= Linear oscillation
For the prescribed displacement along $z$, the source <acceleration> is $\mathbf a(t_r)=-a\omega^2\cos(\omega t_r)\widehat{\mathbf z}$. If $\theta$ is the angle from the $z$ axis, $|\mathbf a|^2-(\mathbf n\cdot\mathbf a)^2=a^2\omega^4\cos^2(\omega t_r)\sin^2\theta$. Averaging $\cos^2$ over one <wave period> gives
$$
\boxed{\frac{d\langle P\rangle_{\rm linear}}{d\Omega}
=\frac{q^2a^2\omega^4}{32\pi^2\epsilon_0c^3}\sin^2\theta.}
$$
The <angular distribution of electric dipole radiation> has zero intensity on the oscillation axis and maximum intensity in the equatorial plane, with no azimuthal dependence. Its total <power> is
$$
\boxed{\langle P\rangle_{\rm linear}=\frac{q^2a^2\omega^4}{12\pi\epsilon_0c^3}.}
$$
The condition $a\omega\ll c$ is the required nonrelativistic condition. The plotted radial coordinate below is angular <power>, rather than field <wave amplitude>.
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