\pi ={\eta }_0/\sqrt{{\epsilon }_r}=\eta$.

$E$ and $H$ are in phase. With $a_x\times a_y=a_z$ and $a_z\times a_y=-a_x$,

Through $1{\mathrm{m}}^2$ of a plane $z=$ const only $P_z=0.96\pi =3.02W/{\mathrm{m}}^2$ counts.

$|P_{av}|=0.24\pi \times 5=1.2\pi =\frac{1}{2}|E||H|=\frac{1}{2}\times 12\pi \times 0.2$, and $P_{av}\parallel a_k$.